Two Identical Chaotic Systems

A chaotic system exhibits a sensitivity to initial conditions and will evolve rapidly and deterministically toward different end states if begun in slightly different states. Although the chaos is unpredictable, each possible outcome is deterministic that is, an orderly behavior.

Consider two identical chaotic systems isolated from each other. They will quickly fall out of step because any slight difference between them would be magnified. Assume that these systems have several parts and that at least one of the parts is stable that is, subjected to a perturbation, the part’s behavior changes a little but settles back to its normal operation. Now drive both systems with the same chaotic signal applied to the same stable part. Can the two systems be synchronized?

Answer

Yes, the two identical chaotic systems described can be synchronized. Chaotic systems are very useful for several reasons: (1) Chaotic systems are a collection of many regular, ordinary behaviors, none of which dominate. (2) The proper perturbation can encourage the chaotic system to follow one of its many regular behaviors. (3) Chaotic systems are very flexible because they can rapidly switch among different behaviors. (4) Chaotic systems are deterministic and, although no one can say which output will result, two identical chaotic systems of the appropriate type will produce the same output in response to the same signal input.

To synchronize two identical chaotic systems each with the stable subpart behavior, one can apply the appropriate pseudoperiodic signal (one type is called a Rossler signal) to coax them into step. For the reasons listed above, the outputs will be the same. The details can be learned in the reference below, where the chaotic attractor and the Poincare section are discussed. Applications to secure communications and to biological systems are included also.