Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 4.4, Problem 3E
Interpretation Introduction

Interpretation:

To sketch θ˙ vs t for θ' = γ - sinθ at different values of γ including limiting cases γ1, γ  >> 1.

Concept Introduction:

Overdamped pendulum is an example of a non-uniform oscillator.

The motion of an overdamped pendulum is governed by θ' = γ - sinθ, where θ is the angle between the pendulum and the downward vertical and γ is the ratio of applied torque to the maximum gravitational torque.

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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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