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71. Popov G (2004) KAM theorem for Gevrey Hamiltonians. Ergod Books and Reviews Theor Dynam Syst 24:1753–1786 Braaksma BLJ, Stolovitch L (2007) Small divisors and large multi- 72. Ramis JP (1994) Séries divergentes et théories asymptotiques. pliers (Petits diviseurs et grands multiplicateurs). Ann l’institut Panor Synth pp 0–74 Fourier 57(2):603–628 73. Ramis JP, Schäfke R (1996) Gevrey separation of slow and fast Broer HW, Levi M (1995) Geometrical aspects of stability theory for variables. Nonlinearity 9:353–384 Hill’s equations. Arch Rat Mech An 131:225–240 74. Roussarie R (1987) Weak and continuous equivalences for fam- Gaeta G (1999) Poincaré renormalized forms. Ann Inst Henri ilies of line diffeomorphisms. In: Dynamical systems and bifur- Poincaré 70(6):461–514 cation theory, Pitman research notes in math. Series Longman Martinet J, Ramis JP (1982) Problèmes des modules pour les èqua- 160:377–385 tions différentielles non linéaires du premier ordre. Publ IHES 75. Sanders JA, Verhulst F, Murdock J (1985) Averaging methods 5563–164 in nonlinear dynamical systems. Revised 2nd edn, Appl Math Martinet J Ramis JP (1983) Classification analytique des équations Sciences 59, 2007. Springer différentielles non linéaires résonnantes du premier ordre. Ann 76. Siegel CL (1942) Iteration of analytic functions. Ann Math Sci École Norm Suprieure Sér 416(4):571–621 43(2):607–612 Vanderbauwhede A (2000) Subharmonic bifurcation at multiple 77. Siegel CL, Moser JK (1971) Lectures on celestial mechanics. resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh Springer, Berlin M (eds) Proceedings of the mathematics conference, Birzeit, 78. Simó C (1994) Averaging under fast quasiperiodic forcing. August 1998. World Scientific, Singapore, pp 254–276 In: Seimenis J (ed) Hamiltonian mechanics, integrability and chaotic behaviour. NATO Adv Sci Inst Ser B Phys 331:13–34, Plenum, New York 79. Simó C (1998) Effective computations in celestial mechanics Numerical Bifurcation Analysis and astrodynamics. In: Rumyantsev VV, Karapetyan AV (eds) Modern methods of analytical mechanics and their applica- HIL MEIJER 1, F ABIO DERCOLE 2, B ART OLDEMAN 3 tions. CISM Courses Lectures, vol 387. Springer, pp 55–102 1 Department of Electrical Engineering, Mathematics 80. Simó C (2000) Analytic and numeric computations of exponen- and Computer Science, University of Twente, Enschede, tially small phenomena. In: Fiedler B, Gröger K, Sprekels J (eds) Proceedings EQUADIFF 99, Berlin. World Scientific, Singapore, The Netherlands pp 967–976 2 Department of Electronics and Information, Politecnico 81. Spivak M (1970) Differential Geometry, vol I. Publish or Perish di Milano, Milano, Italy Inc 3 Department of Computer Science and Software 82. Sternberg S (1959) On the structure of local homeomorphisms Engineering, Concordia University, Montreal, Canada of Euclidean n-space, vol II. Amer J Math 81:578–605 83. Takens F (1973) Forced oscillations and bifurcations. Applica- tions of global analysis, vol I, Utrecht. Comm Math Inst Univ Article Outline Utrecht, pp 31–59 (1974). Reprinted In: Broer HW, Krauskopf Glossary B, Vegter G (eds) (2001) Global analysis of dynamical systems. Deﬁnition of the Subject Festschrift dedicated to floris takens for his 60th birthday Lei- den. Inst Phys Bristol, pp 1–61 Introduction 84. Takens F (1974) Singularities of vector fields. Publ Math IHÉS Continuation and Discretization of Solutions 43:47–100 Normal Forms and the Center Manifold 85. Takens F, Vanderbauwhede A (2009) Local invariant manifolds Continuation and Detection of Bifurcations and normal forms. In: Broer HW, Hasselblatt B, Takens F (eds) Branch Switching Handbook of dynamical systems, vol 3. North-Holland (to appear) Connecting Orbits 86. Thom R (1989) Structural stability and morphogenesis. An out- Software Environments line of a general theory of models, 2nd edn. Addison-Wesley, Future Directions Redwood City (English, French original) Bibliography 87. Vanderbauwhede A (1989) Centre manifolds, normal forms and elementary bifurcations. Dyn Rep 2:89–170 Glossary 88. van der Meer JC (1985) The hamiltonian Hopf bifurcation. LNM, vol 1160. Springer Dynamical system A rule for time evolution on a state 89. Varadarajan VS (1974) Lie groups, Lie algebras and their repre- space. The term system will be used interchangeably. sentations. Englewood Cliffs, Prentice-Hall Here a system is a family given by an ordinary diﬀer- 90. Wagener FOO (2003) A note on Gevrey regular KAM theory and ential equation (ODE) depending on parameters. the inverse approximation lemma. Dyn Syst 18(2):159–163 Equilibrium A constant solution of the system, for given 91. Wagener FOO (2005) On the quasi-periodic d-fold degenerate bifurcation. J Differ Eqn 216:261–281 parameter values. 92. Yoccoz JC (1995) Théorème de Siegel, nombres de Bruno et Limit cycle An isolated periodic solution of the system, polynômes quadratiques. Astérisque 231:3–88 for given parameter values. 6330 N Numerical Bifurcation Analysis

Bifurcation A qualitative change in the dynamics of a dy- ods for numerical bifurcation analysis are reviewed, an ap- namical system produced by changing its parameters. proach to investigate the dynamic behavior of nonlinear Bifurcation points are the critical parameter combina- dynamical systems given by tions at which this happens for arbitrarily small param- x˙ D f (x; p) ; x 2 Rn ; p 2 Rn p ; (1) eter perturbations. Normal form A simplified model system for the analysis where f : Rn Rn p ! Rn is generic and sufficiently of a certain type of bifurcation. smooth. In particular, x(t) represents the state of the sys- Codimension The minimal number of parameters tem at time t and its components are called state (or phase ) needed to perturb a family of systems in a generic variables , x˙(t) denotes the time derivative of x(t), while p manner. denotes the parameters of the system, representing exper- Defining system A set of suitable equations so that the imental control settings or variable inputs. zero set corresponds to a bifurcation of a certain type In many instances, solutions of ( 1), starting at an or to a particular solution of the system. Also called initial condition x(0), appear to converge as t ! 1 to defining function or equation. equilibria (steady states) or limit cycles (periodic orbits). Continuation A numerical method suited for trac- Bounded solutions can also converge to more complex at- ing one-dimensional manifolds, curves (here called tractors , like tori (quasi-periodic orbits) or strange attrac- branches) of solutions for a defining system while one tors (chaotic orbits). The attractors of the system are in- or more parameters are varied. variant under time evolution, i. e., under the application of Test function A function designed to have a regular zero the time- t map ˚ t, where ˚ denotes the flow induced by at a bifurcation. During continuation a test function the system ( 1). Solutions attracting all nearby initial condi- can be monitored to detect bifurcations. tions, are said to be stable , and unstable if they repel some Branch switching Several branches of different codimen- initial conditions. sion can emanate from a bifurcation point. Switching Generally speaking, it is hard to obtain closed formu- from the computation of one branch to an other re- las for ˚ t as the system is nonlinear. In some cases, one quires appropriate procedures. can compute equilibria analytically, but this is often not the case for limit cycles. However, numerical simulations Definition of the Subject of ( 1) easily give an idea of how solutions look, although The theory of dynamical systems studies the behavior one never computes the true orbit due to numerical errors. of solutions of systems, like nonlinear ordinary differen- One can verify stability conditions by linearizing the flow tial equations (ODEs), depending upon parameters. Us- around equilibria and cycles. In particular, an equilibrium ing qualitative methods of bifurcation theory, the behavior x0 is stable if the eigenvalues of the linearization ( Jacobi ) of the system is characterized for various parameter com- matrix A D fx (x0; p) (where the subscript denotes differ- binations. In particular, the catalog of system behaviors entiation) all have a negative real part. Similarly, for a limit showing qualitative differences can be identified, together cycle x0(t) with period T, one defines the Floquet multipli- with the regions in parameter space where the different ers (or simply multipliers) as the eigenvalues of the mono- T behaviors occur. Bifurcations delimit such regions. Sym- dromy matrix M D ˚x (x0(0)). The cycle is stable if the bolic and analytical approaches are in general infeasible, nontrivial multipliers, there is always one equal to 1, are all but numerical bifurcation analysis is a powerful tool that within the unit circle. Equilibria and limit cycles are called aids in the understanding of a nonlinear system. When hyperbolic if the eigenvalues and nontrivial multipliers do computing power became widely available, algorithms for not have a zero real part or modulus one, respectively. this type of analysis matured and the first codes were For any given parameter combination p, the state space developed. With the development of suitable algorithms, representation of all orbits constitutes the phase portrait of the advancement in the qualitative theory has found its the system. In practice, one draws a set of strategic orbits way into several software projects evolving over time. The (or finite segments of them), from which all other orbits availability of software packages allows scientists to study can be intuitively inferred, as illustrated in Fig. 1. and adjust their models and to draw conclusions about Points X0; X1; X2 are equilibria, of which X0 and X1 their dynamics. are unstable and X2 is stable. In particular, X0 is a repellor , i. e., nearby orbits do not tend to remain close to X0, Introduction while X1 is a saddle , i. e., almost all nearby orbits go away Nonlinear ordinary differential equations depending upon from X1 except two, which tend to X1 and lie on the so- parameters are ubiquitous in science. In this article meth- called stable manifold ; the two orbits emanating from X1 Numerical Bifurcation Analysis N 6331

is hardly effective and accurate in practice, because bifurcations of equilibria and cycles are associated with a loss of hyperbolicity, e. g., stability, so that one should dramat- ically increase the length of simulations while approach- ing the bifurcation. In particular, saddle sets are hard to find by simulation, but play a fundamental role in bifurcation analysis, since they, together with attracting and repelling sets, form the skeleton of the phase portrait. This is why numerical bifurcation analysis does not rely on simulation, but rather on continuation , a numerical method suited for computing (approximating through a discrete sequence of points) one-dimensional manifolds (curves, “branches” in regular) implicitly defined as the zero set of Numerical Bifurcation Analysis, Figure 1 a suitable defining function . Phase portrait of a two-dimensional system with two attractors The general idea is to formulate the computation of (the equilibrium X2 and the limit cycle ), a repellor ( X0), and equilibria and their bifurcations as a suitable algebraic a saddle ( X1) problem (AP) of the form compose the unstable manifold . There are therefore two F(u; p) D 0 ; (2) attractors, the equilibrium X2 and the limit cycle , whose where u 2 Rnu is composed of x and possibly other vari- basins of attraction consist of the initial conditions in the ables characterizing the system, see, e. g., defining func- shaded and white areas, respectively. Note that while at- tions as in Sect. “ Continuation and Detection of Bifurca- tractors and repellors can be easily obtained through simu- tions ”. Here, however, for simplicity of notation, u will be lation, forward and backward in time, saddles can be hard considered as in Rn, but the actual dimension of u will al- to find. ways be clear from the context. Similarly, limit cycles and The analysis of system ( 1) becomes even more diffi- their bifurcations are formulated in the form of a bound- cult if one wants to follow the phase portrait under vari- ary-value problem (BVP) ation of parameters. Generically, by perturbing a parameter slightly the phase portrait changes slightly as well. u˙ f (u; p) D 0 ; Namely, if the new phase portrait is topologically equiva- g(u(0) ; u(T); p) D 0 ; (3) 8 T lent to the original one, then nothing changed from a qual- < 0 h(u(t); p)d t D 0 ; itative point of view, i. e., all attracting, repelling, and sad- R Rn Rn Rn p dle sets are still present with unchanged stability prop- with :nb boundary conditions, i. e., g : ! Rnb Rn Rn p Rn i erties, though slightly perturbed. By contrast, the critical , ni integral conditions, i. e., h : ! , and points in parameter space where arbitrarily small param- u in a proper function space. In other words, a list of eter perturbations give rise to nonequivalent phase por- defining functions is formulated, in the form ( 2) or ( 3), traits are called bifurcation points, where bifurcations are to cover all cases of interest. For example, u D x and said to occur. Bifurcations therefore result in a partition F(x; p) D f (x; p) is the AP defining equilibria of ( 1). of parameter space into regions: parameter combinations The commonly used cycle BVP, with the time-rescaling in the same region correspond to topologically equivalent t D T, is dynamics, while nonequivalent phase portraits arise for x˙ T f (x; p) D 0 ; parameter combinations in neighboring regions. Most of- x(0) x(1) D 0 ; (4) ten, this partition is represented by means of a two-dimen- 8 1 > k 1 < 0 x() x˙ ()d D 0 ; sional bifurcation diagram , where the regions of a parameter plane are separated by so-called bifurcation curves . Bi- where: R from here on > denotes the transpose and, for furcations are said to be local if they occur in an arbitrarily simplicity, ˙ for d/d is used. The integral condition is small neighborhood of the equilibrium or cycle; otherwise, the so-called phase condition , which ensures that x() they are said to be global . is the 1 periodic solution of ( 1) closest to the refer- Although one might hope to detect bifurcations by ence solution x k 1() (typically known from the previous simulating system ( 1) for various parameter combinations point along the continuation), among time-shifted orbits and initial conditions, a “brute force” simulation approach x( 0); 0 2 [0 ; 1]. 6332 N Numerical Bifurcation Analysis

