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Research Papers

# Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

[+] Author and Article Information
Robert A. Van Gorder

Department of Mathematics, University of Central Florida, P.O. Box 161364, Orlando, FL 32816-1364rav@knights.ucf.edu

S. Roy Choudhury

Department of Mathematics, University of Central Florida, P.O. Box 161364, Orlando, FL 32816-1364choudhur@cs.ucf.edu

J. Comput. Nonlinear Dynam 6(2), 021013 (Nov 15, 2010) (6 pages) doi:10.1115/1.4002685 History: Received December 15, 2009; Revised April 26, 2010; Published November 15, 2010; Online November 15, 2010

## Abstract

We study the chaotic behavior of the T system, a three dimensional autonomous nonlinear system introduced by Tigan (2005, “Analysis of a Dynamical System Derived From the Lorenz System,” Scientific Bulletin Politehnica University of Timisoara, Tomul, 50, pp. 61–72), which has potential application in secure communications. Here, we first recount the heteroclinic orbits of Tigan and Dumitru (2008, “Analysis of a 3D Chaotic System,” Chaos, Solitons Fractals, 36, pp. 1315–1319), and then we analytically construct homoclinic orbits describing the observed Smale horseshoe chaos. In the parameter regimes identified by this rigorous Shil’nikov analysis, the occurrence of interesting behaviors thus predicted in the T system is verified by the use of numerical diagnostics.

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## Figures

Figure 3

Autocorrelation function for the time series for the T system when a=4, b=2, c=30, x(0)=4, y(0)=4, and z(0)=8

Figure 4

Plot of converged fractal dimension for the T system (given by the best fit horizontal line) when a=4, b=2, c=30, x(0)=4, y(0)=4, and z(0)=8. The best-fit horizontal straight line reveals a converged fractal dimension of about 2.3.

Figure 1

Phase portrait for the T system when a=4, b=2, c=30, x(0)=4, y(0)=4, and z(0)=8. Here, we observe the behavior of the T system about a heteroclinic orbit, and the Smale horseshoe chaos appears.

Figure 2

Time series plots for the T system when a=4, b=2, c=30, x(0)=4, y(0)=4, and z(0)=8, where we observe the behavior of the T system about a heteroclinic orbit, and the Smale horseshoe chaos appears.

Figure 5

Phase portrait for the T system when a=3, b=−2, c=2, x(0)=1, y(0)=1, and z(0)=10. Here, we observe the behavior of the T system about a homoclinic orbit.

Figure 6

Phase portrait for the T system when a=−4, b=−2, and c=−16. A homoclinic orbit, not of the type satisfying Theorem 3.1, is evident.

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