the center manifold theorem l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The Center Manifold Theorem PowerPoint Presentation
Download Presentation
The Center Manifold Theorem

Loading in 2 Seconds...

play fullscreen
1 / 30

The Center Manifold Theorem - PowerPoint PPT Presentation


  • 749 Views
  • Uploaded on

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).

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The Center Manifold Theorem' - arleen


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
the center manifold theorem
The Center Manifold Theorem
  • The Center Manifold Theorem

- Motivation

lower dimensional part

Step 1 :

Step 2 :

k

m

k

m

Lower Dimensional part

Question : How do we isolate this lower dimensional part ?

( times continuously differentiable)

lower dimensional part continued

k

m

Lower Dimensional part (Continued)

Step 3 : Change of variables

i.e.,

Then

center manifold theorem
Center Manifold Theorem

Theorem (Center Manifold Theorem)

Consider the system defined by .

center manifold theorem continued
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
Example (Continued)

Step 2 :

Step 3 :

center manifold equation
Center Manifold Equation

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

Try :

pick out the second order terms
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.,

input output stability

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
Input-Output Stability (Continued)

How can this tuning be generalized for nonlinear systems ?

Another representation

normed linear space

input output stability continued11
Input-Output Stability (Continued)

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

examples of gains
Examples of Gains

A few examples of gains

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

Thus

examples of gains14
Examples of Gains

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

or equivalently

proof
Proof

Proof:

orbital stability
Orbital Stability
  • Orbital Stability
    • Periodic
asymptotically orbitally stable25
Asymptotically Orbitally Stable

Theorem:

Proof : See Nonlinear systems : vol. I

bifurcation
Bifurcation
  • Bifurcation
pitchfork bifurcation

stable

unstable

stable

stable

Pitchfork bifurcation
  • Pitchfork bifurcation

bifurcation

point

transcritical bifurcation
Transcritical bifurcation
  • Transcritical bifurcation
hopf bifurcation
Hopf bifurcation
  • Hopf bifurcation