As will be discussed in Sect. “ Discretization of BVPs ”, with an overview of existing implementations of the de- a proper time-discretization of u() allows one to approx- scribed algorithms in Sect. “ Software Environments ”. Pre- imate any BVP by a suitable AP. Thus, equilibria, limit vious reviews [ 7,9,22 ,43 ] have similar contents. This re- cycles and their bifurcations can all be represented by an view however, focuses more on the principles now under- algebraic defining function like ( 2), and numerical con- lying the most frequently used software packages for bifur- tinuation allows one to produce one-dimensional solu- cation analysis and the algorithms being used. tion branches of ( 2) under the variation of strategic components of p, called free parameters. With this approach, Continuation and Discretization of Solutions equilibria and cycles can be followed without further diffi- culty in parameter regimes where these are unstable. Then, The continuation of a solution u of ( 2) with respect to one during continuation, the stability of equilibria and cycles is parameter p is a fundamental application of the Implicit determined through linearization. Moreover, the charac- Function Theorem (IFT). terization of nearby solutions of ( 1) can be done using nor- Generally speaking, to define one-dimensional solu- mal forms , i. e., the simplest canonical models to which the tion manifolds (branches), the number of unknowns in system, close to a bifurcation, can be reduced on a lower- (2) should be one more than the number of equations, dimensional manifold of the state space, the so-called cen- i. e., np D 1. However, during continuation it is better not ter manifold . While a branch of equilibria or cycles is fol- to distinguish between state variables and parameters as lowed, bifurcations can be detected as the zero of suitable will become apparent in Sects. “ Pseudo-Arclength Contin- test functions. Upon detection of a bifurcation, the defin- uation ” and “ Moore–Penrose Continuation ”. Therefore, nC1 ing function can be augmented by this test function or an- write y D (u; p) 2 Y D R for the continuation vari- other appropriate function, and the new defining function ables in the continuation space Y and consider the con- can then be continued using one more free parameter. tinuation problem An analytical bifurcation study is feasible for simple systems only. Numerical bifurcation analysis is one of the F(y) D 0 ; (5) few but also very powerful tools to understand and de- RnC1 Rn scribe the dynamics of systems depending on parameters. with F : ! . Some basic steps while performing bifurcation analysis Let F be at least continuous differentiable, will be outlined and software implementations of contin- y0 D (u0; p0) be a known solution point of ( 5), and the uation and bifurcation algorithms discussed. matrix Fy(y0) D [Fu(u0; p0)jFp(u0; p0)] be full rank, i. e., First a few standard and often used approaches for the computation and continuation of zeros of a defin- (i) rank( Fu(u0; p0)) D n ; or ing function are reviewed in Sect. “ Continuation and rank( Fy(y0)) D n , (ii) rank( Fu(u0; p0)) D n 1 and 8 R Discretization of Solutions ”. The presentation starts with < Fp(u0; p0) … (Fu(u0; p0)) ; the most obvious, but also naive, approaches to contrast (6) : these with the methods employed by software packages. In Sect. “ Normal Forms and the Center Manifold ” several where R(Fu) denotes the range of Fu. Then the IFT states possible scenarios for the loss of stability of equilibria and that there exists a unique solution branch of ( 5) locally limit cycles are discussed. Not all bifurcations are charac- to y0. Introducing a scalar coordinate s parametrizing the terized by linearization and for the detection and analy- branch, e. g., the arclength positively measured from y0 in sis of these bifurcations, codimension 1 normal forms are one of the two directions along the solution branch, then mentioned and a general method for their computation one can represent the branch by y(s) D (u(s); p(s)) and on a center manifold is presented. Then, a list of suitable the IFT guarantees that F(y(s)) D 0 for jsj is sufficiently test functions and defining systems for the computation small. Moreover, y(s) is continuous differentiable and the of bifurcation branches is discussed in Sect. “ Continua- vector (s) D ys (s) D (us (s); ps (s)) D (v(s); q(s)), tan- tion and Detection of Bifurcations ”. In particular, when gent to the solution branch at y(s), exists and is the unique a system bifurcates new solution branches appear. Tech- solution of niques to switch to such new branches are described in Sect. “ Branch Switching ”. Finally, the computation and Fy(y(s)) (s) D Fu(u(s) ; p(s)) v(s) C Fp(u(s) ; continuation of global bifurcations characterized by orbits p(s)) q(s) D 0 ; 8 > > 2 connecting equilibria is presented, in particular homoclinic < (s) (s) D v(s) v(s) C q(s) D 1 : orbits, in Sect. “ Connecting Orbits ”. This review concludes (7) : Numerical Bifurcation Analysis N 6333

In other words, the matrix Fy(y0) has a one-dimensional of Newton steps is bounded, in order to guarantee termi- nullspace N (Fy(y0)) spanned by (0) and y0 is said to be nation. If this maximum is reached before convergence, a regular point of the continuation space Y. the computation is restarted with a smaller (typically Below, several variants of numerical continuation will halved) stepsize. However, in case of quick convergence, be described. The aim is to produce a sequence of points e. g., after only a few iterations, the stepsize is multiplied yk ; k 0 that approximate the solution branch y(s) in one (1.3 is a typical factor). In any case, the stepsize is varied direction. Starting from y0, the general idea is to make between two assigned limits hmin and hmax , so that contin- 0 a suitable prediction y1, typically along the tangent vector, uation cannot proceed when convergence is not reached from which the Newton method is applied to find the new even with minimum stepsize. When h is chosen too small, point y1. The predictor-corrector procedure is then iter- too much computational work may be performed, while ated. First, the simplest implementation is presented and for h that is chosen too large, little detail of the solution it is shown where it might fail. Many continuation pack- branch is obtained. ages for bifurcation theory use an alternative implementa- As a first improved predictor, note that the IFT sug- tion of which two variants are discussed. Many more ad- gests to use the tangent prediction for the state variables 0 vanced predictor-corrector schemes have been designed, u1 D u0 C hv 0, where v0 is obtained from ( 7) with s D 0 see [ 1,18 ,46 ] and references therein. and as v(0)/ q(0). Indeed, the tangent vector can be approximated by the difference vk D (uk uk 1)/ hk or, even Parameter Continuation better, computed at negligible cost, since the numerical de- composition of the matrix Fu(uk ; pk ) is known from the Parameter continuation assumes that the solution branch last Newton iteration. R of ( 5) can be parameterized by the parameter p 2 . In- These methods are illustrated in Fig. 2. Note in partic- deed, if Fu has full rank, i. e., case (i) in ( 6), then this is ular the folds in the sketch. Here parameter continuation possible by the IFT. Starting from ( u0; p0) and perturbing does not work, since exactly at the fold Fu(u; p) is singu- the parameter a little, with a stepsize h , the new parameter lar such that Newton’s method does not converge and be- is p1 D p0 C h and the most simple predictor for the state yond, for larger p, there is no local solution to ( 5). 0 variable is given by u1 D u0. Application of Newton’s method to find u1 satisfying (5) leads to Pseudo-Arclength Continuation Near folds, the solution branch is not well parameterized jC1 j j 1 j u1 D u1 Fu(u1; p1) F(u1; p1) ; j D 0; 1; 2;::: by the parameter, but one can use a state variable for the parametrization. In fact, the fold is a regular point (case (ii) The iterations are stopped when a certain accuracy in ( 6)) at which the tangent vector D (v; q) has no pa- jC1 j is achieved, i. e., kuk D k u1 u1k < " u and/or rameter component, i. e., q D 0. So, without distinguish- j kF(u1; p1)k < " F . In practice, also the maximum number ing between parameters and state variables, one takes the

Numerical Bifurcation Analysis, Figure 2 Parameter continuation without ( a) and with ( b) tangent prediction. The dotted lines indicate subspaces where solutions are searched 6334 N Numerical Bifurcation Analysis

0 tangent prediction y1 D y0 C h0, as long as the starting Each correction is therefore required to be orthogonal to solution y is a regular point. Since now both p and u j 0 the nullspace of Fy(y1), i. e., are corrected, one more constraint is needed. Pseudo-arclength continuation uses the stepsize h as an approxima- jC1 F(y1 ) D 0 ; tion of the required distance, in arclength, between y0 and j > jC1 j ( (1) (y1 y1) D 0 : the next point y1. This leads to the so-called pseudo-ar- > clength equation (y1 y0) D h. In this way, solution 0 0 Starting with D 0 the Newton iterations are given by branches can be followed past folds. The idea for this con- 1

tinuation method is due to Keller [ 50 ]. j 1 j jC1 j Fy(y ) F(y ) The Newton iteration, applied to D 1 1 ; y1 y1 j > 8 (1) ! 0 ! F(y ) D 0 ; ˆ (9) 1 ˆ jC1 1 > ˆ jC1 Fy(y ) 0 0 (y1 y0) h D 0 ; < D 1 ; j D 0; 1; 2;::: 1 ( j)> 1 ˆ 1 ! is given by ˆ ˆ As illustrated: in Fig. 3b, the Moore–Penrose continuation

j 1 j can be interpreted as a variant of Keller’s method in which jC1 j cF y(y1) cF (y1) y1 D y1 > ; j D 0; 1; 2;::: ; the tangent vector is updated at every Newton step. When 0 ! 0 ! the new point y1 is found, the tangent vector 1 is imme- (8) jC1 jC1 diately obtained as 1 /k1 k from the last Newton it- jC1 j eration, since j does not necessarily have unit length. where y D y1 y1 is forced to lie in the hyperplane 1 orthogonal to the tangent vector, as illustrated in Fig. 3a. Finally, for both the pseudo-arclength and Moore- Upon convergence, the new tangent vector 1 is obtained Penrose continuation methods one can prove that they by solving ( 7) at y1. converge (with superlinear convergence), provided that y0 is a regular point and the stepsize is sufficiently small. Moore–Penrose Continuation This continuation method is based on optimization. Start- Discretization of BVPs 0 ing with the tangent prediction y1 D y0 C h0, a point y1 In this section, orthogonal collocation [3,15 ] is described, 0 with F(y1) D 0 nearest to y1 is searched, so the following a discretization technique to approximate the solution of is optimized a generic BVP ( 3) by a suitable AP. Let u be at least in the space C1([0 ; 1] ; Rn ) of continuous differentiable vector- 0 min fk y1 y1kj F(y1) D 0g : valued functions defined on [0 ; 1]. For BVPs the rescaled y1

Numerical Bifurcation Analysis, Figure 3 Pseudo-arclength ( a) and Moore–Penrose ( b) continuation. Searching a solution in hyperplanes without ( a) and with ( b) updating the tangent vector. The open dots correspond to Newton iterations, full dots to points on the curve. The dotted lines indicate subspaces where solutions are searched Numerical Bifurcation Analysis N 6335

time t D T is used, so that the period T becomes a pa- sion as H0 given by the number nc of critical eigenval- rameter and in the sequel T will be addressed as such. ues or nontrivial multipliers (counting multiplicity), and Introduce a time mesh 0 D 0 < 1 < ::: < N D 1 and, is transformed to a normal form. The power of this ap- on each interval [ j 1; j], approximate the function u by proach is that the bifurcation scenario of the normal form a vector-valued polynomial }j of degree m, j D 1;:::; N. is preserved in the original system. The normal form for The polynomials }j are determined by imposing the ODE a speciﬁc bifurcation is usually studied only up to a ﬁnite in ( 3) at m collocation points z j;i , i D 1;:::; m, i. e., order, i. e., truncated, and many diagrams for bifurcations with higher codimension are in principle incomplete due

