The Center Manifold Theorem

1. The Center Manifold Theorem • The Center Manifold Theorem - Motivation

2. Step 1 :  Step 2 : k m k m Lower Dimensional part Question : How do we isolate this lower dimensional part ? ( times continuously differentiable)

3. k m Lower Dimensional part (Continued) Step 3 : Change of variables i.e., Then

4. Center Manifold Theorem Theorem (Center Manifold Theorem) Consider the system defined by .

5. Center Manifold Theorem (Continued) Step 4 : Solve for h and evaluate the stability of the reduced system. Remark : One does not need an exact solution of the P.D.E. in order to perform the procedure. Ex: Step 1 :

6. Example (Continued) Step 2 : Step 3 :

7. Center Manifold Equation Too hard to solve, in general, so seek an approximated solution. Try :

8. Pick out the second order terms Plug this into the reduced order equation Then the stability of the above system In general, one would seek to add more terms to h(y), i.e.,

9. u y System good (BI) good (BO) Input-Output Stability • Input-Output Stability Lyapunov Stability w.r.t. perturbation in initial condition. Input-Output Stability w.r.t. perturbation in input. For linear system, asymptotical stability  BIBO

10. Input-Output Stability (Continued) How can this tuning be generalized for nonlinear systems ? Another representation normed linear space

11. Input-Output Stability (Continued) H may be unstable, so y(t) might not have the same norm.

12. Note : very close to linearity Input-Output Stability (Continued) Definition:

13. Examples of Gains A few examples of gains 1) H: linear time invariant system described by G(s). Thus

14. Examples of Gains 2) H: static nonlinearity in the sector [a, b] or equivalently

15. + - + + Examples of Gains

16. Examples of Gains

17. Examples of Gains

18. Lyapunov Stability & Lp

19. Theorem (Continued)

20. Proof Proof:

21. Example Ex:

22. Orbital Stability • Orbital Stability • Periodic

23. Stability of a Trivial Solution

24. Asymptotically Orbitally Stable

25. Asymptotically Orbitally Stable Theorem: Proof : See Nonlinear systems : vol. I

26. Example Ex:

27. Bifurcation • Bifurcation

28. stable unstable stable stable Pitchfork bifurcation • Pitchfork bifurcation bifurcation point

29. Transcritical bifurcation • Transcritical bifurcation

30. Hopf bifurcation • Hopf bifurcation