Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 9.6, Problem 3E
Interpretation Introduction

Interpretation:

  • a) To plot y(t) and yr(t) on the same graph and state whether two time series are merging or not even if both are chaotic.

  • b) To plot (y, z) projection of both trajectories.

Concept Introduction:

  • ➢ Lorenz equations:-

    x˙=σ(yx)y˙=rxyxzz˙=xybzHere σ, r, b > 0

    The solution of the Lorenz equations oscillates irregularly for a wide range of parameters, never exactly repeating, but always remaining in the bounded region of the phase space.

  • ➢ The main idea behind sending the secret message using chaos is while transmitting the message, it is masked by a louder chaos. The eavesdropping person only hears chaos, but the receiver perfectly reproduces chaos and subtracts it from the masked message and hears the original message sent by the transmitter.

  • ➢ The response equations are:-

    x(t) = x(t)y˙r= rx(t)yrx(t)zrz˙r = x(t)yrbzr

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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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