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 2.4, Problem 4E
Interpretation Introduction

Interpretation:

Fixed points of the system x˙ = x2(6 - x) are to be classified using linear stability analysis, and if it fails, by using graphical argument.

Concept Introduction:

Fixed points of a system are the points where x˙ = 0, and are denoted by x

When f '(x)<0, the fixed point is stable.

When f '(x)>0, the fixed point is unstable.

The stability of the fixed point can’t be determined by linear analysisif f '(x)=0, and graphical argument should be used.

Semi stable points are the points at which the local flow is towards them as well as away from them.

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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.
matrix 4
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