Deterministic

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Steady State -
Dependent Upon Initial Conditions

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Stochastic/Random

Random - no pattern

Chaotic - a pattern at some values / under some conditions

Jensen Chaos

A demonstration of a simple model where chaos may be seen

The rate (DELTA X) =

 A*X(t)*(1 - X(t)) - X(t)

Instructions:

1) Initial Run

set X = .1
set DT = 1
set A = 1

2) Subsequent Runs

set A to 1,2,3,4
Plot X

3) Using the sensitivity feature, lower the time step to .5, and find the value at which chaos begins.

4) Keep shortening DT, what happens to the value you must give to A to exhibit unstable behavior?

5) Return and set DT = .5. Can you narrow/identify the critical values for A where the behavior of X changes / bifurcates?

Refer to the table of Lyapunov exponents in Section 4.3 of the Chaos Hypertextbook. Are these close to the values you have identified?

6) How do you tell if the output of this or any model is chaotic or just random? Does it exhibit behavior/ a pattern identified previously?

Plot X against LAG X.
Plot RAND against DELAY RAND.

What do you see?
 

Lorenz Chaos
 
 
 

Richardson Chaos