Phase Space & Chaos

Phase Space

Phase Space

A phase space is a space in which all possible “states” of a dynamical system are represented, with each possible state corresponding to one unique point in the phase space. Specifically, the phase space usually consists of all possible values of position and momentums, which is a 6-dimensional space.

Proposition

Any two distinct trajectories in the phase space will not cross with each other.

Proof Assume two trajectories cross with each other at point , then they will be same as at time they have exactly same and , thus all states can be determined.

Def Attractor In a phase space, an attractor is a point towards which neighbouring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. That is system values that get close enough to the attractor values remain close even if slightly disturbed. In general, an attractor is a region in the space of possible states which the system can enter but not leave.

Phase Flow & Liouville’s Theorem

Phase Flow

The phase flow is the one-parameter family of transformations of the phase space: where and are the solutions to the Hamilton’s equations.

Proposition

The set of phase flows is a group under composition.

Liouville's Theorem

The volume of a region in the phase space remains same under the phase flow.

Proof

e.g. In a Hamiltonian system, it is impossible to have asymptotically stable equilibrium points in the phase space, because the phase flow preserves the volume.

Poincare Recurrence Theorem

Let be a volume preserving, continuous one-to-one mapping which maps a bounded region of Euclidean space onto itself: . Then in any neighborhood of any point of , there exists a point such that for some .

Identify Chaos

Def Poincare Section Suppose is some hyperplane in the phase space, and is some trajectory. Points of intersection of and are denotes as . The Poincare section is a continuous mapping of into itself:Since the dynamic system is deterministic, determines , determines , etc.

Def Lyapunov Exponent Lyapunov exponent is a quantity that characterises the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation diverge at a rate given by where is the Lyapunov exponent. Alternatively, we can say

Algorithm Determine Lyapunov Exponent For with considerably larger , in each iteration, calculate Finally output the average of all .