}˙ j(z j;i ) D f (}j(z j;i ); p); j D 1;:::; N; i D 1;:::; m: to global phenomena, such as connecting orbits. Also, H0 is not necessarily unique or smooth [ 75 ], but fortunately, (10) one can still draw some useful qualitative conclusions. One usually chooses the so-called Gauss points as the The codimension (codim) of a bifurcation is the min- collocation points, the roots of the mth order Legendre imal number of parameters needed to encounter the bifurcation and to unfold the corresponding normal form polynomials. Moreover, }1(0) and }N (1) must satisfy the boundary conditions and the whole piecewise polynomial generically. Therefore, in practice, one finds codim 1 phe- must satisfy the integral conditions. nomena when varying a single parameter, and continues Counting the number of unknowns, the discretization them as curves in two-parameter planes. Codim 2 phe- of ( 3) leads to nN polynomials, each with ( m C 1) de- nomena are found as isolated points along codim 1 bifurcation curves. Still, codim 2 bifurcations are important grees of freedom, plus np free parameters, so there are as they are the roots of codim 1 bifurcations, in particu- nN (m C 1) C np continuation variables. These variables are matched by nmN collocation equations from ( 10 ), lar of global phenomena. For this reason they are called organizing centers as, around these points in parameter n(N 1) continuity conditions at the mesh points, nb space, one-parameter bifurcation scenarios change. For boundary conditions, and ni integral conditions, for a total parameter-dependent systems the center manifold H0 can of nN (m C 1) C nb C ni n algebraic equations. Thus, in order for these equations to compose an AP, the number be extended to a parameter-dependent invariant manifold H(p); H0 D H(p0), so that the bifurcation scenario of free parameters is generically np D nb Cni nC1 and, typically, one is the period T. on H(p) is preserved in the original system for kp p0k The collocation method yields high accuracy with su- sufficiently small. perconvergence at the mesh points [ 15 ]. The mesh can In the following, the normal forms for all codim 1 bi- also be adapted during continuation, for instance to min- furcations of equilibria and limit cycles are presented and imize the local discretization error [ 68 ]. The equations their bifurcation scenarios discussed. Then a general com- can be solved efficiently by exploiting the particular spar- putational method for the approximation, up to a finite or- sity structure of the Jacobi matrix in the Newton itera- der, of the parameter-dependent center manifold H(p) is tion. In particular, a few full but essentially smaller sys- presented. The method gives, as a by-product, explicit for- tems are solved instead of one sparse but large system. mulas for the coefficients of a given normal form in terms During this process one finds two nonsingular ( n n) - of the vector field f of system ( 1). submatrices M0 and M1 such that M0u(0) C M1u(1) D 0, 1 i. e., the monodromy matrix M D M1 M0 is found as Normal Forms a by-product. In the case of a periodic BVP ( u(1) D u(0)) the Floquet multipliers are therefore computed at low Bifurcations can be defined by certain algebraic condi- computational cost. tions. For instance, an equilibrium is nonhyperbolic if <() D 0 holds for some eigenvalue. The simplest possibilities are D 0 (limit point bifurcation or branch point, Normal Forms and the Center Manifold though the latter is nongeneric, see Sect. “ Branch Switch- Local bifurcation analysis relies on the reduction of the dy- ing ”) and 1;2 D ˙ i!0; ! 0 > 0 (Hopf bifurcation). Bifur- namics of system ( 1) to a lower-dimensional center mani- cations of limit cycles appear if some of the nontrivial mul- fold H0 near nonhyperbolic equilibria or limit cycles, i. e., tipliers cross the unit circle. The three simplest possibilities when they bifurcate at some critical parameter p0. The are D 1 (limit point of cycles), D 1 (period-dou- ˙i existence of H0 follows from the Center Manifold Theo- bling) or 1;2 D e 0 ; 0 < 0 < (Neimark–Sacker). rem (CMT), see, e.g., [ 11 ], while the reduction principle At the bifurcation ( p D p0), linearization of system ( 1) is shown in [ 72 ]. The reduced ODE has the same dimen- near the equilibrium x0 or around a limit cycle, does not 6336 N Numerical Bifurcation Analysis

result in any stability information in the center manifold. Poincaré map P : Rn 1 ! Rn 1 maps each point close In this case, nonlinear terms are also necessary to obtain enough to z D 0 to the next return point on ˙, i. e., such knowledge. This is provided by the critical normal P : z 7! Z(˚ T(z)(Z 1(z))). Thus, bifurcations of limit cy- form coefficients as discussed below. The state variable in cles turn into bifurcations of fixed points of the Poincaré the normal form will be denoted by w and the unfolding map which can be easily described using local bifurcation parameter by ˛ 2 R, with w D 0 at ˛ D 0 being a non- theory. Moreover, it can be shown that the n 1 eigenval- hyperbolic equilibrium. Bifurcations are labeled in accor- ues of the linearization Pz (0) are the nontrivial eigenval- T dance with the scheme of [ 38 ]. ues of the monodromy matrix M D ˚x (x0(0)), which also has a trivial eigenvalue equal to 1 (the vector f (x0(0) ; p), Codimension 1 Bifurcations of Equilibria tangent to the cycle at x0(0), is mapped by M to itself). Limit point bifurcation (LP ): The equilibrium has a sim- The eigenvalues of Pz (0) are therefore the nontrivial mul- ple eigenvalue D 0 and the restriction of ( 1) to a one-di- tipliers of the cycle. Although the Poincaré map and its mensional center manifold can be transformed to the nor- linearization can also be computed numerically through mal form suitably organized simulations (so-called shooting techniques [ 17 ]), it is better to handle both the cycle multipli- 2 3 R w˙ D ˛ C aLP w C O(jwj ) ; w 2 ; (11) ers and normal form computations associated to nonhyperbolic cycles using BVPs [ 9,23 ,57 ]. Here, however, the where generically aLP ¤ 0 and O denotes higher order normal forms on a Poincaré section are presented, where terms in state variables depending on parameters too. w D 0 at ˛ D 0 is the fixed point of the Poincaré map cor- When the unfolding parameter ˛ crosses the critical value responding to a nonhyperbolic limit cycle. (˛ D 0), two equilibria, one stable and one unstable in the Limit point of cycles (LPC ): The fixed point has one sim- center manifold, collide and disappear. This bifurcation is ple nontrivial multiplier D 1 on the unit circle and the also called saddle-node , fold or tangent bifurcation. Note restriction of P to a one-dimensional center manifold has that this bifurcation occurs at the folds in Figs. 2 and 3. the form Hopf bifurcation (H): The equilibrium has a complex pair 2 3 of eigenvalues 1 D 2 D i!0 and the restriction of ( 1) w 7! ˛ C w C aLPC w C O(w ) ; w 2 R ; to the two-dimensional center manifold is given by where aLPC ¤ 0. As for the LP bifurcation two fixed points 2 4 w˙ D (i!0 C ˛)w C cHw w¯ C O(jwj ) ; w 2 C ; (12) collide and disappear when ˛ crosses the critical value, provided aLPC ¤ 0. This implies the collision of two limit where generically the first Lyapunov coefficient dH D cycles of the original vector field f . <(cH) ¤ 0. When ˛ crosses the critical value, a limit cy- Period-doubling (PD ): The fixed point has one simple cle is born. It is stable (and present for ˛ > 0) if dH < 0 multiplier D 1 on the unit circle and the restriction and unstable if dH > 0 (and present for ˛ < 0). The case of P to a one-dimensional center manifold can be trans- dH < 0 is called supercritical or “soft”, while dH > 0 is formed to the normal form called subcritical or “hard” as there is no (local) attrac- 3 4 tor left after the bifurcation. This bifurcation is most of- w 7! (1 C ˛)w C bPD w C O(w ) ; w 2 R ; ten called Hopf, but also Poincaré–Andronov–Hopf as this where bPD ¤ 0. When the parameter ˛ crosses the critical was also known to the first two. value and bPD ¤ 0, a cycle of period 2 for P bifurcates from the fixed point corresponding to a limit cycle of period 2 T Codimension 1 Bifurcations of Limit Cycles Bifurca- for the original system ( 1). This phenomenon is also called tions of limit cycles are theoretically very well under- the flip bifurcation. If bPD is positive [negative], the bifur- stood using the notion of a Poincaré map. To define cation is supercritical [subcritical] and the double period this map, choose a ( n 1)-dimensional smooth cross-sec- cycle is stable [unstable] (and present for ˛ > 0[ ˛ < 0]). tion ˙ transversal to the cycle and introduce a local coor- NS n 1 Neimark–Sacker ( ): The fixed point has simple critical dinate z 2 R such that z D Z(x) is defined on ˙ and ˙i multipliers 1;2 D e 0 and no other multipliers on the invertible. For example, one chooses a coordinate plane unit circle. Assume that e ik 0 ¤ 1 for k D 1; 2; 3; 4, i.e., x D 0 such that f (x)j ¤ 0. Let x (t) be the cycle j j x jD0 0 there are no strong resonances. Then, the restriction of P with period T, so that z D Z(x (0)) is the cycle intersec- 0 0 to a two-dimensional center manifold can be transformed ˙ D tion with , where z 0 can always be assumed with- to the normal form out loss of generality. Denote by T(z) the return time i(˛) 2 4 to ˙ defined by the flow ˚ with T(z0) D T. Now, the w 7! e (1 C ˛)w C cNS w w¯ C O(jwj ) ; w 2 C ; Numerical Bifurcation Analysis N 6337

where cNS is a complex number and (0) D 0. Provided To verify nondegeneracy conditions, only an approxima- dNS D < (e i 0 cNS ) ¤ 0, a unique closed invariant curve tion to the solution of the homological equation is re- for P appears around the fixed point, when ˛ crosses the quired. To this end, G; H and V are expanded in Taylor critical value. In the original vector field, this corresponds series: to the appearance of a two-dimensional torus with (quasi-) 1 periodic motion. This bifurcation is also called secondary G(w; ˛ ) D g w ˛ ; !! Hopf or torus bifurcation. If dNS is negative [positive], jjCj j 1 X the bifurcation is supercritical [subcritical] and the in- 1 (17) H(w; ˛ ) D h w ˛ ; variant curve (torus) is stable [unstable] (and present for !! ˛ > 0[ ˛ < 0]). jjCjXj 1 2 V(˛) D v10 ˛1 C v01 ˛2 C O(k˛k ) ; Center Manifolds where g are the desired normal form coefficients and Generally speaking, the CMT allows one to restrict the dy- ; are multi-indices. For a multi-index one has namics of ( 1) to a suspended system D (1; 2; : : : ; n ) for nonnegative integers i, ! D ! ! : : : !; jj D C C : : : C , ˜ if Rnc Rn p Rnc 1 2 n 1 2 n w˙ D G(w; ˛ ); G : ! ; (13) 1 n ˜i i for all i D 1;:::; n and w D w1 : : : wn . When dealing with just the critical coefficients, i. e., ˛ D 0, on the center manifold H and here np is typically 1 or 2 depending on the codimension of the bifurcation. Although the index is omitted. Substitution of this ansatz into the normal forms ( 13 ) to which one can restrict the system (16 ) gives a formal power series in w and ˛. As both sides near nonhyperbolic equilibria and cycles are known, these should be equal for all w and ˛, the coefficients of the cor- results are not directly applicable. Thus, efficient numeri- responding powers should be equal. For each vector h , cal algorithms are needed in order to verify the nondegen- (16 ) gives linear systems of the form eracy conditions in the normal forms listed above. L h D R ; (18) Here, a powerful normalization method due to Iooss and coworkers is reviewed, see [ 9,14 ,29 ,37 ,55 ,59 ]. This where L D A In ( is a weighted sum of the method assumes very little a priori information, actually critical eigenvalues) and R involves known quantities only the type of bifurcation such that the form, i. e., the of G and H of order less than or equal to jj C j j. This nonzero coefficients of G, is known. This fits very well in leads to an iterative procedure, where, either system ( 18 ) is a numerical bifurcation setting where one computes fam- nonsingular, or the required coefficients g are obtained ilies of solutions and monitors and detects the occurrence by imposing solvability, i. e., R lies in the range of L of bifurcations with higher codimension during the con- and is therefore orthogonal to the eigenvectors of L> as- tinuation. sociated to the zero eigenvalue. In the second case, the so- Without loss of generality it is assumed that x D 0 at 0 lution of ( 18 ) is not unique, and one typically selects the the bifurcation point p0 D 0. Expand f (x; p) in Taylor se- h without components in the nullspace of L . How- ries ever, the nonuniqueness of the center manifold does not 1 1 affect qualitative conclusions. The parameter transforma- f (x; p) D Ax C 2 B(x; x) C 6 C(x; x; x) C J1 p (14) tion, i. e., the v, is obtained by imposing certain condi- C A1(x; p) C ::: ; tions on some normal form coefficients, leading to a solv- parametrize, locally to ( x; p) D (0 ; 0), the parameter-de- able system. pendent center manifold by One can perform an analogous procedure for the Poincaré map by using a Taylor expansion x D H(w; ˛ ) ; H : Rnc Rn p ! Rn ; (15) 1 1 P(z; p) D Az C 2 B(z; z) C 6 C(z; z; z) C : : : (19) and define a relation p D V(˛) between the original and unfolding parameters. The invariance of the center man- and the homological equation for maps ifold can be exploited by differentiating this parametrization with respect to time to obtain the so-called homologi- P(H(w; ˛ ); V(˛)) D H(G(w; ˛ ); ˛ ) : (20) cal equation The detailed derivation of the formulas for all codim 1 f (H(w; ˛ ); V(˛)) D Hw (w; ˛ )G(w; ˛ ) : (16) and 2 cases for equilibria and cycles can be found in [ 9,55 , 6338 N Numerical Bifurcation Analysis

Numerical Bifurcation Analysis, Table 1 Critical normal form coefficients for generic codim 1 bifurcations of equilibria and fixed points. Here, A, B and C refer to the expansion (14 ) for equilibria, while for fixed points they refer to ( 19 )

Eigenvectors Critical normal form coefficients Av D 0 1 > LP aLP D w B(v; v) A>w D 0 2 1 > Av D i!0v cH D w¯ C(v; v; v¯) C 2B(v; h ) C B(v¯; h ) H 11 20 > 2 A w D i!0w 1 1 h11 D A B(v; v¯); h20 D (2 i!0In A) B(v;v) Av D v 1 > LPC aLPC D w B(v; v) A>w D w 2 1 > Av D v bPD D w C(v; v; v) C 3B(v; h ) PD 2 > 6 A w D w 1 h2 D (In 1 A) B(v; v) Av D i0 v e 1 > cNS D w¯ C(v; v; v¯) C 2B(v; h11 ) C B(v¯; h20 ) NS A>w D e i0 w 2 1 2i 1 ik h D In A B v; v¯ ; h D 0 In A B v; v e 0 ¤ 1; k D 1; 2; 3; 4 11 ( 1 ) ( ) 20 (e 1 ) ( )

