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MAT332H5 F - 2021 - PROBLEM SET 3


Problem 1. (20 pts) Consider the system of differential equations given in polar coordinates by

Define

(1) Does the system have sensitive dependence on initial conditions in or ? (3 pts)

(2) Does the system have sensitive dependence on initial conditions when restricted to or ? (3 pts)

(3) Is or B an attractor? (4 pts)

(4) Is or B a strange attractor? (5 pts)

(5) Let P(x, y) : Σ → Σ be the Map from the semi-hyperplane

Denote ((x), 0) = P(x, 0). Discuss if (x) is increasing or decreasing as x varies. (5 pts)





Problem 2. (20 pts) Consider the system

(1) Find a non-zero fixed point (, ) of the system. (5 pts)

(2) By studying the linearised system, show that the fixed point is a sink. (You might want to first show that the fixed point is hyperbolic) (5 pts)

(3) We now show the system has no periodic orbit intersecting with Σ defined below. Otherwise, consider the  Map P(x, y) : Σ → Σ from the section

(a) Denote ((x), ) = P(x, ). Suppose that there is a fixed point of the map, i.e.,

with x0 > . Calculate the derivative (5 pts)

(b) Conclude that the system has no periodic orbit. (5 pts)





Problem 3. (10 pts) Consider the following 3D system

(1) Suppose the system has a T-periodic orbit ((t), (t), (t)), then show that one must have that:

(3 pts)

(2) Find a periodic orbit of the system. (3 pts)

(3) What are the characteristic multipliers of this periodic orbit found above? (4 pts)