Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 7.5, Problem 4E
Interpretation Introduction

Interpretation:

The system x¨+μf(x)x˙+x = 0 is equivalent to x˙ = μ(y-F(x)),  y˙ = -xμ where F(x) is piecewise linear function is to be shown, and the nullclines are to be graphed. The system exhibits relaxation oscillations for μ>>1 is to be shown as well. Also, the limit cycle in the (x,y) plane is to be plotted, and the period of this limit cycle is to be estimated.

Concept Introduction:

The van der pol non - linear dynamic equation is x¨ + μx˙(x2 - 1) + x = 0

If oscillations contain repetition of extremely slow build up, followed by sudden discharge, then such oscillation are called relaxation oscillations.

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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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