Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 10.4, Problem 11E
Interpretation Introduction

Interpretation:

By considering xn+1= axn analyze long-term behavior of the sequence { xn} as n and check how long-term behavior depends on a.

Concept Introduction:

  • ➢ The period-3 window occurs near 3.8284...r3.8415...

  • ➢ The notation for the logistic map is

    Let, f(x) = rx(1x)

    Then the logistic map is

    xn+1 = f( xn), xn+2 = f(f( xn))f2(xn),xn+3 = f2(f( xn))f3(xn)

    The third iteration map f3(xn) is important to understand the birth of the period-3 cycle.

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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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