Maintaining Chaos
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THEORY An ad hoc method for increasing the system complexity (which they termed anticontrol of chaos) was implemented by Schiffet al.7 Subsequent to this Yang et al. 2 proposed a method of anticontrol based on the observation that a map-based system in a regime of transient chaos, such as that near a transition from periodicity into chaos, has special regions in its phase space that they call "loss regions". If the system enters such a region, it immediately ceases its chaotic motion. Yang et al. identified these regions as well as n preiterates of each loss region. If the system enters a preiterate, they apply a small perturbation to an accessible system parameter in order to interrupt the progression of the system toward a loss region. This interruption makes use of the sensitivity of chaotic systems to small perturbations and places the system in a region of phase space that is neither a loss region nor a preiterate of one. Their method requires explicit knowledge of the map of the system and is accordingly difficult to accomplish experimentally. The lack of a systematic implementation of the Schiff method and the difficulty of experimental implementation of the Yang method are the motivation for the present work. We describe a general anticontrol method that is more readily applicable to experiment and that relies only on experimentally measured quantities for its implementation. To start, only the following assumptions are made about the system: (1) the dynamics of the system can be represented as an n-dimensional nonlinear map (e.g., by a surface of section or a return map) such that points or iterates on such a map are given by !" = f( 1,p), where p is some accessible system parameter; (2) there is at least one specific region of the map (termed a "loss region") that lies on the attractor and into which the iterates will fall when making the transition from chaos to periodicity; and (3) the structure of the map does not change significantly with small changes 3p =_p - p 0 in the control parameterp about some initial value p 0 . On the return map derived from such a system, the locations of loss regions are determined by observing immediate preiterates of points which correspond to periodic orbits. Clusters of these preiterates are then identified as the loss regions. The extent of each loss region is determined by the experimentally observed distribution of points in that region (Fig. 1). The time evolution of each region can be traced back through m preiterates, as desired. Next, in a fashion similar to the OGY chaos control method, the parameter p is changed slightly; one then observes the resulting change in each loss region's location and estimates the local shift of the attractor it for each loss region with respect to a change in p as: -=
-A-( , dp
(1)
Ap
As an approximation, j is taken to be constant for all loss regions on the attractor for sufficiently small parameter changes 6p (otherwise calculation of j, for each loss region would be required). This is not strictly necessary in order to impl
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