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K. Zhou Menton Professor

Nonlinear Systems. K. Zhou Menton Professor. Introduction. General nonlinear systems Automomous (Time Invariant) Systems (does not depend explicitly on time): Equilibrium Points: start from x*, remain at x* for all t>0. Introduction (cont.).

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K. Zhou Menton Professor

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  1. Nonlinear Systems K. Zhou Menton Professor

  2. Introduction • General nonlinear systems • Automomous (Time Invariant) Systems (does not depend explicitly on time): • Equilibrium Points: start from x*, remain at x* for all t>0

  3. Introduction (cont.) • Isolated Equilibrium: no other one in a neighborhood • Continuum of Equilibrium Points: • Linear Models: (a) superposition principle (b) all results are global

  4. Introduction: Nonlinear Phenomena • Finite Escape Time: • Linear: |x(t)|  as t  • Nonlinear: |x(t)| may   for some finite t<  • Multiple Isolated Equilibria: • Linear: only one isolated equilibria • Nonlinear: can have more than one. • Limit Cycles:

  5. Introduction: Nonlinear Phenomena • Subharmonic, Harmonic, or Almost-periodic Oscillations: A nonlinear system under periodic excitation can oscillate with frequencies which are submultiples or multiples of the input frequency • Chaos: • Multiple modes of behavior

  6. Example: Pendulum

  7. Example: Tunnel Diode Circuit

  8. Example: Tunnel Diode Circuit

  9. Example: Mass-Spring System

  10. Example: Mass-Spring System

  11. Negative-Resistance Oscillator

  12. Negative-Resistance Oscillator

  13. Negative-Resistance Oscillator

  14. Phase Plane

  15. Phase Portrait (Isocline Method)

  16. Example (Pendulum)

  17. Linear Systems

  18. Linear Systems

  19. Linear Systems

  20. Linear Systems

  21. Linear Systems

  22. Linear System Under Perturbation • Consider dx/dt=(A+A)x where A is a small perturbation • In case when A has no eigenvalues on the imaginary axis (including origin), the characteristic of the equilibrium is not changed under a sufficiently small perturbation A. I.e., if it is a node, a focus, a saddle, the perturbed system is still a node, a focus, a saddle. • THUS the Node, Saddle, Focus equilibrium points are STRUCTURALLY STABLE. • When A has one zero eigenvalue and a nonzero eigenvalue, the equlibrium point of the perturbed system will be a node (stable or unstable) or a saddle point (since a small perturbation can only result in two real eigenvalues with the nonzero eigenvalue keeping the same sign).

  23. Linear System Under Perturbation • When A has a pair of complex eigenvalues, a small perturbation will result in two stable or unstable complex eigenvalues, so the perturbed equlibrium point is either a stable focus or an unstable focus. • When A has two zero eigenvalues, anything can happen: two (stable or unstable ) real eigenvalues (node), one stable and one unstable eigenvalues (saddle), two complex (stable or unstable) eigenvalues (focus), two imaginary eigenvalues (center). • HYPERBOLIC EQUILIBRIUM POINT: A has no eigenvalues on the imaginary axis.

  24. Example (one zero eigenvalue)

  25. Multiple Equilibria Examples • tunnel diode circuit: dx1/dt=[-h(x1)+x2]/C dx2/dt=[-x1-Rx2+u]/L • With numbers: dx1/dt=0.5[-h(x1)+x2] dx2/dt=0.2(-x1-1.5x2+1.2) h(x1)=17.76x1-103.79x12+229.62x13-226.31x14+83.72x15 • Equilibria: (0.063, 0.758), (0.285,0.61), (0.884,0.21)

  26. Separatrix: the curve that separates the plane into two regions of different qualitative behavior • Q1 and Q3 are stable, Q2 is not stable.

  27. Example (pendulum) dx1/dt=x2 dx2/dt=-gsinx1/l-kx2/m

  28. Linearization (local behavior) Let p=(p1,p2) be an equilibrium point: f1(p1,p2)=0 and f2(p1,p2)=0 dx1/dt=f1(x1,x2)=f1(p1,p2)+a11(x1-p1)+a12(x2-p2)+h.o.t. dx2/dt=f2(x1,x2)=f2(p1,p2)+a21(x1-p1)+a22(x2-p2)+h.o.t. Let y1=x1-p1, y2=x2-p2 Then approximately we have dy1/dt=a11y1+a11y2 dy2/dt=a21y1+a22y2

  29. The behavior of the nonlinear system near the equilibrium will be similar to the behavior of the linearized system if the linear model has no eigenvalues on the imaginary axis (including the origin). • We shall call an equilibrium point of the nonlinear system a stable (respectively, unstable) node, a stable (respectively, unstable focus) focus, or a saddle point if its linearized system has the same behavior.

  30. Example

  31. Example

  32. Example

  33. Limit Cycle • A limit cycle is a isolated periodic solution. • Some limit cycles are stable while others are unstable.

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