The Center Manifold Theorem

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# The Center Manifold Theorem - PowerPoint PPT Presentation

The Center Manifold Theorem. The Center Manifold Theorem. - Motivation. Step 1 : . . Step 2 : . k. m. k. m. Lower Dimensional part. Question : How do we isolate this lower dimensional part ?. ( times continuously differentiable). k. m. Lower Dimensional part (Continued).

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Presentation Transcript
The Center Manifold Theorem
• The Center Manifold Theorem

- Motivation

Step 1 :

Step 2 :

k

m

k

m

Lower Dimensional part

Question : How do we isolate this lower dimensional part ?

( times continuously differentiable)

k

m

Lower Dimensional part (Continued)

Step 3 : Change of variables

i.e.,

Then

Center Manifold Theorem

Theorem (Center Manifold Theorem)

Consider the system defined by .

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 :

Example (Continued)

Step 2 :

Step 3 :

Center Manifold Equation

Too hard to solve, in general, so seek an approximated solution.

Try :

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.,

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

Input-Output Stability (Continued)

How can this tuning be generalized for nonlinear systems ?

Another representation

normed linear space

Input-Output Stability (Continued)

H may be unstable, so y(t) might not have the same norm.

Examples of Gains

A few examples of gains

1) H: linear time invariant system described by G(s).

Thus

Examples of Gains

2) H: static nonlinearity in the sector [a, b]

or equivalently

Proof

Proof:

Orbital Stability
• Orbital Stability
• Periodic
Asymptotically Orbitally Stable

Theorem:

Proof : See Nonlinear systems : vol. I

Bifurcation
• Bifurcation

stable

unstable

stable

stable

Pitchfork bifurcation
• Pitchfork bifurcation

bifurcation

point

Transcritical bifurcation
• Transcritical bifurcation
Hopf bifurcation
• Hopf bifurcation