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 5.3, Problem 2E
Interpretation Introduction

Interpretation:

The romantic styles of Romeo and Juliet are to be characterized for the affair described by R˙ = J, J˙ = -R + J. Fixed points at the origin are to be classified, and its implication is to be written. Also, assuming  R(0) = 1,  J(0) = 0,  R(t) and J(t) are to be sketched.

Concept Introduction:

Love affairs represent the simple model to classify linear systems.

It is governed by a general linear system R˙ = aR + bJ, J˙ = cR + dJ where a, b, c and d are parameters and their signs characterize the romantic styles of Romeo and Juliet.

 R(t)  represents Romeo’s love or hate for Juliet at time t, and  J(t)  represents Juliet’s love or hate for Romeo at time t.

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Question 1. Prove that the function f(x) = 2; f: (2,3] → R, is not uniformly continuous on (2,3].
Consider the cones K = = {(x1, x2, x3) | € R³ : X3 ≥√√√2x² + 3x² M = = {(21,22,23) (x1, x2, x3) Є R³: x3 > + 2 3 Prove that M = K*. Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula (AL)* = (AT)-¹L*, for any cone LC R³ and any 3 × 3 nonsingular matrix A with real entries, where AL = {Ax = R³ : x € L}, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix A.
I am unable to solve part b.
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