Complexity Explorer Santa Few Institute

Nonlinear Dynamics: Mathematical and Computational Approaches (2024)

Lead instructor: Liz Bradley

This course is no longer in session.

7.1 Dynamics and state-space deformation » Quiz

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Question 1

Smale's horseshoe is a map of the unit square into itself (experts: assume that any region of the plane can be continually deformed to the unit square).

Question 2

Smale's horseshoe maps some close points far apart (& vice-versa).

Question 3

The following picture shows a ball of initial conditions (blue) around the saddle point Θ = π in the undamped pendulum.  The stable and unstable manifolds are drawn in.  (Experts: assume this ball starts in a very tight cluster around Θ = π.)


Which of the following sketches below shows the general shape of that ball of initial conditions, evolved forward in time?

Question 4

Horseshoes:

(a)

Only turn up in the dynamics of the Smale horseshoe map

(b)

are important because they play a role in proofs of chaos

(c)

turn up in the dynamics of the pendulum

Question 5

Dissipation is a necessary condition for the existence of attractors.