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 $t\to \mathrm{\infty }$. For the purposes of this definition, a trajectory which approaches a limit of $\mathrm{\infty }$ as $t\to \mathrm{\infty }$ should be considered to have a fixed point at $\mathrm{\infty }$.

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

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.

0.1 Further reading

1. B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science: Similarities and Dissimilarities. http://pespmc1.vub.ac.be/Einmag_Abstr/BCodenotti.html

0.2 References

1. Steven H. Strogatz, ”Nonlinear Dynamics and Chaos”. Westview Press, 1994.