EBK NONLINEAR DYNAMICS AND CHAOS WITH S
EBK NONLINEAR DYNAMICS AND CHAOS WITH S
2nd Edition
ISBN: 9780429680151
Author: STROGATZ
Publisher: VST
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Chapter 7.2, Problem 10E
Interpretation Introduction

Interpretation:

To show thesystem x˙ = y - x3y˙ = - x - y3 has no closed orbits by constructing a Liapunov function V = ax2 + by2 with the suitable a, b.

Concept Introduction:

Construct Liapunov function V = ax2 + by2

Find V˙ by taking derivative.

Using system equation, determine V˙ and V

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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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