Reduction of Dimension for Nonlinear Dynamical Systems 3 Equations for the Reduced State Variable

arXiv:1508.05921v1 [nlin.CD] 24 Aug 2015 oi otaesc as sym- such in software implemented bolic is analytical, and While algorithmic, is system. approach reduced the elimi- the differential obtain employ to we nation possible, where case is us- the reduction when In this approaches. solution analytical the of variety computing a and for ing variables, useful state the be equations of can of one this for system equation nonlinear single a a into reduce can one some in cases, that demonstrate We systems. dynamical linear Abstract date Accepted: / date Received: Gorder Van A. Robert and Harrington systems A. dynamical Heather nonlinear for dimension of Reduction -al [email protected] Kingdom E-mail: Road, United 6GG Woodstock OX2 Quarter, Oxford Observatory Wile Andrew Radcliffe Oxford, Building, of University Institute, Mathematical Gorder Van R.A. 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Harrington and Robert A. Van Gorder of dimension, may prove useful. We then outline the ODE is possible, the need to solve a nonlinear alge- paper. braic system for all of the fixed points simultaneously Often times, if one is trying to approximate the so- is eliminated, resulting in what is often a far less com- lution to a nonlinear system through some sort of ana- putationally demanding problem. lytical approximation, via series, perturbation, or more Another topic is great recent interest in nonlinear complicated approaches, one quickly finds that the cou- science has been both the synchronization of chaos [10, pled equations require balancing many terms coming 11,12] and the control of chaos. In situations where one from the expansion for each of the state variables. In the is interested in mitigating the possibility of emergent case of a single state variable governed by a higher-order chaos, one can couple a chaotic system to various con- ODE, one need only track terms in a single asymptotic trol terms, or indeed to additional dynamical systems, expansion. This approach has been applied when using which may lend a degree to stability. Under such ap- Taylor series, approximate Fourier series, and asymp- proaches, one often increases the complexity or even totic expansions in other types of basis functions to the dimension of the dynamical system being solved. the solution of a system of nonlinear ODE. In hybrid As such, methods to reduce the dimension of such sys- analytic-numeric methods, such as the homotopy anal- tems could improve compatibility. Furthermore, since ysis method [7,8,9], such reductions of a system to a the control of chaos is often linked to a control term single equation also simplifies the optimization problem which itself is determined by a Lyapunov function, the which is solved to obtain the error-minimizing solution construction of contraction maps through the reduction (see, for instance, [2], where the present approach is approach outlined here could be of great use. used in such a capacity). Therefore, the reduction of As stated before [13], the competitive modes anal- order can greatly reduce the complexity of analytical ysis gives an interesting link between the geometry of calculations under several frameworks. phase space possibly yielding chaotic trajectories (recall Contraction maps or Lyapunov functions are useful that the competitive modes requirements appear to be tools for discussing the convergence of solutions to non- a necessary, albeit not sufficient, condition for chaos linear dynamical systems to large-time steady or quasi- [14,15,16,17,18,19]). Conversely, the differential elimi- steady dynamics. In situations where contraction maps nation may cast light into the geometry of solutions in or Lyapunov functions are known for a given dynamical the space of derivations. Since this result of the differen- system, the state variable governed by a single higher- tial elimination is a single higher-order ODE, and since order ODE necessarily results in a contraction map in any chaos emergent from the nonlinear system should the single state variable. However, as is well known to be encoded in the single higher-order ODE, the differ- those studying stability of nonlinear systems, it is not ential algebraic structure of such equations may cast often easy to obtain contraction maps for complicated light into practical geometric tools by which one may systems. As we shall show here, it is possible to use study systems in which chaos is observed. In particu- the reduction of a system to a single higher-order ODE lar, through this reduction approach, the calculation of in order to construct a contraction map for the state mode frequencies in the standard competitive modes variable governed by the aforementioned higher-order analysis becomes much simpler. ODE. The existence of such a map can then be used to This paper is outlined as follows. In Section 2, we deduce the large-time dynamics of the state variable, provide an algorithmic approach, to differential elim- as well as for the other state variables in the original ination for nonlinear dynamical systems based upon system. One example of this is given in [1], and other differential algebra. First laying out the general the- examples are provided in Section 5. ory, we then give specific MAPLE code for performing