# chaotic dynamical system

As Strogatz says in reference [1], “No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition”.

Chaos is the aperiodic long-term in a deterministic system that exhibits sensitive dependence on initial conditions.

Aperiodic long-term means that there are trajectories which do not settle down to fixed points, periodic orbits (http://planetmath.org/Orbit), or quasiperiodic as $t\to\infty$. For the purposes of this definition, a trajectory which approaches a limit of $\infty$ as $t\to\infty$ should be considered to have a fixed point at $\infty$.

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast; i.e. (http://planetmath.org/Ie), the system has a positive Liapunov exponent.

Strogatz notes that he favors additional constraints on the aperiodic long-term , but leaves open (http://planetmath.org/OpenQuestion) what form they may take. He suggests two alternatives to fulfill this:

1. 1.

Requiring that there exists an open set of initial conditions having aperiodic trajectories, or

2. 2.

If one picks a random initial condition $x(0)$ then there must be a nonzero chance of the associated trajectory $x(t)$ being aperiodic.