Computational Methods for Nonlinear Systems

Physics 7682 - Fall 2014

Instructor: Chris Myers

Mondays & Fridays 1:30-3:30, Rockefeller B3 (directions)

Invarient Measure


We have five exercises on discrete maps. Before working on this one, you should first do the Logistic Map exercises.

Chaotic dynamical systems will stretch and fold the state space. In Hamiltonian systems, this stretching and folding preserves volume in phase space, and the natural weighting of all points in phase space is equal (Liouville's theorem). In more general chaotic dynamical systems with dissipation and forcing, the long-time behavior lies on an attractor: points off of the attractor do not contribute to the time-average behavior, and points on the attractor are weighted by an invariant measure or probability density in state space.

In this problem we explore the invariant measure for the logistic map, which goes chaotic through a period doubling cascade. The chaotic region of this map exhibits a complex, textured structure with a series of sharp internal boundaries. In this exercise, we attempt to understand this structure.

Learning Goals

Science: You will learn about Chaos, discrete maps, and invarient measures.

Computation: You will learn about iterative algorithms, and the evolution of probability measures under such maps.