Related to both the topic of analytical approxima- the differential elimination in a systematic manner. The tions and Lyapunov functions would be long time dy- algorithmic approach is useful in the case where the namics and equilibrium behavior of nonlinear dynami- dynamical system can be reduced to a single ODE in cal systems. Indeed, in order to study the equilibrium terms of only one of the state variables. In Section 3 we structure of a high-order system of ODEs, one must implement the approach in order to reduce a variety solve a coupled system of nonlinear algebraic equations of chaotic and hyperchaotic systems, finding that the in order to recover the fixed points for the state vari- form of the nonlinearity in the dynamical system will ables. First reducing the system to a single ODE allows strongly influence the reducibility properties. However, one to obtain a single nonlinear algebraic equation for in cases where the dynamical system is not reducible us- the fixed point of a single state variable, which can then ing differential elimination, one may still obtain more be used to recover the fixed points of the other state complicated reductions, for instance in terms of inte- variables. Therefore, when such a reduction to a single grals, resulting in more complicated integro-differential Reduction of dimension for nonlinear dynamical systems 3 equations for the reduced state variable. The possible 2.1 Algebra preliminaries results are illustrated through concrete examples for the Rössler system (which is completely reducible), the Lorenz system (which is partially reducible - that is, Here we briefly review concepts from algebra and dif- reducible in some but not all state variables), and the ferential algebra. For reference books in differential al- Qi-Chen-Du-Chen-Yuan (which is irreducible under dif- gebra, see [20,21]. If I is a subset of a ring R then (I) is ferential elimination, but which can be reduced to an the (algebraic) ideal generated by I. Let I be an ideal of integro-differential equation). We give summarizing ob- R. Then √I denotes the radical of I. A derivation over servations regarding the reducibility of dynamical sys- a ring R is a map R R which satisfies (we writea ˙ is 7→ tems in Section 4. the derivative of a), for every a,b R, (a +˙ b) =a ˙ + b˙ ˙ ˙ ∈ The remainder of the paper is devoted to applica- and (a b) = (˙a)b + a(b). The field of differential alge- tions of reduction of dimension for dynamical systems. bra is based on the concept of a differential ring (resp. In Section 5, we demonstrate that reduction of dimen- field), which is a ring (resp. field) R endowed with a set sion can be useful for obtaining contraction maps and of derivations that commutes pairwise. A differential Lyapunov functions, which in turn may be used to de- ideal [I] of a differential ring R is an ideal of R stable termine asymptotic stability of dynamical systems and under the action of derivation. also to control chaos in such systems. In Section 6 we demonstrate that reduction of dimension can be used Differential algebra is more similar to commutative to simplify calculations involved in certain techniques algebra than analysis. In commutative algebra, Buch- for studying the solutions of nonlinear dynamical sys- berger solved the membership problem (tests whether a tems. Indeed, when applicable, we find that the ap- given polynomial is contained in a given ideal) through proach greatly reduces the number of nonlinear alge- the theory of Gröbner bases [30]. From algebraic geom- braic equations required to be solved when constructing etry, we know the set of polynomials which vanish over trajectories in state space via undetermined coefficient the solutions of a given polynomial system form an ideal methods by a factor of 1/n, where n is the dimension and even a radical ideal [31]. In the case of differential of the dynamical system, meaning that the number of equations, the set of differential polynomials which van- equations needing to be solved will not increase with ish over the analytic solutions of a given system of dif- the size of the system. Furthermore, when applying the ferential polynomial equations form a differential ideal competitive modes analysis (which is a type of diagnos- and even a radical differential ideal [20]. Ritt solved the tic criteria for finding chaotic trajectories in nonlinear theoretical problem (of membership for radical differ- dynamical systems), we find that only one binary com- ential ideals) and developed algorithmic tools to solve parison is needed of one first reduces the dimension of systems of polynomial ODE and PDEs; however, Ritt’s the dynamical system so that there is a single equation algorithm requires factorization. for one state variable. In contrast, there are normally of order 2n−1 comparisons needed for an n-dimensional Due to the complexity of factorization, Boulier and dynamical system. In Section 7 we provide a discussion co-authors avoided it by developing the Rosenfeld-Gröbner and possible avenues for future work. algorithm, based on the work of Seidenberg and Rosen- feld, and incorporating Gröbner bases [27,25,24]. Since then, the algorithm has been improved both theoreti- cally and practically [23,22,26] and it no longer requires Gröbner bases. It is available in the DifferentialAlgebra 2 Algebraic approach to