Kernel: SageMath (stable)
Chaos
We have learned Two Stable System Behaviors
Converge to Equilibrium
Stable Limit Cycle
Is there any other type?
Chaos System
Romeo Juliet System (Continuous Time Example)
Let
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It do not converge to any equilibrium point, nor stable limit cycle
However, it is still bounded!
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Initial Values Matters a Lot
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Review: Jacobian
If you are at equilibrium point
For a point nearby , e.g.,
Partial Derivative
Find:
Two Rules:
Product Rule:
But you can also do in this way:
Optimization
Suppose the number of sparrows is defined by two parameters:
At which point in the state space will the population be stable?
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Highest point: Local maximum
Eigen values of Hessian Matrix <0
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