Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 9.4, Problem 2E
Interpretation Introduction

Interpretation:

  • a) To state why it is called as the “tent map”.

  • b) To find all fixed points and classify their stability.

  • c) To show map has period -2 orbit and classify the stability of the orbit.

  • d) To find period -3 and period -4 points and classify their stability.

Concept Introduction:

  • ➢ Lorenz equations

    x˙=σ(yx)y˙=rxyxzz˙=xybzHere σ, r, b > 0

    The solution of Lorenz equations oscillates irregularly for a wide range of parameters, never exactly repeating but always remains in a bounded region of phase space.

  • ➢ Strange attractor: It is not same as a fixed point, limit cycle, a point, a curve or surface. It is a fractal with fractional dimension between 2 and 3.

  • ➢ Chaos: It is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

  • ➢ The plot of function zn+1=f(zn) is called a Lorenz map. It gives information about dynamics on the attractor.

  • ➢ If |f(z)|>1 then the limit cycle exists and it is unstable.

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