differential elimination package in MAPLE [22] and SageMath as an interface for the BLAD and BMI libraries [32,33]. Systems of differential equations are ubiquitous and widely studied. Ritt [20] and Kolchin [21] pioneered Algorithmically, differential elimination involves ma- the field of differential algebra, an algebraic theory for nipulation of finite subsets of a differential polynomial studying solutions of ordinary and partial differential ring R = K U where K is the differential field of coef- { } equations. We are particularly interested in differential ficients (i.e. K = Q), and U is a finite set of dependent elimination, an algorithmic subtheory that can simplify variables. The elements of R are differential polynomi- systems of parameterized algebraic differential equa- als, which are polynomials built over the infinite set tions. This permits one to reduce the dimension of a of all derivatives ΘU, of the dependent variables. Con- dynamical system so that one is left with a single ODE sider a system Σ of polynomial differential equations, in the state variable. here, we consider the Lorenz system of three ordinary 4 Heather A. Harrington and Robert A. Van Gorder differential equations: package enables differential elimination of PDEs by in- cluding additional inputs for the derivations (e.g., x˙ = a(x x ) , 1 2 − 1 derivations=[u,x,t]. Note, sys is assumed to have x˙ = x (b x ) x , (1) coefficients in the field Q[x , x , x ] obtained by adjoin- 2 1 − 3 − 2 1 2 3 x˙3 = x1x2 cx3 . ing the independent variables to the field of rationals − and symbolic parameters a,b,c are considered arbitrary The Lorenz system can be re-written as: in the coefficient field. To form the differential ring, we input: x˙ + a(x x )=0 , − 1 2 − 3 R := DifferentialRing(blocks = [x3, x2, x1], Σ = x˙ + x (b x ) x =0 , (2) derivations = [t]) − 2 1 − 3 − 2   x˙3 + x1x2 cx3 =0 . Note that x1 stands to the rightmost place on the list − − which identifies that we are attempting to reduce the  The Rosenfeld-Gröbner algorithm takes as an input a differential equation to only one variable, i.e. x1(t). This differential system Σ and a ranking. A ranking > is any ranking eliminates x3 with respect to x2, and then elim- total ordering over the set ΘU of all derivatives of the el- inates x2 with respect to x1. We now call the Rosenfeld ements of U which satisfies the following axioms: a< a˙ Gröbner algorithm for our system and differential poly- and a

t satisfy −at as x(t)= x0 + ae e y(s)ds . (12) 0 t Z 2at 2 2 2as e (x(t)) = x0 +2 e (a + z(s))(z ˙(s)+ cz(s))ds . Here x is the initial value of the state x(t), that is 0 0 Z x(0) = x0. Yet, from the second equation in the Lorenz (16) system, we have z = b (y ˙ + y)/x, which yields − where x(0) = x0. The first equation in the QCDCY y˙ + y z(t)= b . (13) system has not been used, and we place this relation − −at t as x0 + ae 0 e y(s)ds into that equation to obtain a single equation for the state variable z(t). After several algebraic and differ- Placing the representationsR for x(t) and z(t) into the ential manipulations, we arrive at the single equation third equation of the Lorenz system, and performing algebraic manipulations to simplify the resulting expression, we obtain z˙ 2e2at 1 a (z ˙ + cz)(a + z) x2 + J[z, z˙] t − − a + z 0 −at as (ÿ +(1+ c)y ˙ + cy) x0 + ae e y(s)ds d 0 2 2at Z +2 x0 + J[z, z˙] e (z ˙ + cz)(a + z) t dt + (y ˙ + y) ay a2e−at easy(s)ds 2e4at(z ˙ + cz)2(a +z)2 − 0 − Z 2 3 (14) 2 t 2(b z)(a + z) x0 + J[z, z˙]s =0 , −at as − − + y x0 + ae e y(s)ds (17) Z0 t −at as cb x0 + ae e y(s)ds =0 . where we have defined the integral operator − 0 Z t Note that the equation both involves an integral and is J[z, z˙]=2 e2as(a + z(s))(z ˙(s)+ cz(s))ds . (18) non-autonomous. Z0 Similar results can be obtained for the other state 3.3 Qi-Chen-Du-Chen-Yuan System variables. The fact that the obtained equations involve an integral operator which cannot simply be differen- We now consider the Qi-Chen-Du-Chen-Yuan (QCDCY) tiated away demonstrates why the differential elimina- system [54], which is given by tion algorithm was not useful for this case. Still, performing the manipulations by hand, we have reduced x˙ = a(y x)+ yz , the fairly complicated QCDCY system into a single − y˙ = bx y xz , (15) integro-differential equation, thereby reducing the di- − − mension of the original system. z˙ = xy cz . − The differential elimination algorithm indicates there is no reduction to a single ODE in any of the three state 4 Reductions of n-dimensional dynamical variables. This system has a quadratic nonlinearity in systems each equation, and this added complication is behind the difficulties in obtaining such a reduction. However, We now give some summarizing remarks based on what we may still obtain an equation for a single state vari- we have seen in the previous sections. We shall assume able, if we are willing to include integral terms. Due to that each system is coupled through at least one state the complexity in obtaining such an equation, we shall variable (otherwise the state variables naturally sepa- restrict our attention to finding a single equation for rate into distinct lower-dimension equations, and the the state variable z(t), noting that similar approaches approach is not needed). 8 Heather A. Harrington and Robert A. Van Gorder

