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Structural Stability and Bifurcations

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Structural Stability and Bifurcations Lezione 5 Structural stability and bifurcations Pier Luca Maffettone - Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 2 /61 References Very instructive with simple approach Strogatz, S. H., Nonlinear dynamics and chaos: with application to physics, biology, chemistry and engineering, Addison Wesley, New York 1994. A complete overview of the problems Wiggins S., Introduction to applied nonlinear dynamical systems and chaos, Springer Verlag, New York 1990 (2nd Ed. 2003) Kuznetsov Y. A., Elements of applied bifurcation theory, Springer Verlag, New York 2004 (3rd Rev. Ed.) Very detailed on some aspects Carr J., Applications of centre manifold theory, Springer Verlag, New York, 1981 AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 3 /61 Previous lectures - Nonlinear dynamical systems Stability of hyperbolic equilibrium points and periodic orbits Stability of nonhyperbolic situations Center manifold theorem Dynamics on the center manifold The linearized dynamics does not give information on stability AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 4 /61 Motivations - Theory Models contain parameters What happens to the geometry of the phase space when a parameter changes? Quantitative changes Qualitative changes Implication on the safety Qualitative changes will be called bifurcations. When can we observe qualitative changes? Can we a priori know the possible scenarios for different dynamical systems? Dynamical systems can be quite “large”: do we need to account for details of their “largeness” or we can limit to something simpler? AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 5 /61 Motivations - Applications Nonlinear models of engineering systems exhibit instabilities: These phenomena must be understood for a Multiplicity, Ignitions correct design and Symmetry breakings optimization Phase transitions... Software for the stability analysis Automatic Reliable Large scale systems AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 6 /61 Outline Introduction Topological equivalence Structural stability Bifurcations Local Global Simplifications on the center manifold Normal forms AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 7 /61 General frame In this lecture we will only consider systems of the types: Continuous dx n r n n m = f(x, µ),x R ,f C (R ,R ) with r 1,µ R with m 1 dt ∈ ∈ ! ∈ ! Discrete n r n n m xk+1 = F (xk,µ),x R ,F C (R ,R ) with r 1,µ R with m 1 ∈ ∈ ! ∈ ! The parameters µ are now explicitly considered to change for a change in the operating conditions to account for uncertainties in time AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 8 /61 A very simple linear example We start from a very simple linear example x˙ = µx x=0 is an equilibrium point It is asymptotically stable if µ<0, the phase portrait does not change if the parameter is perturbed a little bit. It is unstable if µ>0, the phase portrait does not change if the parameter is perturbed a little bit. It is stable if µ=0. The latter condition merits some attention: it is a nonhyperbolic equilibrium point Undecided Any perturbation of the parameter value determines a qualitative change in the phase portrait of the system AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 9 /61 Motivation Two important considerations: Hyperbolic points seems to be “indifferent” to small parameter changes, while the nonhyperbolic point is strongly affected from them Is this a kind of stability with respect to parameter changes? NB: the stability we know was related to changes in the state of the system. It seems that a qualitative change occurs when the system passes through a nonhyperbolic point: Bifurcations? These two aspects will be addressed in detail in this lecture. AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 10 /61 Topological equivalence AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 11 /61 Topological equivalence of linear systems Let’s reconsider the case of a 2D linear system It is apparent that we can describe the hyperbolic systems with just 3 cases: Sources, saddles and sinks In the case of nonhyperbolic systems we can identify 5 situations (not all shown in this figure) Similar conclusions could be drawn for larger dimension systems (the saddles could be of different types in such cases) AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 12 /61 Topological equivalence of linear systems For linear systems one could define an equivalence relation and equivalence classes on the basis of the eigenvalues. For hyperbolic 2D systems for example AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 13 /61 Topological equivalence of linear systems In the case of nonhyperbolic conditions one has to account for algebraic and geometrical multiplicity of the eigenvalues In the 2D case the possible situations are: AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 14 /61 Topological equivalence of systems We are strongly tempted to try to develop an equivalence criterion of the phase portrait for any kind of dynamical system (linear, nonlinear, continuous, discrete) n m x˙ = f(x, α),x R , α R ∈ ∈ n m y˙ = g(y,β),y R , β R ∈ ∈ DEFINITION Two systems SA1 (phase space X) and SA2 (phase space Y) of the same order n share the same qualitative behavior and are topologically equivalent if and only if there exists a homeomorphism that maps orbits of SA1 on those of SA2 preserving the direction of time. A homeomorphism is an invertible map such that both the map and its inverse are 0 1 0 continuous ψ C ( X, Y ) ψ − C ( Y, X ) . ∈ ∈ It is important to remark that the two systems could be the same system for a different value of the parameters. AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 15 /61 Why homeomorphism? AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 16 /61 Topological equivalence of systems The definition fulfills the following three properties 1. Reflection: SA1 is Topologically Equivalent to SA1 2. Symmetry: SA1 Topologically Equivalent to SA2 implies SA2 Topologically Equivalent to SA1 3. Transitivity: If SA1 is Topologically Equivalent to SA2 and SA2 is Topologically Equivalent to SA3 then SA1 is Topologically Equivalent to SA3 With this definition: two hyperbolic continuous linear systems with stable and unstable eigenspaces of the same dimensions are topologically equivalent. The case of maps: two linear maps with stable and and unstable eigenspaces of the same dimensions and the matrices characterizing the dynamics on both eigenspaces have the same determinant are topologically equivalent AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 17 /61 Topological equivalence of nonlinear systems In the case of nonlinear systems the phase portrait are topologically equivalent if: 1. They have the same number of equilibrium points with the same stability properties; 2. They have the same number of periodic orbits with the same stability properties 3. They have the same invariant in one to one correspondence For nonlinear systems a local equivalence is of course useful DEFINITION The system SA1 in the subset U of the phase space X is said topologically equivalent to the system SA2 in the subset V of the phase space Y if there exists a homeomorphism that transforms the orbits (or pieces of orbits) of the first system into orbits (or pieces of orbits) of the second system by preserving the time direction of points corresponding orbits. AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 18 /61 Topological equivalence of systems An important case: if U includes an equilibrium point xE of the first system and V includes an equilibrium yE point of the second system In such a case the system SA1 is said to be close to xE topologically equivalent to SA2 in yE An example: the nonlinear system in U is topologically equivalent to the associate linearized system in V if the latter is hyperbolic (another way to state the Hartman-Grobmann theorem) U V AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 19 /61 Bifurcation conditions AA 2008/2009 Pier Luca Maffettone Nonlinear Dynamical Systems I AA 2008/09 Lezione 5 Structural stability and bifurcations 20 /61 Structural stability DEFINITION A system x ˙ = f ( x, µ 0 ) , or x k +1 = F ( x K ,µ 0 ) is structurally stable (with respect to the parameter) if and only if there exists an ε>0 such that its phase portrait is topologically equivalent to that of the system x˙ = f ( x, µ ) , or x k +1 = F ( x K ,µ ) for any µ such that ||µ-µ0||<ε. A hyperbolic equilibrium is structurally stable under smooth perturbations A sufficiently small perturbation of the vector field or of the map does not induce a qualitatively change of a structurally stable dynamical system A local version of the definition is of course available The global and local definitions of the structural stability are the basis of the definitions of global and local bifurcations.
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