Stability and instability in nonlinear dynamical systems Martin Hasler School of Computer and Communication Sciences Swiss Federal Institute of Technology Lausanne (EPFL) Switzerland Review of the stability / instability in linear dynamical systems Notion of stability / instability in nonlinear dynamical systems Criterion for the stability / instability of - equilibrium / fixed points - periodic solutions Lyapunov exponents and criterion for chaos Course in Kiev, Sept. 20-22, 2011

Linear autonomous dynamical systems Solutions: For each initial state x(0) there exists exactly one solution: Course in Kiev, Sept. 20-22, 2011

Stability / instability of linear autonomous dynamical systems System can be: stable: solutions are bounded, close initial states  solutions stay close asymptotically stable: solutions converge to zero close initial states  solutions converge to each other unstable: almost all solutions diverge to infinity close initial states  almost always solutions drift apart Main criteria for stability / instability: If all eigenvalues of A satisfy then system is asymptotically stable If at least one eigenvalue of A satisfies then system is unstable Course in Kiev, Sept. 20-22, 2011

Nonlinear autonomous dynamical systems Course in Kiev, Sept. 20-22, 2011

Nonlinear autonomous dynamical systems Solutions: No explicit expressions for the solutions Under weak hypotheses, for each initial state exactly one solution exists for For fixed t, x(t) depends continuously on x(0) and on parameters. Stability is property for Special solutions: constant solutions - : equilibrium point (continuous time) - : fixed point (discrete time) Course in Kiev, Sept. 20-22, 2011

Van der Pol oscillator Course in Kiev, Sept. 20-22, 2011

Definition of (small scale) stability Solution x(t) is stable, if for each e > 0 there is a d > 0 such that Solution x(t) is asymptotically stable, if for each e > 0 there is a d > 0 s.t. Basin of attraction of an asymptotically stable solution x(t): Globally asymptotically stable solution x(t) : Basin of attr. Course in Kiev, Sept. 20-22, 2011

Iterations of the logistic map Course in Kiev, Sept. 20-22, 2011

Stability of equilibrium / fixed point Theorem: If the eigenvalues of satisfy for all i, then the equilibrium (fixed) point is asymptotically stable. If the eigenvalues of satisfy for at least one i, then the equilibrium (fixed) point is unstable. Course in Kiev, Sept. 20-22, 2011

Stability of a periodic solution of a discrete- time dynamical system Theorem: If the eigenvalues of satisfy for all i, then the periodic solution is asymptotically stable. If the eigenvalues of satisfy for at least one i, then the equilibrium (fixed) point is unstable. Course in Kiev, Sept. 20-22, 2011

Variational equations around a periodic solution of a continuous-time system Course in Kiev, Sept. 20-22, 2011

Stability of a periodic solution of a continuous-time dynamical system Theorem: If the eigenvalue 1 of M(T) is simple and all other eigenvalues satisfy , then the T-periodic solution is stable. Furthermore, solutions y(t) starting close to the converge to a phase-shifted version of : If there is an eigenvalue of M(T) with , then is unstable. Course in Kiev, Sept. 20-22, 2011

Variational equations around any solution Arbitrary solution x(t). Nearby solution y(t). Increments Linear approximation: Variational equations around x(t): Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents If eigenvalues of L: Lyapunov exponents of x(t) (Oseledec) : For almost all solutions x(t) If system is ergodic (single attractor)  for almost all solutions x(t), L is the same Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents Lyapunov exponents , eigenvectors xi. For almost all initial increments:  Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents of one-dimensional discrete-time systems (Birkhoff) : For almost all solutions x(t), L exists If system is ergodic (single attractor)  for almost all solutions x(t), L is the same Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents of constant solutions in one-dimensional discrete-time systems All solutions starting in the basin of attraction of have Lyap.exp. L Solutions starting close to have a different Lyapunov exponent Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents of periodic solutions in one-dimensional discrete-time systems All solutions starting in its basins of attraction have Lyap.exp. L Solutions starting close to it have a different Lyapunov exponent Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents of one-dimensional discrete-time systems If all solutions are asymptotically periodic, but various asymptotically stable and unstable periodic solutions and fixed points are present:  - solutions converging to the same periodic solution or fixed point have the same (negative) Lyapunov exponent. - unstable periodic solutions or fixed points have different (positive) Lyapunov exponents (exceptional solutions) - if only a single periodic solution is aymptotically stable, almost all solutions have the same (negative) Lyapunov exponent If the system has a single attractor which is chaotic, almost all solutions have the same (positive) Lyapunov exponent. There are in general, however, infinitely many unstable periodic solutions, which have different Lyapunov exponents. Course in Kiev, Sept. 20-22, 2011

Lyapunov exponents for constant and periodic solutions: general case Discrete time systems: Lyapunov exponents of constant solution at fixed point : Lyapunov exponents of T-periodic solution : Continuous time systems: Lyapunov exponents of constant solution at fixed point : Lyapunov exponents of T-periodic solution Course in Kiev, Sept. 20-22, 2011