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 7.6, Problem 7E
Interpretation Introduction

Interpretation:

To calculate the averaged equations and to analyze the long-term behavior of the system x¨ + x + ε h(x, x˙) = 0, where h(x, x˙) = (x4- 1)x˙. To find the amplitude and frequency of any limit cycles for the original system. If possible, to solve the averaged equations explicitly for x(t, ε), given the initial conditions x(0) = a, x˙(0)=0.

Concept Introduction:

The averaged or slow-time equations are given by r = h sin θ, rϕ = h cos θ.

The common solution of the system is given by x0 = r(T) cos(ϕ(T)+τ).

The explicit solution is specified as x(t, ε) = x0+ O(ε).

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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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