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

Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as . For the purposes of this definition, a trajectory which approaches a limit of as should be considered to have a fixed point at .

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast; i.e., the system has a positive Liapunov exponent.

Strogatz notes that he favors additional constraints on the aperiodic long-term behavior, but leaves open 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 then there must be a nonzero chance of the associated trajectory being aperiodic.

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

References

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