COMBINING COMPETITIVE MODES AND SINGULARITY THEORY TO MAP HIGHER-ORDER AND HIDDEN ATTRACTORS

The theory of competitive modes is used on a variety of higher-order and hidden attractor systems as necessary conditions for which multiparameter systems exhibit bounded aperiodic behavior. Parameter sets for which the frequency components are nearly competitive are treated first, thereby demonstrating the utility of the method in determining regimes resulting in bounded aperiodic behaviors, either chaotic or non-chaotic. Sufficient conditions for the absence of bounded aperiodic solutions are also discussed and illustrated. Next, a recasting of the competitive modes analysis into geometric criteria for higher-order systems is used to accurately map out the location and spatial extent, as well as the approximate outline, of the attractor. These techniques are then further extended to hidden attractor systems, which are otherwise very difficult to locate in phase space. Finally, singularity theory is used in preliminary fashion to track changes in attractor cross-sections as system parameters are varied.