Using chaos to direct trajectories to targets

Troy Shinbrot, Edward Ott, Celso Grebogi, James A. Yorke
1990 Physical Review Letters  
A method is developed which uses the exponential sensitivity of a chaotic system to tiny perturbations to direct the system to a desired accessible state in a short time. This is done by applying a small, judiciously chosen, perturbation to an available system parameter. An expression for the time required to reach an accessible state by applying such a perturbation is derived and confirmed by numerical experiment. The method introduced is shown to be eA'ective even in the presence of
more » ... itude noise or small modeling errors. PACS numbers: 05.45.+b Chaotic systems exhibit extreme sensitivity to initial conditions. This characteristic is often regarded as an annoyance, yet it provides us with an extremely useful capability without a counterpart in nonchaotic systems. In particular, the future state of a chaotic system can be substantially altered by a tiny perturbation. If we can accurately sense the state of the system and intelligently perturb it, this presents us with the possibility of rapidly directing the system to a desired state. As we will show below for a particular example, an initial condition for a chaotic system which, if left on its own, would require more than 6000 time steps to reach a small target region can be directed with small perturbations to the desired region in only 12 time steps. In this paper, we report a method for achieving such targeting, and we show that the method can be effective even in the presence of small-amplitude noise and small modeling errors. For simplicity, we consider a three-dimensional
doi:10.1103/physrevlett.65.3215 pmid:10042812 fatcat:ztlwbjryvfcivbpg5rwsg4l6z4