Nonchaotic Multidimensional Flows

Chaos and Time-Series Analysis

9/19/00 Lecture #3 in Physics 505

* Comments on Homework #1 (The Logistic Equation)

* Review (last week) - One Dimensional Maps (Logistic Map)

* Maps versus Flows

Maps: Flows:
Discrete time Continuous time
Variables change abruptly Variables change smoothly
Described by algebraic equations Described by differential equations
Complicated dynamics (in 1-D) Simple dynamics (in 1-D)
Xn+1 = f(Xn) dx/dt = f(x)
Example: Xn+1 = AXn Example: dx/dt = ax
Solution: Xn = AnX0
Solution: x = x0eat
Growth for A >1 
Decay for A < 1
Growth for a > 0 
Decay for a < 0
We call this an "orbit" We call this a "trajectory"
n --> t,   A --> ea t --> n,   a --> loge A

* Logistic Differential Equation (1-D nonlinear flow)

* Circular Motion (2-D linear flow)

* Mass on a Spring (frictionless)

* Nonautonomous Equations

* Damped Harmonic Oscillator

* Van der Pol Equation (2-D nonlinear ODE)

* Numerical Methods for solving ODEs


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