4.1 Linear Systems 4.3 Diﬀerentially Irreducible Nonlinear Systems

For first order linear systems of dimension n, there is We have observed that for more complicated nonlin- always a reduction into a single higher-order ODE. This ear dynamical systems, there is no reduction to a single follows from the process of Gaussian elimination. In the ODE in one state variable. While it may be the case case that the matrix of coefficients for such a first-order that differential elimination does not pick up an ODE system is full rank, the resulting higher-order ODE will that does exist, it seems as though the failure of dif- be of order n. If the matrix of coefficients is singular, ferential elimination is a sign that integrations will be then the resulting higher-order ODE will be of order needed in order to reduce the dimension of such sys- less than n. tems. Indeed, when integrations of this kind are called for, the manipulations are no longer confined to the specified differential ring, and the differential elimina- 4.2 Reducible Nonlinear Systems tion cannot be performed. While one can attempt these integrations manually, as opposed to algorithmically, For first order nonlinear systems of dimension n, there obviously it would be desirable to have some kind of are multiple possibilities, owing to the structure of the algorithmic approach. Perhaps one may adjoin inte- nonlinearity. gral operators to the differential ring, in order to per- In cases where the system permits the complete form reductions for more complicated nonlinear sys- differential elimination (an example being the Rossler tems. This would likely work in cases like the Lorenz equation), all state variables in a first order nonlin- system, for which there is partial reducibility under dif- ear system can be expressed in terms of a higher-order ferential elimination. For instance, if one were to define t as ODE. Note, however, that it is possible for the order a new variable Y (t) = 0 e y(s)ds, then one would of the single ODE to be different from the dimension n obtain a non-autonomous ODE for Y (t) from equation R of the first order system. As an example of this point, (14). Therefore, this fairly simple integral transforma- consider the system tion, in addition to differential operators, can reduce the dimension of the Lorenz system with respect to x˙ = x y z , − − the state variable y(t). However, in cases like that of y˙ = x2 , (19) the QCDCY system, note that the form of the integral z˙ = x x3 . operator given in (18) is rather complicated, depend- − ing nonlinearly on both the state variable z(t) and its Clearly, differentiation of the first equation givesx ¨ = derivativez ˙(t). For such cases, there is no combination x˙ y˙ z˙ =x ˙ x2 + x x3. So, we obtain of elementary integral transforms that can be adjoined − − − − to the differential ring which would permit reduction x¨ x˙ x + x2 + x3 =0 , (20) − − of dimension to a single ODE. As such, it appears as though reduction of dimension for certain more com- which is a second order equation for the state variable plicated systems will result in reductions to integro- x(t), even though the original system was first order. A differential equations, rather than ODEs, for some fun- similar example can be found in [1], where a fourth or- damental reason related to how complicated the orig- der nonlinear dynamical system was reduced to a single inal dynamical system is. Therefore, the study of pos- second order nonlinear ODE. sible algorithmic methods for the reduction of dimen- It is possible for a system to be reduced to a single sion for dynamical systems into single scalar integro- equation, which is not an ODE. This was evident even differential equations would be an interesting and po- for the Lorenz equation, where an equation for one of tentially very useful area of future work. the state variables involves an integral term in addition to derivative terms. Note that the equation was not closed under any number of differentiations, due to the 5 Contraction Maps and Lyapunov Functions form of the intergral terms. As such, the single reduced equation for the state variable could never be expressed Turning out attention now to practical applications for strictly as an ODE of any finite order. Note that this reduction of dimension, recall that contraction maps can occur for one of the state variables, while a dif- and Lyapunov functions are useful tools for studying ferent state variable might satisfy a finite order ODE. the asymptotic stability of nonlinear dynamical sys- For such cases, the nonlinearity in the system results in tems. In this section, we use the three examples worked their being certain favored state variables with which explicitly in Section 3 in order to demonstrate the util- to perform the reduction to a single ODE. ity of reduction of dimension for finding Lyapunov func- Reduction of dimension for nonlinear dynamical systems 9 tions. Using these results, we can recover stability re- a < 2 (this will simplify the mathematics, and is con- | | sults for these dynamical systems which were obtained sistent with the physics of the Rössler system). Then, through other approaches, and which agree with exist- we obtain ing results in the literature. √4 a2 √4 a2 Y (t)= ǫeat/2 cos − t + C sin − t ( 2 ! 2 !) 