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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Establishing the Onset, and Parameter and Spatial Domains, of Chaos in a Smooth Quadratic Autonomous System Using Sil’nikov Criteria and Competitive Modes Analysis

Discontinuity, Nonlinearity, and Complexity 2(1) (2012) 1--19 | DOI:10.5890/DNC.2012.10.003

Thomas K. Vogel$^{1}$; S. Roy Choudhury$^{2}$

$^{1}$ Department of Mathematics and Computer Science, Stetson University, 421 N.Woodland Blvd., Unit 8332–DeLand, FL 32763,

$^{2}$ Department of Mathematics, University of Central Florida, PO Box 161364, Orlando, FL 32816-1364

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In this paper we examine the chaotic regimes of a recently discovered hyperchaotic system in greater depth. Towards that end, we first analyze the numerical observations of horseshoe-type chaotic behavior in this system in detail by the use of Shilnikov analysis. Subsequently,we also employ the technique of Competitive Modes analysis to identify “possible chaotic parameter regimes” for this multi-parameter system. We find that the Competitive Modes conjectures may in fact be interpreted and employed slightly more generally than has usually been done in recent investigations, with negative values of the squared mode frequencies in fact being admissible in chaotic regimes, provided that the competition among them persists. This is somewhat reminiscent of, but of course not directly correlated to, “stretching (along unstable manifolds) and folding (due to local volume dissipation)” on chaotic attractors. This new feature allows for the system variables (which also define the position ON the attractor) to grow exponentially during time intervals when mode frequencies are imaginary and comparable, while oscillating at instants when the frequencies are real and locked in or entrained. Finally, in a novel twist, we re-interpret the components of the Competitive Modes analysis as simple geometric criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of the Competitive Modes conjectures.


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