Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.4, Problem 3E
Interpretation Introduction
Interpretation:
The
Concept Introduction:
Supercritical pitchfork bifurcation occurs when the single fixed points are present it is stable, and after parameters change, this fixed point becomes unstable, and two new symmetric fixed points which appear are stable.
Subcritical pitchfork bifurcation occurs when there is a single fixed point present which is unstable, and after parameters change, this fixed point becomes stable and two new symmetric fixed points which appear become unstable.
Pitchfork bifurcation is the bifurcation in which, when parameters are varied then the fixed points tend to appear and disappear in symmetrical pairs.
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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
Chapter 3 Solutions
Nonlinear Dynamics and Chaos
Ch. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Prob. 3ECh. 3.1 - Prob. 4ECh. 3.1 - Prob. 5ECh. 3.2 - Prob. 1ECh. 3.2 - Prob. 2ECh. 3.2 - Prob. 3ECh. 3.2 - Prob. 4ECh. 3.2 - Prob. 5E
Ch. 3.3 - Prob. 1ECh. 3.3 - Prob. 2ECh. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - Prob. 7ECh. 3.4 - Prob. 8ECh. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Prob. 14ECh. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Prob. 3ECh. 3.5 - Prob. 4ECh. 3.5 - Prob. 5ECh. 3.5 - Prob. 6ECh. 3.5 - Prob. 7ECh. 3.5 - Prob. 8ECh. 3.6 - Prob. 1ECh. 3.6 - Prob. 2ECh. 3.6 - Prob. 3ECh. 3.6 - Prob. 4ECh. 3.6 - Prob. 5ECh. 3.6 - Prob. 6ECh. 3.6 - Prob. 7ECh. 3.7 - Prob. 1ECh. 3.7 - Prob. 2ECh. 3.7 - Prob. 3ECh. 3.7 - Prob. 4ECh. 3.7 - Prob. 5ECh. 3.7 - Prob. 6ECh. 3.7 - Prob. 7ECh. 3.7 - Prob. 8E
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