1. 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 , 2011

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

3. 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 , 2011

4. Nonlinear autonomous dynamical systems
Course in Kiev, Sept , 2011

5. 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 , 2011

6. Van der Pol oscillator
Course in Kiev, Sept , 2011

7. 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 , 2011

8. Iterations of the logistic map
Course in Kiev, Sept , 2011

9. 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 , 2011

10. 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 , 2011

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

12. 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 , 2011

13. 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 , 2011

14. 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 , 2011

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

16. 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 , 2011

17. 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 , 2011

18. 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 , 2011

19. 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 , 2011

20. 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 , 2011