56 ,59 ,60 ]. The formulas for the critical normal form coef- ues of the Jacobi matrix. For large systems and stiff prob- ficients for codim 1 bifurcations are presented in Table 1. lems this is prohibitive as it is numerically expensive and Note once more that for limit cycles a numerical more not always accurate. Instead, one can base test functions appropriate method exists [ 57 ] based on periodic normal on determinants. forms [ 47 ,48 ]. Test Functions for Limit Point Bifurcations Along an Continuation and Detection of Bifurcations equilibrium curve the product of the eigenvalues changes sign at a limit point. Recall that the determinant of A is Along a solution branch one generically passes through bi- the product of its eigenvalues. Therefore, the following test furcation points of higher codimension. To detect such an function can be computed event, a test function ' is defined, where the event cor-

responds to a regular zero. If at two consecutive points 'LP D det( fx (x; p)) (21) yk 1; yk along the branch the test function changes sign, i. e., '(yk )'(yk 1) < 0, then the zero can be located more without computing the eigenvalues explicitly. precisely. Usually, a one-dimensional secant method is For the LP bifurcation the pseudo-arclength or used to find such a point. Now, if system ( 1) has a bifur- Moore–Penrose continuation methods provide an excel- cation at y0 D (x0; p0), then there is generically a curve lent test function as a by-product of the continuation. Note y D y(s) where the system displays this bifurcation. In or- that while passing through the fold, the last component der to find this curve, one starts with a known point y 0 of the tangent vector changes sign as the continuation and formulates a defining system and then continue that direction in the parameter reverses. The test function is solution in one extra free parameter. therefore defined as

'LP D : (22) Test Functions for Codimension 1 Bifurcations nC1 An equilibrium may lose stability through a limit point, Test Functions for Hopf Bifurcations Denote the a Hopf bifurcation or in a branch point. At a limit eigenvalues of A by i (x; p) ; i D 1 :::; n and consider the or branch point bifurcation the Jacobi matrix A D following product fx (x0; p0) has an algebraically simple eigenvalue D 0

(see Sect. “ Branch Switching ” for branch points), while at 'H D (i (x; p) C j(x; p)) : a Hopf point there is a pair of complex conjugate eigenval- Yi

Test Functions for Codimension 1 Cycle Bifurcations Defining Systems for Limit Point Bifurcations The Recall that the nontrivial multipliers 1; : : : ; n 1 deter- first defining system is minimally extended adding the test mine the stability of the cycle and can be efficiently com- function ( 21 ) puted as the nontrivial multipliers of the monodromy matrix M, see Sect. “ Discretization of BVPs ”. Now the follow- f (x; p) D 0 ; ing two sets of test functions can be used to detect LPC , PD (24) det( fx (x; p)) D 0 : and NS bifurcations

n 1 This system consists of n C 1 equations in n C 2 un- 'LPC D (i 1) ; ' LPC D p ; knowns ( x; p). One problem is that the computation of the D determinant can lose accuracy for large systems. This can Yi 1 n 1 be avoided in two ways, by augmenting the system with 'PD D (i C 1) ; ' PD D det( M C In) ; the eigenspaces or using a bordering technique. iD1 Fully extended systems include the eigenvectors and Y n 1 for a LP bifurcation this leads to 'NS D (i j 1) ; ' NS D det( M ˇ M f (x; p) D 0 ; 1DYi where p denotes the parameter component of the tangent < v0 v 1 D 0 ; vector similar to ( 22 ). It should also be checked that a zero i : of 'NS corresponds to nonreal multipliers e 0 , similar to where v0 is a vector not orthogonal to N ( fx (x; p)). This the test function to detect the Hopf bifurcation. system consists of 2 n C 1 equations in 2 n C 2 unknowns There are alternatives for these test functions. One (x; p; v). can deﬁne bordered systems using the monodromy ma- The bordering technique uses the bordering trix [ 9,42 ] or a BVP formulation [ 23 ]. lemma [ 41 ]. Let A 2 Rnn be a singular matrix and let 6340 N Numerical Bifurcation Analysis

B; C 2 Rnm such that the system An alternative to ( 28 ) is now formulated as

A B V 0nm f (x; p) D 0 ; > D (26) 2 C 0m g Im fx (x; p) C In v D 0 ; 8 > (29) ˆ v v 1 D 0 ; <ˆ > is nonsingular ( V 2 Rnm, g 2 Rmm ). Typically B and C w v D 0 ; are associated to the eigenspaces of A> and A correspond- ˆ :ˆ ing to the zero eigenvalue, respectively or, during continu- where w is not orthogonal to the two-dimensional real ation, approximated by their values computed at the previ- eigenspace of the eigenvalues ˙i!. It has 2 n C 2 equa- ous point along the branch. It follows from the bordering tions for 2 n C 3 unknowns ( x; p; v; ). However, w needs lemma that A has rank deficiency m if and only if g has to be updated during continuation, e. g., as the solution 2 > > rank deficiency m. of [ fx (x; p) C In] w; v w D (0 ; 0) computed at the previous continuation point. With A D fx (x; p) and m D 1, one has g D 0 if and A further reduction is obtained exploiting the rank de- only if det( fx (x; p)) D 0. A modified and minimally extended system for limit points is thus given by ficiency. Consider the system

2 ; D ; fx (x; p) C In B V 0n2 f (x p) 0 > D g(x; p) D 0 ; C 02 g I2 and it follows from the bordering lemma that g vanishes where g is defined by ( 26 ) with A>B D AC D 0 at a pre- at Hopf points and any two components of g, e. g., g and viously computed point. During the continuation the 11 g22 , see [ 42 ], can be taken to augment Eq. ( 2) to obtain the derivatives of g w.r.t. to x and p are needed. They can either following minimally augmented system be approximated by finite differences, or explicitly (and efficiently) obtained from the second-derivatives of the vec- f (x; p) D 0 ; tor field f , see [ 42 ]. g D 0 ; (30) 8 11 < g22 D 0 ; Defining Systems for Hopf Bifurcations Defining systems for Hopf bifurcations are formulated analogously to which: has n C 2 equations for n C 3 unknowns ( x; p; ). the LP case. Adding the test function ( 23 ) creates a minimally extended system Defining Systems for Codimension 1 Bifurcations of Limit Cycles f (x; p) D 0 ; (27) det(2 f (x; p) ˇ I ) D 0 ; In principle, to study bifurcations of limit cycles one can x n compute numerically the Poincaré map and study bifur- while the fully extended system is given by cations of fixed points. If system ( 1) is not stiff, then the Poincaré map and its derivatives may be obtained with sat- f (x; p) D 0 ; isfactory accuracy. In many cases, however, continuation fx (x; p)v1 C !v2 D 0 ; using BVP formulations is much more efficient. 8 ˆ fx (x; p)v2 !v1 D 0 ; (28) Suppose a cycle x bifurcates at p D p0, then the BVP ˆ > > <ˆ w1 v1 C w2 v2 1 D 0 ; (4) defining the limit cycle must be augmented with suit- > > w v2 w v1 D 0 ; able extra functions. As for codim 1 branches of equilib- ˆ 1 2 ˆ ria, one can define either fully extended systems by includ- :ˆ where w D w1 C iw 2 is not orthogonal to the eigenvector ing the relevant eigenfunctions in the computation [ 9], or v D v1 C iv 2 corresponding to the eigenvalue i!. The vec- minimally extended systems using bordered BVPs [ 23 ,39 ]. tor w D vk 1 computed at the previous point is a suitable The regularity of these defining systems is also discussed in choice during continuation. System ( 28 ) is expressed us- these references. Since the discretization of the cycle leads ing real variables and has 3 n C 2 equations for 3 n C 3 un- to large APs, here the minimally extended approach can knowns ( x; p; v1; v2; ! ). lead to faster results even though some more algebra is in- A reduced defining system can be obtained from ( 28 ) volved, see the comparison in [ 57 ]. Below only the equa- 2 by noting that the matrix fx (x; p) C In has rank defi- tions are presented, which are added to the defining system ciency two at a Hopf bifurcation point with D !2 [67 ]. (4) for the continuation of limit cycles. Numerical Bifurcation Analysis N 6341

Fully Extended Systems The following equations can be Minimally Extended Systems For limit cycle continua- used to augment ( 4) and continue codim 1 bifurcations of tion the discretization of the fully extended BVP ( 4) with limit cycles. The eigenfunctions v need to be discretized in (31 ), ( 32 ) or ( 33 ) may lead to large APs to be solved. a similar way to that in Sect. “ Discretization of BVPs ”. The In [ 39 ] a minimally extended formulation is proposed to previous cycle x k 1 and eigenfunction vk 1 are assumed augmenting ( 4) with a function g with only a few compo- to be known. nents. The corresponding function g is defined using bor- LPC : For the limit point of cycles bifurcation, the BVP dered systems. (4) is augmented with the equations LPC : For this bifurcation, one uses suitable bordering functions v1; w1 and vectors v2; w2; w3 such that the fol- v˙() T f x (x(); p)v() f (x(); p) D 0 ; lowing system linear in ( v; ; g) is regular v(1) v(0) D 0 ; 8 1 (31) v>()x˙k 1()d D 0 ; ˆ 0 v˙() T f x (x(); p)v f (x(); p) C w1g D 0 ; < 1 v>()v()k 1d C k 1 D 1 ; 0 R v(1) v(0) C w2g D 0 ; 8 1 ˆ f (x(); p)>v()d C w g D 0 ; for the: variablesR ( x; p; T; v; ). Note that ˆ 0 3 < 1 v>v()d C v D 1 : R 0 1 2 v˙() T f x (x(); p)v() T f p(x(); p)q ˆ ˆ R (34) f (x(); p) D 0 ; : 8 LPC ˆ v(1) v(0) D 0 ; The function g D g(x; T; p) vanishes at a point. The ˆ 1 > k 1 <ˆ 0 v ()x˙ ()d D 0 ; bordering functions v1; w1 and vectors v2; w2; w3 can be 1 > k 1 k 1 k 1 k 1 0 v ()v() d C q q C D 1 ; updated to keep ( 34 ) nonsingular, in particular, v1 D v ˆ R k 1 ˆ and v2 D from the previously computed point ˆ R defines: the tangent vector D (v; q; ) to the solution are used. It is convenient to introduce the Dirac oper- branch, so that ( 31 ) simply imposes q D 0, i. e., the limit ator ıi f D f (i) and the integral operator Int v() f D point. Together with ( 4), they compose a BVP with 2 n 1 v()> f ()d and to rewrite ( 34 ) in operator form ODEs, 2 n boundary conditions, and 2 integral conditions, 0 i. e., n D 2n C 2 2n C 1 D 3, namely T and two free R p D T f (x(); p) f (x(); p) w c0 parameters. Similar dimensional considerations hold for x 1 v ı0 ı1 0 w2 0 the PD and NS cases below. 0 1 D 0 1 : Int ; 0 w 0 1 0 PD : For the period-doubling bifurcation, the extra f (x( ) p) 3 g B Int v 0 C B 1 C equations augmenting ( 4) are B v1() 2 C @ A B C @ A @ (35)A v˙() T f x (x(); p)v() D 0 ; v(1) C v(0) D 0 ; (32) PD : The same notation as for the minimally extended 8 1 > k 1 LPC defining system is used and suitable bordering func- < 0 v ()v ()d D 1 ; tions v1; w1 and vector w2 are chosen such that the follow- for: the variablesR ( x; p; T; v). Here v is the eigenfunc- ing system is regular tion of the linearized ODE associated with the multiplier D 1. In fact, the second equation in ( 32 ) im- D T f (x(); p) w 0 x 1 v poses v(1) D Mv (0) D v(0), where M is the monodromy ı C ı w D 0 : (36) 0 0 1 21 g 0 1 matrix, while the third equation scales the eigenfunction Int v1() 0 1 against the previous continuation point. @ A @ A NS : For the Neimark–Sacker bifurcation, the BVP ( 4) At a PD bifurcation g(x; T; p) defined by ( 36 ) vanishes. is augmented with the equations NS : Let ˆ D cos( ) denote the real part of the nonhyperbolic multiplier and choose bordering functions ˙ ; D ; v( ) T f x (x( ) p)v( ) 0 v1; v2; w11 ; w12 and vectors w21 ; w22 such that the follow- i v(1) e v(0) D 0 ; (33) ing system is nonsingular and defines the four components 8 1 > k 1 < 0 v¯ ()v ()d D 1 ; of g : R ; ; ; ; 2 C1 ; ; Cn for the variables ( x p T v ) with v ([0 1] ). D T f x (x(); p) w11 w12 Here v is the eigenfunction of the linearized ODE asso- ı2 2ıˆ 1 C ı0 w21 w22 v 0n2 i 0 1 D : (37) ciated with the multiplier D e . Of course, the real for- Int 0 0 g I v1() 2 mulation should be used in practice. B Int 0 0 C B v2() C @ A 6342 N Numerical Bifurcation Analysis

