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 longterm in a deterministic system that exhibits sensitive dependence on initial conditions^{}.
Aperiodic longterm 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 longterm , 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.
Title  chaotic dynamical system 

Canonical name  ChaoticDynamicalSystem 
Date of creation  20130322 13:05:26 
Last modified on  20130322 13:05:26 
Owner  bshanks (153) 
Last modified by  bshanks (153) 
Numerical id  15 
Author  bshanks (153) 
Entry type  Definition 
Classification  msc 37G99 
Synonym  chaotic system 
Synonym  deterministic chaotic system 
Synonym  chaotic behavior 
Related topic  DynamicalSystem 
Related topic  SystemDefinitions 