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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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.
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