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 2.2, Problem 5E
Interpretation Introduction

Interpretation:

The equation x˙ = 1+12cosx is to be analyzed graphically; the vector field is to be sketched on the real line; all fixed points are to be determined; their stability should be classified, and the graph of  x(t) is to be sketched for different initial conditions. Also, if possible, the analytic solution for  x(t) is to be obtained.

Concept Introduction:

For the given equation, agraph is to be plotted with the vector field on the real line.

All the fixed points are to be found. Fixed points are the points where x˙=0.

When the vector field plotted on the real lines flows towards the fixed points, they are called stable points or attractors or sink.

When the vector field plotted on the real lines flows away from the fixed points, they are called unstable points.

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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.
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Please ensure that all parts of the question are answered thoroughly and clearly. Include a diagram to help explain answers. Make sure the explanation is easy to follow. Would appreciate work done written on paper. Thank you.
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