5.1 Rössler System t s aζ −aζ ′ + m0 K(t,s)exp e (e Y (ζ)) dζ ds , The Rössler system has two equilibrium values, y∗, 0 0 ± Z Z for y(t), and the constant y∗ must satisfy the quadratic (26) equation where C is a constant that will depend on the initial a(y∗)2 + cy∗ + b =0 . (21) value of Y˙ (0) (the value of which will not impact our analysis) and K(t,s) is the kernel In order to discover a Lyapunov function for the 2 a 4 a Rössler system, it is tempting to assume a bowl-shaped K(t,s)= e 2 (t−s) sin − (t s) . (27) 2 2 2 2 − map of the form αx + βy + γz , or minor variations on this theme involving higher power polynomials of Observe that for 2 0, for large enough time Therefore, we shall use one of the three equations ob- t˜(ǫ) > 0, the solution Y (t) will lie in a neighborhood tained for the isolated state variables of the Rössler ǫ < Y (t) < ǫ for all t > t˜(ǫ). Therefore, Y 0 as system. − → t . Yet, by definition of Y (t), this implies y Consider equation (4) for the Rössler system (3). → ∞ → y∗ as t . Using this, one may shown that x Let us write Y (t) = y(t) y∗ in the neighborhood of → ∞ → − ay∗ and z y∗ as t . Hence, we have shown either equilibrium value y∗. This transformation will − → − → ∞ that a < 0 gives a stable solution, which was already prove useful, as the Lyapunov function needs to vanish known from different work. The nice thing about this at the equilibrium value selected. (There is therefore the approach is that is allows us to bypass a linear stability need to construct such a function in a neighborhood of analysis involving the calculation of eigenvalues at the each equilibrium point.) Under this transformation, (4) algebraic solution to y∗ found from (21). Indeed, we is put into the form did not even need to calculate the equilibrium value y∗ ... for the present analysis, as the analysis holds for an Y aY¨ + Y˙ Y¨ aY˙ + Y Y˙ aY =0 . (22) − − − − arbitrary equilibrium value satisfying (21). Let us define a function m = Y¨ aY˙ + Y so that (22) − is put into the form 5.2 Lorenz System m˙ (Y˙ aY )m =0 . (23) − − In order to find a Lyapunov function for the Lorenz system in a neighborhood of the zero equilibrium (x,y,z)= Observe that (23) can be written as (0, 0, 0), let us assume a bowl type function of the form ′ m˙ eat e−atY m =0 . (24) − V (x,y,z)= αx2 + βy2 + γz2 , (28) From this, we recover where α > 0, β > 0, and γ > 0 are constant param- t eters to be selected. recall that physically interesting Y¨ aY˙ + Y = m = m exp eaζ (e−aζ Y (ζ))′dζ , model parameters a, b, and c are positive. Then, the − 0 Z0 time derivative of V is given by (25) 1 V˙ = αax2 βy2 γcz2 +(αa+βb)xy +(γ βb)xyz . where m0 is a constant of integration. As we are inter- 2 − − − − ested in recovering information about the asymptotic (29) stability of the Rössler system, let us pick the initial condition Y (0) = ǫ. This corresponds to setting the ini- Clearly, we should take γ = βb. Note that tial condition such that it is contained within a neighborhood of the equilibrium value. Let us also restrict (√αax βy)2 = αax2 βy2 +2 αβaxy . (30) − − − − p p 10 Heather A. Harrington and Robert A. Van Gorder

Then, To remove the first term, which is hyperbolic in nature, we should choose β = α + γ. Meanwhile, the remove 1 V˙ = (√αax βy)2 γcz2 +(αa+βb 2 αβa)xy , the second term, which is also hyperbolic, we should 2 − − − − a set β = b α for non-zero b. Since all other parameters p p (31) − a are positive, we must require b < 0. Then, β = |b| α, a−|b| hence V˙ 0 provided that βb < 2√αβa αa (since and placing this into β = α + γ gives γ = . As ≤ − |b| this would imply αax2 βy2 + (αa + βb)xy < 0). Let we need γ > 0, this gives the added restriction a> b . − − 1 | | us pick β = αa. Then, the condition reduces to b < 1. The parameter α is arbitrary, so we take α = 2 . We 1 As α> 0 was arbitrary, we set α = 2 . This means that therefore obtain whenever a > 0, 0 0, there exists a 1 a a b Lyapunov function V (x,y,z)= x2 + y2 + − | |z2 , (36) 2 2 b 2 b 1 a ab | | | | V (x,y,z)= x2 + y2 + z2 , (32) 2 2 2 and this candidate function is indeed a contraction map satisfying V (0, 0, 0) = 0, V˙ < 0 for (x,y,z) = (0, 0, 0), 6 since V (0, 0, 0) = 0, V as (x,y,z) (ra- and V as (x,y,z) , provided that the | |˙ → ∞ | | → ∞ | | → ∞ | | → ∞ dially unbounded), and V < 0 for (x,y,z) = (0, 0, 0). parameter restrictions a > b , b < 0, and c > 0 hold. 6 | | Interestingly, the condition b , b < 0, 6 | | In order to find a contraction map for the Qi-Chen-Du- and c > 0. Hence, Vˆ (z, z˙) is a contraction map for the Chen-Yuan (QCDCY) system, we begin with the bowl state variable z(t) when a > b , b < 0, and c > 0. | | shaped assumption for a Lyapunov function about the With this, we have determined the stability of the zero equilibrium (x,y,z)=(0, 0, 0), equilibrium for the QCDCY system. V (x,y,z)= αx2 + βy2 + γz2 , (34) where α > 0, β > 0, and γ > 0 are constant parame- 6 Computational considerations for chaotic ters to be selected. Differentiating with respect to t and trajectories using the three constituent equations of the QCDCY system, we have There are a variety of methods available for trying to find chaotic trajectories in nonlinear dynamical sys- V˙ = (α β + γ)xyz + (αa + βb)xy αax2 βy2 γcz2 . tems, and the approach highlighted in this paper does − − − − (35) not add to collection of tools, explicitly. However, the reduction of dimension approach outlined in Section 2 Since we need α > 0, β > 0, and γ > 0, we