Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 4.1, Problem 2E
Interpretation Introduction

Interpretation:

To find and classify all the fixed points of the vector field θ˙ = 1 + 2 cos θ and to sketch the phase portrait on the circle.

Concept Introduction:

The vector field on the circle can be represented by θ˙ = f(θ).

Fixed points are the points where θ˙ = 0.

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1. 2. Show that the following are not logically equivalent by finding a counterexample: (p^q) →r and (db) V (d←d) Show that the following is not a contradiction by finding a counterexample: (pV-q) AqA (pv¬q Vr) 3. Here is a purported proof that (pq) ^ (q → p) = F: (db) v (bd) = (db) v (bd) =(qVp) A (g→p) = (¬¬q V ¬p) ^ (q→ p) (db) V (db) = =¬(a→p)^(a→p) = (gp) ^¬(a → p) =F (a) Show that (pq) ^ (q→p) and F are not logically equivalent by finding a counterex- ample. (b) Identify the error(s) in this proof and justify why they are errors. Justify the other steps with their corresponding laws of propositional logic.
5 Show by multiplying matrices that the following equation represents an ellipse: 5 - -7 I (x)(3)()=30. y) 7 7)
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