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 7.2, Problem 17E
Interpretation Introduction

Interpretation:

Assuming the hypotheses of Dulac’s criteria and R is topologically equivalent to annulus (exactly one hole in it).By using Green’s theorem, showthat there exists at mostone closed orbit in R.

Concept Introduction:

Dulac’s Criterion: For x˙=f(x) is a continuously differentiable vector field on a simply connected subset R of the plane. If there exists a continuously differentiable real valued function g(x) such that .(gx˙) has one sign throughout R, then there are no closed orbits lying entirely in R.

The hypotheses of the Dulac’s criteria are

If the limit cycle doesn’t enclose a hole in the region, then the cycle is not possible.

If the limit cycle encloses a hole in the region, then there is at least one limit cycle.

Green’s theorem: If C is positively oriented, simple curve, piecewise smooth, and D be the region enclosed by the curve. If P and Q have continuous first order partial derivatives on D then

CPdx+Qdy=D(QxPy)dA

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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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