At a NS bifurcation the four components of g(x; T; p) de- in the normal form ( 12 ), also called Bautin or degener- fined by ( 37 ) vanish, and, similar to the Hopf bifurcation, ate Hopf). For cycles, there are strong resonances ( R1 – the BVP ( 4) can be augmented with any two components R4 ), fold-flip ( LPPD ), fold-Neimark–Sacker ( LPNS ), flip- of g. Neimark–Sacker ( PDNS ), and double Neimark–Sacker (NSNS ) among those involving linear terms, while higher order degeneracies lead to cusp ( CP ), degenerate flip Test Functions for Codimension 2 Bifurcations (GPD ), and Chenciner ( CH ) bifurcations. Naturally the normal form coefficients, see [ 9,56 ], are a suitable choice During the continuation of codim 1 branches, one meets for the corresponding test functions. In Tables 2 and 3, generically codim 2 bifurcations. Some of which arise test functions are given which are defined along the corre- through extra instabilities in the linear terms, while other sponding codim 1 branches of equilibrium and limit cycle codim 2 bifurcations are defined through degeneracies in bifurcations, respectively. The functions refer to the cor- the normal form coefficients. For equilibria, codim 2 bifur- responding defining system and to Table 1. Upon detect- BT cations of the first type are the Bogdanov–Takens ( , two ing and locating a zero of a test function it may be neces- zero eigenvalues with only one associated eigenvector), sary to check that a bifurcation is really involved, similar ZH the zero-Hopf ( , also called Gavrilov-Guckenheimer, to the Hopf case where neutral saddles are excluded. For a simple zero eigenvalue and a simple imaginary pair), and details about the dynamics and the bifurcation diagrams HH the double Hopf ( , two distinct imaginary pairs), while at codim 2 points, see [ 2,44 ,56 ]. higher order degeneracies lead to cusp ( CP , aLP D 0 in the normal form ( 11 )) or generalized Hopf ( GH , dH D 0 Branch Switching This section considers points in the continuation space Numerical Bifurcation Analysis, Table 2 from which several solution branches of interest, with Test functions along limit point and Hopf bifurcation curves. The the same codimension, emanate. At these points suit- matrix Ac for the test function of the double Hopf bifurcation can be obtained as the orthogonal complement in Rn of the Jacobi able “branch switching” procedures are required to switch matrix A w.r.t. the two-dimensional eigenspace associated with from one solution branch to another. First, the transver- the computed branch of Hopf bifurcations sal intersection of two solution branches of the same continuation problem is considered, which occurs at so-called Label LP H Branch Points ( BP ) (also called “singular” or “transcriti- cusp CP aLP cal” bifurcation points). Branch points are nongeneric, in generalized Hopf GH dH the sense that arbitrarily small perturbations of F in ( 2) w> v Bogdanov–Takens BT LP LP turn the intersection into two separated branches, which zero-Hopf ZH 'H 'LP come close to the “ghost” of the (disappeared) intersec- double Hopf HH det(2 Ac ˇ In 2)

Numerical Bifurcation Analysis, Table 3 Test functions along LPC , PD and NS bifurcation curves. The matrix Mc along the Neimark–Sacker bifurcation curve is defined similarly as Ac in Table 2 along the Hopf bifurcation as the orthogonal complement of the monodromy matrix M w.r.t. the two-dimensional eigenspace associated with the computed branch of Neimark–Sacker bifurcations

Label LPC PD NS cusp CP aLPC degenerate flip GPD bPD Chenciner CH dNS > resonance 1:1 R1 wLP vLP ˆ 1 > resonance 1:2 R2 wPD vPD ˆ C 1 R3 1 resonance 1:3 ˆ C 2 resonance 1:4 R4 ˆ fold-flip LPPD 'PD 'LP fold-Neimark–Sacker LPNS 'NS 'LP flip-Neimark–Sacker PDNS 'NS 'PD

double Neimark–Sacker NSNS det( Mc ˇ Mc I(n 2)( n 3)/2 ) Numerical Bifurcation Analysis N 6343

Numerical Bifurcation Analysis, Figure 4 BP BP BP (a) Projection of two solution branches of ( 2), intersecting at y , on the null-space of Fy(y ) ( N ), close to y (planar representation in coordinates ( ˛1; ˛ 2)withrespectto agivenbasis).Thetwo solution branchesare approximated by the straight lines in N spanned by their tangent vectors at yBP (thick vectors ). (b) and (c) Projection on N , close to yBP , of the two solution branches of the perturbed problem (see ( 43 ) for b > 0 and b < 0, respectively)

BP tion but then fold ( LP ) and leave as if they follow the other A BP is not a regular point, since rank ( Fy(y )) D branch (see Fig. 4). n 1. Distinguishing between state and parameters, there BPs, however, are very common in applications due are two possibilities to particular symmetries of the continuation problem, like BP BP BP reflections in state space, conserved quantities or the pres- (i) dim N (Fu(y )) D 1; Fp(y ) 2 R(Fu(y )) ence of trivial solutions. This is why BP detection and con- H) 1 D (v1; 0) ; 2 D (v2; q2); tinuation recently received attention [ 16 ,25 ,28 ]. Then, the 8 N BP BP R BP ˆ (ii) dim (Fu(y )) D 2; Fp(y ) … (Fu(y )) <ˆ switch from a codim 0 solution branch to that of a dif- H) 1 D (v1; 0) ; 2 D (v2; 0) ; ferent continuation problem at codim 1 bifurcations is ˆ (38) examined. In particular, the equilibrium-to-cycle switch : at a Hopf bifurcation and the period-1-to-period-2 cy- for suitably chosen v1; v2; q2. In particular, in the first case, cle switch at a flip bifurcation are discussed. Finally, var- BP v1 spans the nullspace of Fu(y ) and 2 is determined ious switches between codim 1 solution branches of dif- BP BP > > 2 by solving Fu(y )v2 C Fp(y )q2; v1 v2; v2 v2 C q2 D ferent continuation problems at codim 2 bifurcations are (0 ; 0; 1). addressed. BPs can be detected by means of the following test function

Fy(y) Branch Switching at Simple Branch Points 'BP D det ; (39) > Simple BPs are points yBP D (uBP ; pBP ), encountered along a solution branch of ( 2), at which the nullspace where is the tangent vector to the solution branch dur- N (Fy(y)) of Fy(y) is two-dimensional, i. e., the nullspace ing continuation, which indeed vanishes when ( 2) admits is spanned by two independent vectors 1; 2 2 Y, with a second independent tangent vector. Note from ( 38 ) that > i i D 1, i D 1; 2. Generically, two solution branches test function ( 21 ) also vanishes at BPs, so that test function of ( 2) pass through yBP , with transversal tangent vectors (22 ) is more appropriate for LPs. given by suitable combinations of 1 and 2. In the follow- The vectors tangent to the two solution branches in- ing, only the case of the AP ( 2), i. e., Y D RnC1 (u 2 Rn tersecting at yBP can be computed as follows. Parametrize and p 2 R) and F : RnC1 ! Rn is considered. Similar one of the two solution branches by a scalar coordi- considerations hold for the BVP ( 3) (see [ 16 ] for details), nate s, e. g., the arclength, so that y(s) and ys (s) denote though, loosely speaking, results for APs can be applied to the branch and its tangent vector locally to y(0) D yBP . BVPs after time discretization. Then, F(y(s)) is identically equal to zero, so taking twice 6344 N Numerical Bifurcation Analysis

n the derivative w.r.t. s one obtains Fyy (y(s))[ ys (s); ys (s)] C Counting equations, 2 n C 2, and variables u; 2 R , BP Fy(y(s)) yss (s) D 0, which at y reads p 2 R, i. e., 2 n C 1 scalar variables, it follows that two extra parameters generically need to be freed. In other words, BP BP Fyy (y )[ ys (0) ; ys (0)] C Fy(y )yss (0) D 0 ; (40) BPs are codim 2 bifurcations, which are not expected along generic solution branches of ( 2). n with ys (0) D ˛11 C ˛22. Let 2 R span the nullspace BP > > BP of Fy(y ) with D 1. Since the range of Fy(y ) is Non-generic Problems BP continuation can be per- BP > orthogonal to the nullspace of Fy(y ) , one can elimi- formed in a single extra free parameter for nongeneric > nate yss (0) in ( 40 ) by left-multiplying both sides by , problems characterized by symmetries that persist for all thus obtaining parameter values. In such cases, the continuation prob- > BP lem ( 2) is perturbed into Fyy (y )[ ˛11 C ˛22; ˛ 11 C ˛22] D 0 : (41)

F(y) C bu b D 0 ; (43) Equation ( 41 ) is called the algebraic branching equa- n tion [50 ] and is often written as where b 2 R and ub 2 R are new variables of the defining system. The idea is that u “breaks the symmetry”, in 2 2 b c11 ˛1 C 2c12 ˛1˛2 C c22 ˛2 D 0 ; (42) the sense that problem ( 43 ) has no BP for small b ¤ 0, and BP continuation can be performed in two extra free with c D >F (yBP )[ ; ]; i; j D 1; 2. At BP detec- ij yy i j parameters, one of which, b, remains zero during the con- tion, the discriminant c c c2 is generically negative 11 22 12 tinuation. The choice of u is not trivial. Geometrically, u (otherwise the BP would be an isolated solution point b b must be such that small values of b perturb Fig. 4a into of ( 2)), so that two distinct pairs ( ˛ ; ˛ ) and ( ˛˜ ; ˛˜ ), 1 2 1 2 Fig. 4b, say for b > 0, and into Fig. 4c for b < 0. It turns uniquely defined up to scaling, solve ( 42 ) and give the di- out (see, e.g., [ 16 ]) that ub D is a good choice, i. e., per- rections of the two emanating branches. BP turbations not in the range of F (y ) break the symmetry, Once the two directions are known, one can easily per- y since close to the BP, they must be balanced by the nonlin- form branch switching by an initial prediction from yBP ear terms of the expansion of F in ( 43 ), and this implies along the desired direction. This, however, requires the significant deviations of the perturbed solution branch y second-order derivatives of F w.r.t. all continuation vari- from the unperturbed y(s). ables. Though good approximations can often be achieved BP continuation for nongeneric problems is therefore by finite differences, an alternative and computationally BP defined by cheap prediction can be taken in the nullspace of Fy(y ) along the direction orthogonal to ys(0). The vector ys(s) is F(x; p) C b D 0 ; in fact known at each point during the continuation of the F (x; p)> D 0 ; 8 x (44) solution branch up to BP detection, so that the cheap pre- F (x; p)> D 0 ; ˆ p diction for the other branch spans the (one-dimensional) < > 1 D 0 : nullspace of ˆ This:ˆ defining system is also useful for accurately comput- BP Fy(y ) ing BPs. In fact, the basin of convergence of the New- > : ys (0) ton iterations in ( 8) or ( 9) shrinks at BPs (recall F (y) y does not have full rank at BPs), while system ( 44 ), in Branch Point Continuation the 2 n C 2 variables ( u; p; b; ), has a unique solution BP BP Generic Problems Several defining systems have been (u ; p ; 0; ) close to the BP. Thus, when the BP test proposed for BP continuation, see [ 16 ,25 ,28 ,63 ,64 ,65 ]. function ( 39 ) changes sign along a solution branch of ( 2), Among fully extended formulations, the most compact Newton corrections can be applied to ( 44 ), starting from the best possible prediction, i. e., with b D 0 and as the one characterizes BPs as points at which the range of Fy(y) > > eigenvector of Fu(u; p) associated with the real eigen- has rank defect 1, i. e., the nullspace of Fy(y) is one-dimensional. BP continuation is therefore defined by value closest to zero.