should can be used to make finding chaos in dynamical systems consider model parameters satisfying a> 0 and c> 0. more efficient. To demonstrate this, we shall consider Reduction of dimension for nonlinear dynamical systems 11 two rather distinct approaches, namely, the undeter- a)B and C = (α2n2 + aα n + 1)B for all n n n − | | n ∈ mined coefficients method for obtaining chaotic trajec- Z. If we were to truncate the expansion for y(t), in tories and the competitive modes analysis for identifi- the manner described above, we would need to solve cation of chaotic parameter regimes. For each of these 2J + 1 nonlinear algebraic equations for B j=J and { j }j=−J approaches, we show that an application of reduction α, while the coefficients for x(t) and z(t) are immedi- of dimension results in a simplification of each test for ately found once we know these parameters. This means chaos. that by first reducing the dimension of the ODE system, we would be able to reduce the computational complexity of the problem by a factor of three. For 6.1 Calculation of trajectories via undetermined higher-dimensional system, the reduction in computa- coefficients tional complexity will scale as the dimension of the system itself. In other words, a solution in term of the un- When attempting to analytically calculate chaotic tra- determined coefficient method will not depend on the jectories, even in an approximate sense, one often re- size of the dynamical system provided that the dynam- duces the dimension of the governing equations. The ical system can be reduced in dimension to a single reason for this lies in the fact that it is easier to con- equation governing one state variable. sider an expansion for one state variable, rather than multiple state variables. To best illustrate this point, let us return to the Rössler equation (3). 6.2 Competitive modes analysis: A check for chaos One popular method for approximating trajectories of chaotic systems analytically is the undetermined co- The method of competitive modes involves recasting efficient method [3,4,5,6]. Since Taylor series expan- a dynamical system as a coupled system of oscillators sions for nonlinear systems often have a finite region [13,14,15,16,17,18]. Consider the general nonlinear au- of convergence centered at the origin, yet the chaotic tonomous system of dimension n given by dynamics remain bounded in space, one often consid- ers non-polynomial base functions. One popular choice x˙i = fi(x1, x2, ..., xn) . (39) would be a function of the form Differentiation of (39) once gives a coupled system of ∞ A e−αjt for t 0 , S(t; A j=∞ , α)= j=0 j ≥ second order equations, { j }j=−∞ ∞ αjt (Pj=0 A−j e for t< 0 . n P (38) ∂fi xï = fj = xigi(x1, x2, ..., xi, ..., xn) ∂xj − In this expression, the A R and the parameter α> 0 j=1 j ∈ X are undetermined parameters which one typically will + hi(x1, x2, ..., xi−1, xi+1, ..., xn) . obtain in an iterative manner. Assuming such an expan- (40) sion in time, it males sense to consider a solution the the Rössler system (3) of the form x(t)= S(t; A j=∞ , α), { j }j=−∞ When a gi is positive, its respective ith equation be- y(t)= S(t; B j=∞ , α), and z(t)= S(t; C j=∞ , α). haves like an oscillator. The following conjecture is posed { j }j=−∞ { j }j=−∞ Placing these equations into (3), one would obtain an in [14]: infinite system of nonlinear algebraic equations for all of the coefficients and the temporal scaling α > 0. In Competitive Modes Requirements: The conditions practice, one would truncate these expansions, taking for dynamical systems to be chaotic are given by: the sum over J j J for some J > 0. As the (A) there exist at least two modes, labeled gi in the − ≤ ≤ solutions may converge slowly - if they converge at all system; (owing to the nonlinearity), one would need to solve (B) at least two g’s are competitive or nearly competi- 6J + 1 nonlinear algebraic equations. tive, that is, for some i and j, g g > 0 at some t; i ≈ j Assume, instead, that we wish to solve (4) by the (C) at least one of the g’s is a function of evolution approach described above. We would then insert the variables such as t; and expansion for y(t) into (4). Assuming that we can solve (D) at least one of the h’s is a function of system vari- the resulting nonlinear algebraic equations for the con- ables. stants B j=∞ and α, we can then recover x(t) and { j }j=−∞ z(t) by recalling x =y ˙ ay and z = y¨+ ay˙ y. From The requirements (A)-(D) essentially tell us that a − − − these expressions, it is simple to show A = (α n + condition for chaos is that two or more equations in (40) n − | | 12 Heather A. Harrington and Robert A. Van Gorder behave as oscillators (gi > 0), and that two of these os- equations are never oscillators. Meanwhile, we can de- cillators lock frequencies at one or more times. In prac- compose the right hand sides of the latter two equa- tice, we find that the frequencies agree at a countably tions, so that infinite collection of time values [13,18]. The frequen- F (y ,...,y )= y g + h (44) cies should be functions of time (i.e., we have nonlinear 1 p − p−1 p−1 p−1 frequencies), and there should be at least one forcing function which depends on a state variable. and In order to consider all possible chaotic dynamics, p−1 ∂F ∂F one would have to compare each pair g = g , i = j, yi+1 + F (y1,...,yp)= ypgp + hp . (45) i j 6 ∂y ∂y − i=1 i p i, j =1, 2, 3,...,n. Accounting for symmetry, this gives X 2n−1 1 matchings to consider. For high-dimensional − Note that there are now exactly two mode frequen- dynamical systems, this number becomes rather large. cies, g and g , and there is always an h depend- As an example, in the case of a ten-dimensional system, p−1 p i ing on state variables. In order to determine if the sys- there will be 511 possible matchings to be checked in tem (42) satisfies the competitiveness conditions (A)- order to ensure one has determined the possible chaotic (D) (and therefore if the original system satisfies these regimes. For such situations, the approach is not par- competitiveness conditions), it is sufficient to check if ticularly efficient, and a competitive modes analysis is g = g > 0 for some collection of time values. This is often considered for systems of dimension three or four p−1 p only one condition to check, rather than 2n−1 1 con- in the literature. − ditions to check from the original system. Therefore, Let us consider dynamical systems (39) which can while the conversion of the system (39) to the equiva- be put into the form of a single equation for one state lent system (42) may seem somewhat roundabout, do- variable. As such an equation will encode the dynam- ing so greatly simplifies the search for possible chaotic ics of the complete system, it is sufficient to consider dynamics under the competitive modes framework. a competitive modes analysis for the resulting equation. Suppose that the resulting equation has maximal derivative of order p> 0 (where p need not be equal to 7 Discussion n, as we have seen in earlier sections). Then, associating y(t) to this single state variable, we have The construction of Lyapunov functions for nonlinear dpy dy dp−1y dynamical systems is often either simple, or quite chal- = F y, ,..., . (41) lenging, with little room in between. Aside from choos- dtp dt dtp−1 ing some standard forms (such as the common bowl Since the competitive modes analysis relies on us ob- shape centered about an equilibrium value), there is taining a system of oscillator equations, let take y = y1 more an art to the selection of such function. However, and write the equation (41) as the system as we have demonstrated for the Rössler system, it is possible to use differential elimination to obtain a con- y˙1 = y2 ..., y˙p−1 = yp , y˙p = F (y1,...,yp) . (42) traction map in a single state variable, which can then be used to obtain a stability result for all state vari- Differentiation of (42) results in the system of second ables. A similar approach was also employed in [1] to order equations given by study Michaelis-Menten enzymatic reactions, and after using a reduced equation, the proof of global asymp- y¨1 = y3 , totic stability for a positive steady state was rather . simple. Differential elimination, and reduction of di- . mension more generally, can therefore be useful in help- y¨p−2 = yp , ing one determine asymptotic properties of solutions to

y¨p−1 =y ˙p = F (y1,...,yp) , nonlinear dynamical systems. On the other hand, it is p p−1 now known that there are autonomous chaotic systems ∂F ∂F ∂F y¨ = y˙ = y + F (y ,...,y ) . which lack any equilibrium [78]. The approach outlined p ∂y i ∂y i+1 ∂y 1 p i=1 i i=1 i p here could still be used to construct maps which demon- X X (43) strate the boundedness of trajectories in time for such systems, even if such trajectories do not approach any The right hand side of the first p 2 second order equa- fixed points. − tions do not depend on the state variable for each re- While useful for obtaining contraction maps, reduc- spective equation, so g = = g = 0. Hence, these tion of dimension is also a promising tool for finding 1 ··· p−2 Reduction of dimension for nonlinear dynamical systems 13 and studying chaotic trajectories in nonlinear dynam- light on the behavior of solutions, and can prove useful ical systems. We were able to show that reduction of in obtaining Lyapunov functions. More generally than dimension can be used to simplify calculations involved for dynamical systems, these inverse operators ∂−n play in using undetermined coefficient methods [3,4,5,6] by a role in the study of operators and integrable hierar- a factor of 1/n, where n is the dimension of the dy- chies arising in nonlinear evolution PDEs [80,81,82,83, namical system, meaning that the number of equations 84,85,86]. Therefore, the extension of the algorithm to needing to be solved will not increase with the size of the ring of formal Laurent series in ∂ would be a fruit- the system - i.e. the computational complexity of the ful area for future work, not only for dynamical systems approach will not scale with the size of the system but but also for integrable partial differential equations. rather will remain fixed. This means that one may approximate chaotic trajectories through such approaches in systems of rather large dimension, without the com- Acknowledgements putational problem becoming unwieldy, provided that HAH gratefully acknowledges the support of EPSRC the system can be reduced to a single ODE governing Fellowship EP/K041096/1. only one state variable. While chaos is often studied numerically, recently analytical approaches have been employed to construct References trajectories approximating chaotic orbits. When