Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 10.5, Problem 7E
Interpretation Introduction

Interpretation:

To show that analytically the graph in Figure 10.5.2 suggest that λ = 0 at each period-doubling bifurcation value is correct.

Concept Introduction:

  • ➢ A logistic map can exhibit aperiodic orbits for certain parameter values.

  • ➢ Given some initial condition x0, consider the nearby point x00, where the initial separation  δ0 is extremely small. Let  δn be the separation after n iterates. If n|  |δ0| en λ , then λ is called the Liapunov exponent.

  • ➢ Apositive Liapunov exponent is a signature of chaos.

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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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