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MAT332H5 F - 2021 - PROBLEM SET 3
发布时间:2021-11-24
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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)