em- ploying these methods, it can be beneficial to consider 1. K. Mallory and R. A. Van Gorder, A transformed time- dependent Michaelis-Menten enzymatic reaction model a single equation rather than a system of equations, and its asymptotic stability, Journal of Mathematical even if the single equation is more complicated. This is Chemistry 52, 222-230 (2014). true for series and perturbation approaches, as the re- 2. R. Li, R. A. Van Gorder, K. Mallory and K. Vajrav- duction of order requires one to track fewer functions, elu, Solution method for the transformed time-dependent Michaelis-Menten enzymatic reaction model, Journal of which is particularly useful when dealing with messy Mathematical Chemistry 52, 2494-2506 (2014). equations. Furthermore, analytical approaches permit- 3. T. Zhou, G. Chen, and Q. Yang, Constructing a new ting the control of error, such as the optimal homotopy chaotic system based on the Silnikov criterion, Chaos, Solitons & Fractals 19, 985-993 (2004). analysis method, rely on assigning an error control pa- 4. Y. Jiang and J. Sun, Si’lnikov homoclinic orbits in a new rameter to each state variable. Reduction of dimension chaotic system, Chaos, Solitons & Fractals 32, 150-159 can allow one to minimize error via a single control pa- (2007). rameter [2], rather than over multiple parameters [79], 5. R.A. Van Gorder and S.R. 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Liao, Homotopy Analysis Method in Nonlinear Dif- the system of a single ODE for one state variable, we ferential Equations. Springer & Higher Education Press, prove that only one comparison would needed. This is Heidelberg, 2012. 9. S. Liao, An optimal homotopy-analysis approach for particularly beneficial when studying large dimensional strongly nonlinear differential equations, Communica- systems, since the number of naive comparisons needed tions in Nonlinear Science and Numerical Simulation 15, scales like 2n−1 in dimension n. 2315-2332 (2010). 10. C.K. Ahn, Neural Network H∞ Chaos Synchronization, In the future, it would be interesting to consider Nonlinear Dynamics 60, 295-302 (2010). an algorithm that considered elimination not only ele- 11. C.K. Ahn, S.T. Jung, S.K. Kang, and S.C. Joo, Adaptive n d H∞ Synchronization for Uncertain Chaotic Systems with ments ∂ (∂ = dt ) of a differential ring R[[∂]], but also integral operators ∂−n (where ∂−n satisfies ∂−n∂n = ∂0 External Disturbance, Communications in Nonlinear Sci- n ence and Numerical Simulation 15, 2168-2177 (2010). and hence is the inversion of the operator ∂ ). Indeed, 12. C.K. Ahn, Output Feedback H∞ Synchronization for De- in cases where the differential elimination algorithm layed Chaotic Neural Networks, Nonlinear Dynamics 59, failed to give a reduction of the system to a single ODE, 319-327 (2010). we found by manual substitutions that one can arrive 13. S. R. Choudhury and R. A. Van Gorder, Competitive modes as reliable predictors of chaos versus hyperchaos at an integro-differential equation. While more compli- and as geometric mappings accurately delimiting attrac- cated, such integro-differential equations can still cast tors, Nonlinear Dynamics 69, 2255 (2012). 14 Heather A. 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Chen system [46,47]: Appendix B: List of Hyperchaotic Systems − x˙ = a(y x) , When testing our approach, we considered a variety of specific y˙ = (b − a)x − xz + by , (51) hyperchaotic systems. Results for these are listed in Table 2. z˙ = xy − cz . As the form and scaling of such equations can vary in the literature, we list the specific form of these equations used in Lüsystem [48,49]: our work. Let x,y,z,w ∈ C4(R) and let a,b,c,d,e,f ∈ R be param- x˙ = a(y − x) , eters. y˙ = by − xz , (52) 4D Rössler flow [67]: z˙ = xy − cz . x˙ = −y − z, y˙ = x + 0.25y + w, T system [50,51,52]: (61) z˙ = 3 + xz , x˙ = a(y − x) , w˙ = −0.5z + 0.05w. y˙ = (b − a)x − axz , (53) z˙ = xy − cz . Hyperchaotic Chen system [68,69]:

4D Qi-Du-Chen-Chen-Yuan system [53]: x˙ = a(y − x) , y˙ = −bx − xz + cy − w, x˙ = a(y − x) + yzw, (62) z˙ = xy − dz, y˙ = b(x + y) − xzw , (54) w˙ = x. z˙ = −cz + xyw , w˙ = −dw + xyz . Hyperchaotic L¨usystem [70]:

Qi-Chen-Du-Chen-Yuan system [54]: x˙ = a(y − x) + w, y˙ = by − xz , x˙ = a(y − x) + yz, (63) z˙ = xy − cz , y˙ = bx − y − xz , (55) w˙ = xz + dw. z˙ = xy − cz . Modiﬁed hyperchaotic L¨usystem [71]: Generalized Lorenz canonical form [55,56,57]: x˙ = a(y − x + yz) , x˙ = ax − (x − y)z, y˙ = by − xz + w, y˙ = −by − (x − y)z, (56) (64) z˙ = xy − cz , z˙ = −cz + (x + y)(x + dy) . w˙ = −dx. Two-parameter model for the blue-sky catastrophe [58, 59,60]: Hyperchaotic Wang-Liu system [72]:

2 2 2 2 − x˙ = 2 + a − 10 x + y x + y + 2y + z , x˙ = a(y x) , 3 2 2 y˙ = bx − cxz + w, y˙ = −z − (1 + y) y + 2y + z − 4x + ay, (57) 2 (65) 2 2 z˙ = −dz + ex , z˙ = (1 + y)z + x − b. w˙ = −fx. 4D Lorenz - Stenﬂo system [61,62,63]: Hyperchaotic Jia system [73,74]: x˙ = a(y − x) + bw , x˙ = a(y − x) + w, y˙ = cx − xz − y, (58) y˙ = bx − xz − y, z˙ = xy − dz, (66) z˙ = xy − cz , w˙ = −x − aw. w˙ = dw − xz . Genesio-Tesi system [64,65]: Hyperchaotic Qi - van Wyk - van Wyk - Chen system [75, x˙ = y, 76]: y˙ = z, (59) x˙ = a(y − x) + yz, z˙ = ax + by + cz + x2 . y˙ = b(x + y) − xz , (67) Arneodo-Coullet-Tresser system [66]: z˙ = −cz − dw + xy , w˙ = ez − fw + xy . x˙ = y, y˙ = z, (60) z˙ = ax − by − cz − x3 .