F(u; p) D 0 ; Minimally Extended Formulation A minimally ex- F (u; p)> D 0 ; tended defining system for BP continuation requires two 8 u F (u; p)> D 0 ; scalar conditions, g1(u; p) D 0, g2(u; p) D 0, to be added ˆ p < > 1 D 0 : to the unperturbed or perturbed problem ( 2) or ( 43 ) ˆ :ˆ Numerical Bifurcation Analysis N 6345 for generic and nongeneric problems, respectively. These start the continuation of the emanating branch of “period- PD functions g1 and g2 are defined in [ 28 ] by solving 2” limit cycles, i. e., those which close to y have approx- imately the double of the period of the bifurcating cycle. k 1 Fy(y)1 C g1 D 0 ; For this, branch switching at simple BPs can be used. In k 1 Fy(y)2 C g2 D 0 ; fact, two solution branches of the limit cycle BVP ( 4) trans- 8 k 1 > PD ˆ (1 ) 1 1 D 0 ; versely intersect at y if one considers T as the doubled ˆ k 1 > ˆ (2 ) 2 1 D 0 ; period: the branch of interest and the branch along which <ˆ k 1 > (1 ) 2 D 0 ; the corresponding period-1 cycle is traced twice. In other k 1 > ˆ ( ) 1 D 0 ; words, one can see the period-doubling bifurcation in the ˆ 2 ˆ period-1 branch as the “period-halving” bifurcation in the :ˆ in the unknowns 1; 2; g1; g2, while is updated by solv- period-2 branch. ing Alternatively, the vector D (v; q) tangent to the period-2 branch at yPD is given by the flip theorem [ 56 ] > Fy(y) C g11 C g22 D 0 ; and does not need to be computed by solving the algebraic > 1 D 0 ; branching Eq. ( 41 ). In particular, the initial solution of the period-2 BVP is x(t) D xPD (2 t), p D pPD , T D 2TPD , in the unknowns , g1, g2, after each Newton conver- while q D (ps ; Ts ) D 0 and gence. w(t) ; 0 t < 1 ; v(t) D Branch Switching at Hopf Points ( w(t 1) ; 1 t < 2 ; H H H At a Hopf bifurcation point y D (x ; p ), one typically where w(t) is the unit-length eigenfunction of the lin- wants to start the continuation of the emanating branch earized (time-dependent) ODE associated with the mul- of limit cycles. For this, one might think of using the tiplier 1, i. e., branch switching procedure described above to switch PD PD PD from a constant to a periodic solution branch of the limit w˙ T fx (x ; p )w D 0 ; cycle BVP ( 4). Unfortunately, yH is not a simple BP for w(1) C w(0) D 0 ; 8 1 > problem ( 4), since the period T is undetermined along < 0 w() w() d D 1 : the constant solution branch, so that, formally, an infinite : R number of branches emanate from yH. Thus, a prediction Branch Switching at Codimension 2 Equilibria in the proper direction, i. e., along the vector D (v; q) n 2 Assume that y2 D (x2; p2), x2 2 R , p2 2 R , identifies tangent to the periodic solution branch, is required. a codim 2 equilibrium bifurcation point of system ( 1), Let y(s) represent the periodic solution branch, with CP GH H either cusp ( ), Generalized Hopf ( ), Bogdanov– y(0) D y . Then, x and v are period-1 vector-valued func- Takens, ( BT ), zero-Hopf ( ZH ), or double Hopf ( HH ). Then, C1 Rn R tions in ([0 ; 1] ; ), p 2 and T are the free parame- according to the analysis of the corresponding normal ters, and q D (ps ; Ts ). The Hopf bifurcation theorem [ 56 ] form ( 13 ) with two parameters and critical dimension ensures that ps D Ts D 0 and that v is the unit-length so- nc D 1; 2; 3; 4 at CP , BT and GH , ZH , HH points, respec- lution of the linearized, time-independent equation v˙ D H H tively, several curves of codim 1 bifurcations of equilib- T(0) fx (x ; p )v, i. e., v() D sin(2 )wr C cos(2 )wi , > > > ria and limit cycles emanate from p2 in the parameter where w D wr C iw i (wr wr C wi wi D 1; wr wi D 0) H H plane [ 56 ]. Here, the problem of how the continuation of is the complex eigenvector of fx (x ; p ) associated to the such curves can be started from y2 is discussed, by restrict- eigenvalue i!, ! D 2/T(0). ing the attention to equilibria bifurcations (an example of The periodic solution branch of the limit cycle BVP ( 4) a cycle bifurcation is given at the end, and see [ 9,61 ]). can therefore be followed, provided the phase condition In general, the normal form analysis also provides an 1 > ˙ D (see the last equation in ( 4)) is replaced by 0 x vd 0 approximation of each emanating codim 1 bifurcation, in at the first Newton correction. Otherwise, x would be un- R the form of a parameterized expansion determined among time-shifted solutions. 1 1 w D w" ; ˛ D ˛" ; (45) ! ! Branch Switching at Flip Points X1 X1 At a flip bifurcation point yPD D (xPD ; pPD ), where xPD 2 for small " > 0 and up to some finite order. Then, the (pa- C1([0 ; 1] ; Rn ), xPD (1) D xPD (0), one typically wants to rameter-dependent) center manifold ( 15 ) maps such an 6346 N Numerical Bifurcation Analysis

approximation back to the solution branch of the orig- center manifold [ 61 ]. However, initial predictions for ho- inal system ( 1), locally to y2 D (H(0 ; 0) ; V(0)), and al- moclinic and heteroclinic bifurcations for both zero-Hopf lows one to compute the proper prediction from y2 along and double Hopf cases are not available in general, but the direction of the desired codim 1 bifurcation. How- see [ 35 ]. ever, as will be concluded in the following, such approx- The double Hopf bifurcation appears when two differ- imations are not needed to start equilibria bifurcations ent branches of Hopf bifurcations intersect. Several bifur- and are therefore not derived (see [ 9] for all available de- cation curves are rooted at the double Hopf point. In par- tails). ticular, it is known that there are generically two branches, two half-lines in the parameter plane, emanating from this Switching at a Cusp Point At a generic cusp point point along which a Neimark–Sacker bifurcation of limit yCP D (uCP ; pCP ), two fold bifurcation curves terminate cycles occurs [ 56 ]. Here it is discussed how to initialize tangentially. Generically, the cusp point is a regular point the continuation of a Neimark–Sacker branch (using ( 37 )) for the fold defining systems ( 24 ) and ( 25 ), where the tan- from a double Hopf point after continuation of a Hopf gent vector has zero p-components. The cusp geometri- branch. The initialization requires approximations of the cally appears once the fold solution branch is projected in cycle x, the period T, the parameters p and the real part of the parameter plane. Thus, the continuation of the two the multiplier ˆ. These can be obtained by reducing the dy- fold bifurcations can simply be started as forward and namics of ( 1) to the center manifold. On the center mani- backward fold continuation from yCP . Since the continu- fold, the dynamics near a HH bifurcation point is governed ation direction is uniquely defined, neither a tangent pre- by the following normal form diction nor a nonlinear expansion of the desired solution branch are necessary. w˙ 1 D w˙2 Switching at a Generalized Hopf Point At a Hopf point (i! (˛) C ˛ )w C g w jw j2 C g w jw j2 with vanishing Lyapunov coefficient yGH D (xGH ; pGH ), 1 1 1 2100 1 1 1011 1 2 (i! (˛) C ˛ )w C g w jw j2 C g w jw j2 a LPC bifurcation terminates tangentially to a Hopf bi- 2 2 2 1110 2 1 0021 2 2 4 furcation, which turns from super- to subcritical, and vice C O(k (w1; w2)k ) ; versa. Thus, yGH is a regular point for the Hopf defining (46) systems ( 27 )–( 30 ). 2 i where ( w1; w2) 2 C . In polar coordinates, w1 D 1e 1 ; i Switching at a Bogdanov–Takens Point At a generic w2 D 2e 2 , the asymptotics from the normal form as in Bogdanov–Takens point yBT D (uBT ; pBT ), a Hopf and (45 ) for the nonhyperbolic cycle on one branch are given a (saddle) homoclinic bifurcation terminate tangentially by to a fold bifurcation, along which a real nonzero eigen- BT 2 2 value of the Jacobi matrix fx (x; p) changes sign at y . (1; 2; ˛ 1; ˛ 2) D "; 0; Re(g2100 )" ; < (g1110 )" ; Generically, yBT is a regular point for the fold defining sys- (47) tems ( 24 ) and ( 25 ) and for the Hopf defining systems ( 27 ), BT (29 ), and ( 30 ). However, y is a simple BP for the Hopf with 1 2 [0 ; 2]; 2 D 0. defining system ( 28 ). In fact, the fold and Hopf branches Although high-dimensional, the computation of the are both solution branches of the continuation problem coefficients and center manifold vectors is relatively ! D D > D > D > (28 ), where 0 and v1 v2, with v1 v1 v2 v2 1/2, straightforward. Introduce Av j D i!jv j; A w j D i!jw j along the fold branch. The branch switch procedures de- > > with v¯j v j D w¯ j v j D 1 and let D (10) ; (01) and intro- scribed in this section readily apply in this case. duce the standard basis vectors e10 D (1 ; 0) ; e01 D (0 ; 1). Using the expansion ( 14 ), the cubic critical normal form Switching at Zero-Hopf and Double Hopf Points coefficients and the parameter dependence are calculated Generically, zero-Hopf and double Hopf points are regular from points for codim 1 equilibria bifurcations (fold and Hopf), > so that the proper initial prediction is uniquely defined. g2100 D w¯ 1 [C(v1; v1; v¯1) C B(h2000 ; v¯1) For limit cycle bifurcations and connecting orbits, nonlin- C 2B(h1100 ; v1)] 2 ; ear expansions of the fold and Hopf branches are needed > to derive initial predictions for the emanating branches. g1011 D w¯ 1 [C(v1; v2; v¯2) CıB(h1010 ; v¯2) For cycles the switching procedure can be set up using the C B(h1001 ; v2) C B(h0011 ; v1)] ; Numerical Bifurcation Analysis N 6347

> g1110 D w¯2 [C(v2; v1; v¯1) C B(h1100 ; v2) C B(h1010 ; v¯1) A homoclinic orbit is an orbit connecting an equilib- ¯ rium point to itself, that is, if xC D x . Similarly there C B(h1001 ; v1)] ; exist heteroclinic connecting orbits between equilibrium > g0021 D w¯2 [C(v2; v2; v¯2) C B(h0020 ; v¯2) points and periodic orbits, and homoclinic and hetero- C 2B(h0011 ; v2)]/2 ; clinic orbits connecting periodic orbits to periodic orbits. > 1 Traditionally, homoclinic orbits to equilibrium points j; D p¯ [A1(v j; e) B(v j; A J1e)] 2 ; (48) j were computed indirectly using numerical shooting or by ı where j D 1; 2 and continuing a periodic solution with a large enough but fixed period, that is close enough to a homoclinic or- 1 bit [ 24 ]. More modern and robust techniques compute h2000 D (2 i!1In A) B(v1; v1) ; connecting orbits directly using projection boundary con- D ! 1 ; ; h0020 (2 i 2In A) B(v2 v2) ditions . Computing connections with and between peri- 1 h1100 D A B(v1; v¯1) ; odic orbits is subject to current research [ 26 ,27 ,54 ]. For (49) 1 a more detailed description of the methods described here, h0011 D A B(v2; v¯2) ; 1 see [ 9]. h1010 D (i(!1 C !2)In A) B(v1; v2) ; h D (i(! ! )I A) 1B(v ; v¯ ) ; 1001 1 2 n 1 2 Formulation as a BVP where h are the vectors in the expansion of the center A heteroclinic orbit can be expressed as a BVP in the fol- manifold ( 17 ). lowing way: Now, to construct a cycle for ( 46 ), a mesh for 1 is de- x˙(t) D f (x(t); p) ; lim x(t) D x ; fined and the asymptotics ( 47 ) are inserted into the polar t! 1 coordinates. This cycle is mapped back to the original sys- 1 > tem by using ( 17 ) with ( 47 ), ( 48 ) and ( 49 ). The transfor- lim x(t) D xC ; (x(t) x0(t)) x˙0(t)d t D 0 ; t!1 1 mation between free system and unfolding parameters is Z R 1 > given by p(") D ( 1 2) (˛1; ˛ 2) using ( 47 ). Finally, where the integral condition is with respect to a reference approximating formulas for the period and the real part of solution x0(t) and fixes the phase, similarly to the phase the multiplier are given by condition for periodic orbits. This BVP, however, operates on an infinite interval, while, numerically, one can only 2 D ; ˆ D ! C ! "2 ; operate on a finite, truncated interval [ T ; TC]. In this T 2 cos( T( 2 d 2 )) !1 C d!1" case the problem can be reformulated as > > 1 > (d !1; d!2) D = ( 1 2) (<( 1 2) ) <(g2100 ; g1110 ) x˙(t) f (x(t); p) D 0 ; C = (g2100 ; g1110 ) ; Ls (p)( x( T ) x (p)) D 0 ; 8 ! ; ! Lu (p)( x(TC) xC(p)) D 0 ; where d 1 d 2 indicate the change in rotation in the an- ˆ < TC > ˙ D ; gles 1; 2 for " ¤ 0. This construction is done up to sec- T (x(t) x0(t)) x0(t)d t 0 ond order in " and leads to an initial approximation for ˆ :ˆ R the continuation of a Neimark–Sacker bifurcation curve where the equations involving Ls(p) and Lu(p) form the starting from a double Hopf point. A similar set up can be projection boundary conditions. Here Ls (p) is an ns n defined for the other branch. matrix where the rows span the ns-dimensional stable > eigenspace of A (x ), and similarly Lu (p) is an nu n matrix where the rows span the nu-dimensional unstable Connecting Orbits > eigenspace of A (xC), where A(x) denotes the Jacobi ma- Connecting orbits, such as homoclinic and heteroclinic or- trix of ( 1) at x. The projection boundary conditions then bits, can be continued using a variety of techniques. To fix ensure that the starting point x( T ) lies in the unstable some terminology: a heteroclinic orbit that connects two eigenspace of x and that the end point x(TC) lies in the equilibrium points x and xC in the ODE system ( 1) is an stable eigenspace of xC (see Fig. 5). orbit for which In parallel, one must also continue of the equilibrium points x (p) and xC(p), unless they are fixed. Then the lim x(t) D x and lim x(t) D xC : eigenspaces can be determined by doing a Schur decom- t! 1 t!1 position of the corresponding Jacobi matrices. These must 6348 N Numerical Bifurcation Analysis

