Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 10.5, Problem 1E
Interpretation Introduction

Interpretation:

To calculate the Liapunav exponent for the linear map xn+1= r xn

Concept Introduction:

  • ➢ Logistic map can exhibit aperiodic orbits for certain parameter values.

  • ➢ Given some initial condition x0, consider nearby point x00, where the initial separation  δ0 is extremely small. Let  δn be the separation after n iterates. If n|  |δ0| en λ , then λ is called the Liapunov exponent.

  • ➢ Apositive Liapunov exponent is a signature of chaos.

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(b) Let I[y] be a functional of y(x) defined by [[y] = √(x²y' + 2xyy' + 2xy + y²) dr, subject to boundary conditions y(0) = 0, y(1) = 1. State the Euler-Lagrange equation for finding extreme values of I [y] for this prob- lem. Explain why the function y(x) = x is an extremal, and for this function, show that I = 2. Without doing further calculations, give the values of I for the functions y(x) = x² and y(x) = x³.
Please use mathematical induction to prove this
L sin 2x (1+ cos 3x) dx 59
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