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Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 27--45 | DOI:10.5890/DNC.2020.03.003

Brayden McDonald$^{1}$, S. Roy Choudhury$^{2}$

$^{1}$ Department of Computer Science, University of Central Florida

$^{2}$ Department of Mathematics, University of Central Florida

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Abstract

Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system, leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed point of the original system.

Acknowledgments

The author wishes to express his very sincere gratitude to the Editor Professor Volchenkov and/or an anonymous referee for several comments which improved the accuracy of the paper, including specifying the time intervals on all the numerical plots, as well as the comments regarding hidden attractors in Section 4 which have, in fact, also triggered several other investigations unrelated to the current work.

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