Homoclinic Branch Switching It is sometimes desirable to do branch switching from a homoclinic orbit to an n-homoclinic orbit, that is, a homoclinic orbit that goes through some neighborhood of the related equilibrium point n 1 times before finally converging to the equilibrium. Such n-homoclinic orbits arise in a number of situations involving the eigenvalues of the equilibrium and the orientation of the flow around the orbit. Suppose that the orbit is homoclinic to a saddle with complex conjugate eigenvalues (a saddle- focus). Let the saddle-quantity be the sum of the real parts of the leading stable and the leading unstable eigenvalue. If this quantity is positive, then a so-called Shil’nikov snake exists, which implies the existence of infinitely many n-periodic and n-homoclinic orbits for nearby parameters. Numerical Bifurcation Analysis, Figure 5 These n-homoclinic orbits also arise from certain codim 2 Projection boundary conditions in two dimensions: the orbit bifurcations: homoclinic to x0 is approximated by the truncated orbit from u x( T ) on the unstable eigenspace T to x(TC) on the stable eigenspace Ts. The unstable and stable manifolds are denoted 1. Belyakov bifurcations: Either the saddle-quantity of the u s by W and W , respectively: this figure shows that the homo- saddle-focus goes through zero, or there is a transition clinic orbit is also approximated in parameter space,because the two manifolds do not coincide between a saddle-focus and a real saddle (here the equilibrium has a double leading eigenvalue). 2. Homoclinic flip bifurcations: The inclination flip and be subsequently scaled to ensure continuity in the param- the orbit flip, where the flow around the orbit changes eter p [6]. Alternatively, one can construct smooth projec- between orientable and twisted. tors using an algorithm for the continuation of invariant 3. The resonant homoclinic doubling: A homoclinic orbit subspaces which includes the Ricatti equation in the defin- for which the flow is twisted connected to a real saddle ing system, see [ 33 ] for this new method. All these condi- where the saddle quantity goes through zero. tions taken together give a BVP with two free parameters, since, in general, a homoclinic or heteroclinic connection is a codim 1 phenomenon. Case 1. produces infinitely many n-homoclinic orbits, whereas cases 2. and 3. only produce a two-homoclinic orbit. Detecting Homoclinic Bifurcations By breaking up a homoclinic orbit globally into two It is then possible to detect codim 2 bifurcations of ho- parts where the division is in a cross-section away from moclinic orbits by setting up test functions and monitor- the equilibrium, somewhere “half-way”, and then gluing ing those, detecting when they cross zero. For the contin- pieces together, it is possible to construct n-homoclinic uation of these codim 2 bifurcations in three parameters orbits from 1-homoclinic orbits. The gaps between to- such test functions can then be kept constant equal to zero, be-glued pieces can be well-defined using Lin’s method, which provides an extra boundary condition. Simple test which leaves the gaps in the direction of the adjoint vec- functions involve the values of leading eigenvalues of the tor. This vector can be found by solving the adjoint varia- equilibrium point, i. e., the stable and unstable eigenval- tional equation mentioned above at the “half-way” point. ues closest to the imaginary axis. Some other bifurcations Combining gap distances and times taken for the flow in such as the inclination flip involve solving the so-called ad- pieces, one can construct a well-posed BVP. This BVP can joint variational equation , which can detect whether the then be continued in those quantities so that if the gap flow around the orbit is orientable or twisted like a Möbius sizes go to zero, one converges to an n-homoclinic or- strip. Homoclinics to a saddle-node, that is, where one bit [ 66 ]. of the eigenvalues of the equilibrium is zero, can be de- Lin’s method was also applied in [ 54 ] to compute tected and followed similarly, by constructing appropriate point-to-cycle connections by gluing a piece from the test functions. For details, see [ 9] and the references men- equilibrium to a cross-section to a piece from the cross sec- tioned therein. tion to the cycle. Numerical Bifurcation Analysis N 6349

Software Environments eling. In contrast to AUTO it uses Moore–Penrose contin- This review has outlined necessary steps and suitable uation and minimally extended systems to compute equi- methods to perform numerical bifurcation analysis. Sum- libria, cycles and connecting orbits. Since MATLAB is an marizing, the following subsequent tasks can be recog- interpreted language, it is slower than the other packages, nized. although a considerable speedup is obtained as the code for the Jacobi matrices for limit cycles and their bifurcations are written in C-codes and compiled. MATCONT Initial procedure 1. Compute the invariant solution 2. Characterize the linearized behavior has, however, much more functionality. It computes nor- Continuation 3. Variation of parameters mal form coefficients up to codim 2 bifurcations of equi- 4. Monitor dynamical indicators libria [ 20 ] and for codim 1 of limit cycles normal form co- Automated analysis 5. Detect special points and compute efficients are computed using a BVP-algorithm [ 28 ]. A re- normal forms cent addition is to switch branches from codim 2 bifur- 6. Switch branches cation points of equilibria to codim 1 bifurcations of cycles [ 61 ]. Ideally, these computations are automatically performed Finally PyDSTOOL supports similar functionality for by software and indeed, many efforts have been spent ODEs as CONTENT . on implementing the algorithms mentioned in this review and related ones. With the appearance of comput- Future Directions ers at research institutes the first codes and noninter- active packages were developed. For a recent historical This review focuses on methods for equilibria and limit cy- overview of these and their capabilities and algorithms, cles to gain insight into the dynamics of a nonlinear ODE see [ 38 ]. Here it is worthwhile to mention AUTO 86 [ 24 ], depending on parameters, but, generally speaking, other LINBLF [51 ] and DSTOOL [4] as these are the predeces- characteristics play a role too. For instance, a Neimark– sors to the packages AUTO -07 P [21 ], CONTENT [58 ], MAT - Sacker bifurcation leads to the presence of tori. The com- CONT [19 ], PyDSTOOL [13 ] discussed here. In particular, putation and continuation of higher dimensional tori has AUTO 86 is very powerful and in use to date, but not al- been considered in [ 49 ,70 ,71 ]. The generalization of the ways easy to handle. Therefore, several attempts have been methods for equilibria and cycles is, however, not straight- made to make an interactive graphical user interface. One forward. Stable and unstable manifolds are another aspect. example is XPPAUT [32 ] which is a standard tool in neuro- In particular their visualization can hint at the presence of science. global bifurcations. A review of methods for computing The latest version of AUTO is AUTO -07 P and written such manifolds is presented in [ 52 ]. Connecting orbits can in Fortran and supports parallelization. It uses pseudo-ar- be calculated, but initializing such a continuation is a non- clength continuation and performs a limited bifurcation trivial task. This may be started from certain codim 2 bi- analysis of equilibria, fixed points, limit cycles and con- furcation points as in [ 8], but good initial approximations necting orbits using HOMCONT [12 ]. A recent addition is are not available for other cases. the continuation of branch points [ 16 ]. It uses fully ex- This review is also restricted to ODEs, but one can de- tended systems, but has specially adapted linear solvers so fine other classes of dynamical systems. For instance, if the that it is still quite fast. system is given by an explicitly defined map, i. e., not im- An interactive package written in C++ is CONTENT , plicitly as a Poincaré map, the described approach can also where the Moore–Penrose continuation was first imple- be carried out [ 10 ,37 ,40 ,59 ]. Another important and re- mented. It supports bifurcation analysis of equilibria up to lated class is given by delay equations, and the algorithms the continuation of codim 2 and detection of some codim for equilibria and periodic orbits in the ODE case can 3 bifurcations. It uses a similar procedure as AUTO to con- be applied with suitable modifications, see [ 5,74 ] and the tinue limit cycles and detects codim 1 bifurcations of limit software implementations DDE -BIFTOOL [31 ] and PDDE - cycles, but does not continue these. It also handles bifur- CONT . The computation of normal forms and connecting cations of cycles of maps up to the detection of codim 2 orbits for this class is not yet thoroughly investigated or bifurcations. Normal forms for all codim 1 bifurcations supported. are computed. Interestingly, both fully and minimally ex- For large systems, e. g., discretizations of a partial tended systems are implemented. differential equation (PDE), the algebra for some algo- A new project MATCONT emerged out of CONTENT . It rithms in this review becomes quite involved and numer- is written in MATLAB , a widely used software tool in mod- ically expensive. These problems need special treatment, 6350 N Numerical Bifurcation Analysis

see LOCA [69 ], PDE -CONT [30 ,62 ] and related references. 17. Deuflhard P, Bornemann F (2002) Scientific Computing With These packages focus on computing (periodic) solutions Ordinary Differential Equations. In: Marsden JE, Sirovich L, and bifurcation curves. Good algorithms for analyzing bi- Antman S (eds) Texts in applied mathematics, vol 42. Springer, New York furcations of PDEs will be a major research topic. 18. Deuflhard P, Fiedler B, Kunkel P (1987) Efficient numerical Finally, there are also slow-fast (stiﬀ) systems or sys- pathfollowing beyond criticalpoints. SIAMJ NumAnal 24:912– tems with a particular structure such as symmetries. For 927 these classes, many questions remain open. 19. Dhooge A, Govaerts W, Kuznetsov YA (2003) Matcont: A Mat- While on most of the mentioned topics pioneering lab package for numerical bifurcation analysis of ODE’s. ACM TOMS 29, pp 141–164. http://sourceforge.net/projects/ work has been done, the methods are far from as “com- matcont . Accessed 9 Sep 2008 plete” as for ODEs. One overview of current research top- 20. Dhooge A, Govaerts W, Kuznetsov YA, Meijer HGE, Sautois ics in dynamical systems can be found in [ 53 ]. B (2008) New features of the software Matcont for bifurcation analysis of dynamical systems. Math Comp Mod Dyn Syst 14:147–175 Bibliography 21. Doedel EJ (2007) AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, User’s Guide. Primary Literature http://cmvl.cs.concordia.ca/auto . Accessed 9 Sep 2008 22. Doedel EJ (2007) Lecture notes on numerical analysis of non- 1. Allgower EL, Georg K (2000) Numerical Continuation Methods: linear equations. In: Krauskopf B, Osinga HM, Galan-Vioque An Introduction. Springer, Berlin J (eds) Numerical continuation methods for dynamical sys- 2. Arnold VI (1983) Geometrical Methods in the Theory of Ordi- tems: Path following and boundary value problems. Springer- nary Differential Equations. Springer, Berlin Canopus, Dordrecht, pp 1–49 3. Ascher UC, Mattheij RMM, Russell B (1995) Numerical Solution 23. Doedel EJ, Govaerts W, Kuznetsov YA (2003) Computation of of Boundary Value Problems for Ordinary Differenital Equa- periodic solution bifurcations in ODEs using bordered systems. tions. SIAM, Philadelphia SIAM J Numer Anal 41:401–435 4. Back A, Guckenheimer J, Myers M, Wicklin F, Worfolk P (1992) 24. Doedel EJ, Kernevez JP (1986) AUTO, Software for Continua- DsTool: Computer assisted exploration of dynamical systems. tion and Bifurcation Problems in Ordinary Differential Equa- Notices Amer Math Soc 39:303–309 tions. Applied Mathematics, California Institute of Technology, 5. Barton DAW, Krauskopf B, Wilson RE (2007) Homoclinic bifurca- Pasadena tions in a neutral delay model of a transmission line oscillator. 25. Doedel EJ, Romanov VA, Paffenroth RC, Keller HB, Dichmann Nonlinearity 20:809–829 DJ, Vioque GJ, Vanderbauwhede A (2007) Elemental periodic 6. Beyn WJ (1990) The numerical computation of connecting or- orbits associated with the liberation points in the circular re- bits in dynamical systems. IMA J Numer Anal 10:379–405 stricted 3-body problem. Int J Bif Chaos 17:2625–2677 7. Beyn WJ (1991) Numerical methods for dynamical systems. In: 26. Doedel EJ, Kooi BW, Kuznetsov YA, van Voorn GAK (2008) Con- Light W (ed) Advances in numerical analysis, Vol. I (Lancaster, tinuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle 1990). Clarendon, Oxford, pp 175–236 connections. Int J Bif Chaos 18:1889–1903 8. Beyn WJ (1994) Numerical analysis of homoclinic orbits em- 27. Doedel EJ, Kooi BW, Kuznetsov YA, van Voorn GAK (2008) Con- anating from a Takens–Bogdanov point. IMA J Numer Anal tinuation of connecting orbits in 3D-ODEs: (II) Cycle-to-cycle 14:381–410 connections. arXiv.org:0804.0179. Accessed 9 Sep 2008 9. Beyn WJ, Champneys A, Doedel EJ, Govaerts W, Kuznetsov YA, 28. Doedel EJ, Kuznetsov YA, Govaerts W, Dhooge A (2005) Nu- Sandstede B (2002) Numerical continuation, and computation merical continuation of branch points of equilibria and peri- of normal forms. In: Fiedler B (ed) Handbook of Dynamical Sys- odic orbits. Int Bif J Chaos 15:841–860 tems, vol 2. Elsevier Science, Amsterdam, pp 149–219 29. Elphick C, Tirapegui E, Brachet M, Coullet P, Iooss G (1987) 10. Beyn WJ, Kleinkauf JM (1997) The numerical computation of A simple global characterization for normal forms of singular homoclinic orbits for maps. Siam J Numer Anal 34:1207–1236 vector fields. Phys D 32:95–127 11. Carr J (1981) Applications of centre manifold theory. Springer, 30. Engelborghs K, Lust K, Roose D (1999) Direct computation New York of period doubling bifurcation points of large-scale systems 12. Champneys AR, Kuznetsov YA, Sandstede B (1996) A numeri- of ODEs using a Newton-Picard method. IMA J Numer Anal cal toolbox for homoclinic bifurcation analysis. Int J Bif Chaos 19:525–547 6:867–887 31. Engelborghs K, Luzyanina T, Roose D (2002) Numerical bi- 13. Clewley R, Sherwood E, LaMar D, Guckenheimer J (2005) PyD- furcation analysis of delay differential equations using DDE- STool, available at http://sourceforge.net/projects/pydstool . BIFTOOL. ACM Trans Math Softw 28:1–21 Accessed 09 Sept 2008 32. Ermentrout B (2002) Simulating, Analyzing, and Animating Dy- 14. Coullet PH, Spiegel EA (1983) Amplitude equations for systems namical Systems: A Guide to XPPAUT for Researchers and Stu- with competing instabilities. SIAM J Appl Math 43:776–821 dents. SIAM, Philadelphia 15. de Boor C, Swartz B (1973) Collocation at Gaussian points. SIAM 33. Friedman M, Govaerts W, Kuznetsov YA, Sautois B (2005) Con- J Num Anal 10:582–606 tinuation of homoclinic orbits in MATLAB. In: Sunderam VS, 16. Dercole F (2008) Bpcont: An auto driver for the continuation van Albada GD, Sloot PMA, Dongarra J (eds) Proceedings ICCS of branch points of algebraic and boundary-value problems. 2005, Atlanta, Part I, Springer Lecture Notes in Computer Sci- SIAM J Sci Comp 30:2405–2426 ence Vol. 3514. Springer, Berlin, pp 263–270 Numerical Bifurcation Analysis N 6351

