A Chaotic System: Discretization and Control
This paper examines chaos in some population models for biological processes. In particular, the paper is concerned with chaos produced by discretization of continuous-time population models, and with the question of whether or not this chaotic behavior can occur when a feedback control management strategy is applied to continuous-time population models. Using two classical population models, it is shown that both exponential discretization and discretization associated with numerical simulation can produce chaos in the resulting discrete-time system. For example, a continuous-time Lotka-Volterra system exhibits periodic trajectories, but a particular discretization procedure is shown to yield discrete-time trajectories that all converge to a very complicated chaotic strange attractor, even if the discretization time steps are small. The results in the paper also show that this chaotic behavior can occur even if a feedback control management strategy is applied to the continuous-time system to stabilize the equilibrium point.
KeywordsEquilibrium Point Chaotic System Feedback Controller Chaotic Motion Strange Attractor
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