### Phase Diagram, Nonlinear Dynamic, and Chaos (Cornell Strogatz)

#### by allenlu2007

Strogatz’s lectures (youtube) on nonlinear dynamic and chaos is excellent!

看完25集的 lectures, 簡短的 summary 如下。

First, some background behind phase diagram for classic mechanics such as Lagrangian/Hamiltonian

1. Conservative system (i.e. no energy loss or gain) phase diagram: (i) dH/dt = 0, phase flow along constant Hamiltoninan curve (energy conservation); (ii) reversible and uncompress flow, meaning the volume/area with different initial conditions is invariant. (div. (p_dot, q_dot) = 0). This is in contrast to lossy/gain system (stable or chaotic) where the volume usually shrinks exponentially.

2. It is equivalent to divergence free (no sink or source, no loss/gain)

3. Only Liaponov stable condition in this case, no exponential stable point (no loss/gain)

volume w.r.t time is equal to div(x_dot). From Hamiltonia

div(x_dot) = runH / runprunq – runH / runqrunp = 0 so it is constant volume for conservative system.

For disspative system, the volume reduced to zero, may be a point, cycle, or plane, or strange attractor.

If the linear/nonlinear dynamic consist of energy dissipation/gain

1. There is sink (exponential stable condition)

2. Hamiltonian is not conserved

3. Compressed flow with sink

In addition to classic mechanics (linear and automous system)

Phase diagram is very import tool for nonlinear (and slowly time-varying, sort of) dynamic system.

Purely from topology/geometry reason assuming it is nonlinear autonomous system (no explicit t in equation)

1st order system: no oscillation, no overshoot!

2nd order system: oscillation possible, but no chaos (because perturbe initial condition is stable, except at unstable point.

3rd order system: possible chaos. perturb initial condition may result to different stable points.

It seems that 1st order system is not interesting, not really! Check Strogatz’s lecture.

When 1st order system couples with sweeping parameter(s), it introduces some interesting phenomena.

First: bifurcation: three types: saddle, pitchfork, transcental (marginal interesting IMO)

Second: Insect outbreak: jump and hysteresis (very interesting behavior IMO, any application in electronics circuit)

2nd order nonlinear dynamic system (most popular since it covers most dynamic system governed by classic mechanics (q_dot_dot ..)

Linearized 2nd order dynamic syste

m: trace and determinant of Jacobain –>

fixed points includes: node, center, spiral, saddle, –> graphic way to remember.

### Conservative System

Conservative system (保守系統: 一般是指能量保守，但可以是其它 quantity). This is the key of classic mechanics (Hamiltonian mechanics, dH/dt = 0)

First observation: undamped (不論是 positive damping, energy gain; or negative damping, energy loss) in the system. 所謂 damping 是 x_dot * damping const 項 in the equation of motion.

所有 linear 1D phase plane 都不是保守系統。linear 2D phase plane 也只有 T=0, D>0 的 x 軸是保守系統，其他的所有系統都是非保守，可見保守系統是非常特例。

How about nonlinear dynamic system? 可能有 energy conservation 嗎?

Yes: undamped and unforced Duffin oscillator.

Second observation: unforced. Forced system 不論是 linear or nonlinear, gain or loss, 都非保守系統。Force 在保守系統是由 potential gradient 產生。

### Limiting Cycle

Nonlinear 2nd order dynamic system: limiting cycle : how to determine if exists

Index theorem (only marginally useful), Dulac theory (not very useful)

Van der pole oscillator: relaxation oscillation, and harmonic oscillation with perturbation.

Nonlinear dynamic: **Energy method! (crude but useful); average theory (perturbation theory)**

**Some Interesting Topics**

*** ESD modeling using nonlinear dynamic system (I think it can be done, but what’s the use?)**

*** Chaotic sigma delta modulator to suppress the limiting cycle hum noise (this is practically useful)**

*** **