34. Fuller AT (1968) Conditions for a matrix to have only charac- methods for computing (un)stable manifolds of vector fields. teristic roots with negative real parts. J Math Anal Appl 23: Int J Bif Chaos 15(3):763–791 71–98 53. Krauskopf B, Osinga HM, Galan-Vioque J (2007) Numerical Con- 35. Gaspard P (1993) Local birth of homoclinic chaos. Phys D tinuation methods for dynamical systems. Springer-Canopus, 62:94–122 Dordrecht 36. Govaerts W, Guckenheimer J, Khibnik A (1997) Defining func- 54. Krauskopf B, Riess T (2008) A Lin’s method approach to finding tions for multiple Hopf bifurcations. SIAM J Numer Anal and continuing heteroclinic connections involvingperiodic or- 34:1269–1288 bits. Nonlinearity 21:1655–1690 37. Govaerts W, Khoshsiar Ghaziani R, Kuznetsov YA, Meijer HGE 55. Kuznetsov YA (1999) Numerical normalization techniques for (2007) Numerical methods for two-parameter local bifurcation all codim 2 bifurcations of equilibria in ODE’s. SIAM J Numer analysis of maps. SIAM J Sci Comput 29:2644–2667 Anal 36:1104–1124 38. Govaerts W, Kuznetsov YA (2007) Numerical continuation 56. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory, tools. In: Krauskopf B, Osinga HM, Galen-Vioque J (eds) Nu- 3rd edn. Springer, Berlin merical continuation methods for dynamical systems: Path fol- 57. Kuznetsov YA, Govaerts W, Doedel EJ, Dhooge (2005) Numeri- lowing and boundary value problems. Springer-Canopus, Dor- cal periodic normalization for codim 1 bifurcations of limit cy- drecht, pp 51–75 cles. SIAM J Numer Anal 43:1407–1435 39. Govaerts W, Kuznetsov YA, Dhooge A (2005) Numerical contin- 58. Kuznetsov YA, Levitin VV (1995) Content: A multiplatform envi- uationof bifurcationsof limitcycles inmatlab.SIAMJSciComp ronment for analyzing dynamical systems. ftp://ftp.cwi.nl/pub/ 27:231–252 CONTENT/ . Accessed 9 Sep 2008 40. Govaerts W, Kuznetsov YA, Sijnave B (1999) Bifurcations of 59. Kuznetsov YA, Meijer HGE (2005) Numerical normal forms for maps in the software package content. In: Ganzha VG, Mayr codim 2 bifurcations of maps with at most two critical eigen- EW, Vorozhtsov EV (eds) Computer algebra in scientific com- values. SIAM J Sci Comput 26:1932–1954 puting–CASC’99 (Munich). Springer, Berlin, pp 191–206 60. Kuznetsov YA, Meijer HGE (2006) Remarks on interacting 41. Govaerts W, Pryce JD (1993) Mixed block elimination for linear Neimark–Sacker bifurcations. J Diff Eqs Appl 12:1009–1035 systems with wider borders. IMA J Num Anal 13:161–180 61. Kuznetsov YA, Meijer HGE, Govaerts W, Sautois B (2008) 42. Govaerts WJF (2000) Numerical Methods for Bifurcations of Dy- Switching to nonhyperbolic cycles from codim 2 bifurcations namical Equilibria. SIAM, Philadelphia of equilibria in ODEs. Physica D 237:3061–3068 43. Guckenheimer J (2002) Numerical analysis of dynamical sys- 62. Lust K, Roose D, Spence A, Champneys AR (1998) An adaptive tems. In: Fiedler B (ed) Handbook of dynamical systems, Vol. Newton-Picard algorithm with subspace iteration for comput- 2. Elsevier Science, North-Holland, pp 346–390 ing periodic solutions. SIAM J Sci Comput 19:1188–1209 44. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dy- 63. Mei Z (1989) A numerical approximation for the simple bifur- namical Systems and Bifurcations of Vector Fields. Springer, cation problems. Numer Func Anal Opt 10:383–400 New York 64. Mei Z (2000) Numerical Bifurcation Analysis for Reaction- 45. Guckenheimer J, Myers M, Sturmfels B (1997) Computing Hopf Diffusion Equations. Springer, Berlin bifurcations I. SIAM J Numer Anal 34:1–21 65. Moore G (1980) The numerical treatment of nontrivial bifurca- 46. Henderson Me (2007) Higher-dimensional continuation. In: tion points. Numer Func Anal Opt 2:441–472 Krauskopf B, Osinga HM, Galen-Vioque J (eds) Numerical 66. Oldeman BE, Champneys AR, Krauskopf B (2003) Homo- continuation methods for dynamical systems: Path following clinic branch switching: a numerical implementation of Lin’s and boundary value problems. Springer-Canopus, Dordrecht, method. Int J Bif Chaos 13:2977–2999 pp 77–115 67. Roose D, Hlavaçek V (1985) A direct method for the computa- 47. Iooss G (1988) Global characterization of the normal form for tion of Hopf bifurcation points. SIAM J Appl Math 45:879–894 a vector field near a closed orbit. J Diff Eqs 76:47–76 68. Russell RD, Christiansen J (1978) Adaptive mesh selection 48. Iooss G, Adelmeyer M (1992) Topics in Bifurcation Theory and strategies for solving boundary value problems. SIAM J Num Applications. World Scientific, Singapore Anal 15:59–80 49. Jorba A (2001) Numerical computation of the normal be- 69. Salinger AG, Burroughs EA, Pawlowski RP, Phipps ET, Romero haviour of invariant curves of n-dimensional maps. Nonlinear- LA (2005) Bifurcation tracking algorithms and software for ity 14:943–976 large scale applications. Int J Bif Chaos 15:1015–1032 50. Keller HB (1977) Numerical solution of bifurcation and non- 70. Schilder F, Osinga HM, Vogt W (2005) Continuation of linear eigenvalue problems. In: Rabinowitz PH (ed) Applica- quasiperiodic invariant tori. SIAM J Appl Dyn Syst 4:459–488 tions of bifurcation theory. Proc Advanced Sem, Univ Wiscon- 71. Schilder F, Vogt W, Schreiber S, Osinga HM (2006) Fourier sin, Madison, (1976), Publ Math Res Center, No. 38. Academic methods for quasi-periodic oscillations. Int J Num Meth Eng Press, New York pp 359–384 67:629–671 51. Khibnik AI (1990) LINBLF: A program for continuation and bi- 72. Shoshitaishvili AN (1975) The bifurcation of the topological furcation analysis of equilibria up to codimension three. In: type of the singular points of vector fields that depend on pa- Roose D, de Dier D, Spence A (eds) Continuation and bifurca- rameters. Trudy Sem Petrovsk, (Vyp. 1):279–309 tions: numerical techniques and applications. Leuven, (1989), 73. Stephanos C (1900) Sur une extension du calcul des substitu- NATO Adv Sci Inst Ser C Math Phys Sci, vol 313. Kluwer, Dor- tions linéaires. J Math Pures Appl 6:73–128 drecht, pp 283–296 74. Szalai R, Stépán G, Hogan SJ (2006) Continuation of bifurca- 52. Krauskopf B, Osinga HM, Doedel EJ, Henderson ME, Gucken- tions in periodic delay-differential equations using character- heimer J, Vladimirsky A, Dellnitz M, Junge O (2005) A survey of istic matrices. SIAM J Sci Comput 28:1301–1317 6352 N Numerical Issues When Using Wavelets

75. van Strien SJ (1979) Center manifolds are not C 1. Math uct of two signals in `2(Z) will be written . For the oc- Zeitschrift 166:143–145 tave band wavelet representation, analysis (respectively, synthesis) ﬁlters are denoted h and g (respectively, h˜ Books and Reviews and g˜). The scaling and wavelet functions used for the Devaney RL (2003) An introduction to chaotic dynamical systems. analysis (respectively, synthesis) are denoted ((x/2) R Z Westview Press, Boulder D k h[k](x k); x 2 and k 2 ) and ( (x/2) D Doedel EJ, Keller HB, Kernevez JP (1991) Numerical Analysis and g[k](x k); x 2 R and k 2 Z) (respectively, ˜ and kP Control of Bifurcation Problems (I): Bifurcation in Finite Dimen- ˜ ). We also deﬁne the scaled dilated and translated sions. Int J Bif Chaos 1:493–520 P Doedel EJ, Keller HB, Kernevez JP (1991) Numerical Analysis and version of at scale j and position k as j;k (x) D j j Control of Bifurcation Problems (II): Bifurcation in Infinite Di- 2 (2 x k), and similarly for , ˜ and ˜ . mensions. Int J Bif Chaos 1:745–772 Hirsch MW, Smale S, Devaney RL (2004) Differential equations, dy- Glossary namical systems, and an introduction to chaos, 2nd edn. Pure and Applied Mathematics, vol 60 Elsevier/Academic Press, Am- WT Wavelet Transform sterdam CWT Continuous Wavelet Transform Murdock J (2003) Normal Forms and Unfoldings for Local Dynami- cal Systems. Springer, New York DWT Discrete (decimated) Wavelet Transform UWT Undecimated Wavelet Transform IUWT Isotropic Undecimated Wavelet Transform

Numerical Issues Definition of the Subject When Using Wavelets Wavelets and related multiscale representations pervade JEAN -L UC STARCK 1, J ALAL FADILI 2 all areas of signal processing. The recent inclusion of 1 DAPNIA/SEDI-SAP, Service d’Astrophysique, wavelet algorithms in JPEG 2000 – the new still-picture CEA/Saclay, Gif sur Yvette, France compression standard – testifies to this lasting and signif- 2 GREYC CNRS UMR 6072, Image Processing Group, icant impact. The reason of the success of the wavelets is Ecole Nationale Supèrieure d’Ingènieurs de Caen, due to the fact that wavelet basis represents well a large Caen Cedex, France class of signals, and therefore allows us to detect roughly isotropic elements occurring at all spatial scales and lo- Article Outline cations. As the noise in the physical sciences is often not Gaussian, the modeling, in the wavelet space, of many Notations kind of noise (Poisson noise, combination of Gaussian and Glossary Poisson noise, long-memory 1/ f noise, non-stationary Definition of the Subject noise, ...)has alsobeenakey step for the use of waveletsin Introduction scientific, medical, or industrial applications [ 41 ]. Exten- The Continuous Wavelet Transform sive wavelet packages exist now, commercial (see for ex- The (Bi-)Orthogonal Wavelet Transform ample [ 29 ]) or non commercial (see for example [ 47 ,48 ]), The Lifting Scheme which allows any researcher, doctor, or engineer to analyze The Undecimated Wavelet Transform his data using wavelets. The 2D Isotropic Undecimated Wavelet Transform Designing Non-orthogonal Filter Banks Introduction Iterative Reconstruction Over the last two decades there has been abundant in- Future Directions terest in wavelet methods. In many hundreds of papers Bibliography published in journals throughout the scientific and engineering disciplines, a wide range of wavelet-based tools Notations and ideas have been proposed and studied. Background For a real discrete-time filter whose impulse response texts on the wavelet transform include [ 11 ,13 ,26 ,41 ,44 ]. is h[n]; h¯[n] D h[ n]; n 2 Z is its time-reversed ver- The most widely used wavelet transform (WT) algorithm sion. The hat ˆ notation will be used for the Fourier is certainly the decimated bi-orthogonal wavelet transform transform of square-integrable signals. For a filter h, (DWT) which is used in JPEG 2000. While the bi-orthog- its z-transform is written H(z). The convolution prod- onal wavelet transform has led to successful implementa-