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

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

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

Example: Pendulum

Example: Tunnel Diode Circuit

Example: Mass-Spring System

Negative-Resistance Oscillator

Phase Plane

Phase Portrait (Isocline Method)

Example (Pendulum)

Linear Systems

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).

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.

Example (one zero eigenvalue)

Multiple Equilibria Examples tunnel diode circuit: dx 1 /dt=[-h(x 1 )+x 2 ]/C dx 2 /dt=[-x 1 -Rx 2 +u]/L With numbers: dx 1 /dt=0.5[-h(x 1 )+x 2 ] dx 2 /dt=0.2(-x x ) h(x 1 )=17.76x x x x x 1 5 Equilibria: (0.063, 0.758), (0.285,0.61), (0.884,0.21)

Separatrix: the curve that separates the plane into two regions of different qualitative behavior Q 1 and Q 3 are stable, Q 2 is not stable.

Example (pendulum) dx 1 /dt=x 2 dx 2 /dt=-gsinx 1 /l-kx 2 /m

Linearization (local behavior) Let p=(p 1,p 2 ) be an equilibrium point: f 1 (p 1,p 2 )=0 and f 2 (p 1,p 2 )=0 dx 1 /dt=f 1 (x 1,x 2 )=f 1 (p 1,p 2 )+a 11 (x 1 -p 1 )+a 12 (x 2 -p 2 )+h.o.t. dx 2 /dt=f 2 (x 1,x 2 )=f 2 (p 1,p 2 )+a 21 (x 1 -p 1 )+a 22 (x 2 -p 2 )+h.o.t. Let y 1 =x 1 -p 1, y 2 =x 2 -p 2 Then approximately we have dy 1 /dt=a 11 y 1 +a 11 y 2 dy 2 /dt=a 21 y 1 +a 22 y 2

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.

Example

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