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.
Issue Section:
Research Papers
1.
Tigan
, Gh.
Analysis of a Dynamical System Derived From the Lorenz System
,” Scientific Bulletin Politehnica University of Timisoara, Tomul
, 50
(64
), pp. 61
–72
.2.
Tigan
, G.
Bifurcation and the Stability in a System Derived From the Lorenz System
,” Proceedings of the Third International Colloquium on Mathematics in Engineering and Numerical Physics
, pp. 265
–272
.3.
Álvarez
, G.
Li
, S.
Montoya
, F.
Pastor
, G.
Romera
, M.
Breaking Projective Chaos Synchronization Secure Communication Using Filtering and Generalized Synchronization
,” Chaos, Solitons Fractals
0960-0779, 24
, pp. 775
–783
.4.
Pecora
, L. M.
Carroll
, T. L.
Synchronization in Chaotic Systems
,” Phys. Rev. Lett.
0031-9007, 64
(8
), pp. 821
–824
.5.
Pecora
, L. M.
Carroll
, T. L.
Driving Systems With Chaotic Signals
,” Phys. Rev. A
1050-2947, 44
(4
), pp. 2374
–2383
.6.
Sprott
, J. C.
Some Simple Chaotic Flows
,” Phys. Rev. E
1063-651X, 50
(2
), pp. R647
–R650
.7.
Shil’nikov
, L. P.
A Case of the Existence of a Countable Number of Periodic Motions
,” Sov. Math. Dokl.
0197-6788, 6
, pp. 163
–166
.8.
Shil’nikov
, L. P.
A Contribution of the Problem of the Structure of an Extended Neighborhood of Rough Equilibrium State of Saddle-Focus Type
,” Math. USSR. Sb.
0025-5734, 10
, pp. 91
–102
.9.
Silva
, C. P.
Shil’nikov Theorem—A Tutorial
,” IEEE Trans. Circuits Syst., I: Regul. Pap.
1549-8328, 40
, pp. 675
–682
.10.
Zheng
, Z.
Chen
, G.
Existence of Heteroclinic Orbits of the Shil’nikov Type in a 3D Quadratic Autonomous Chaotic System
,” J. Math. Anal. Appl.
0022-247X, 315
, pp. 106
–119
.11.
Zhou
, T. S.
Chen
, G.
Celikovsky
, S.
Shil’nikov Chaos in the Generalized Lorenz Canonical Form of Dynamics System
,” Nonlinear Dyn.
0924-090X, 39
, pp. 319
–334
.12.
Sun
, F. -Y.
Shil’nikov Heteroclinic Orbits in a Chaotic System
,” Int. J. Mod. Phys. B
0217-9792, 21
, pp. 4429
–4436
.13.
Wang
, J.
Zhao
, M.
Zhang
, Y.
Xiong
, X.
Shil’nikov-Type Orbits of Lorenz-Family Systems
,” Physica A
0378-4371, 375
, pp. 438
–446
.14.
Wilczak
, D.
The Existence of Shil’nikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof
,” Found Comput. Math.
1615-3375, 6
, pp. 495
–535
.15.
Lamb
, J. S. W.
Teixeira
, M. -A.
Webster
, K. N.
Heteroclinic Bifurcations Near Hopf-Zero Bifurcation in Reversible Vector Fields in R3
,” J. Differ. Equations
0022-0396, 219
, pp. 78
–115
.16.
Corbera
, M.
Llibre
, J.
Teixeira
, M. -A.
Symmetric Periodic Orbits Near a Heteroclinic Loop in R3 Formed by Two Singular Points, a Semistable Periodic Orbit and Their Invariant Manifolds
,” Physica D
0167-2789, 238
, pp. 699
–705
.17.
Krauskopf
, B.
Rieß
, T.
A Lin’s Method Approach to Finding and Continuing Heteroclinic Connections Involving Periodic Orbits
,” Nonlinearity
0951-7715, 21
, pp. 1655
–1690
.18.
Wagenknecht
, T.
Two-Heteroclinic Orbits Emerging in the Reversible Homoclinic Pitchfork Bifurcation
,” Nonlinearity
0951-7715, 18
, pp. 527
–542
.19.
Jiang
, Y.
Sun
, J.
Shil’nikov Homoclinic Orbits in a New Chaotic System
,” Chaos, Solitons Fractals
0960-0779, 32
, pp. 150
–159
.20.
Wang
, X.
Shil’nikov Chaos and Hopf Bifurcation Analysis of Rucklidge System
,” Chaos, Solitons Fractals
0960-0779, 42
, pp. 2208
–2217
.21.
Zhou
, L.
Chen
, Y.
Chen
, F.
Stability and Chaos of a Damped Satellite Partially Filled With Liquid
,” Acta Astronaut.
0094-5765, 65
, pp. 1628
–1638
.22.
Wang
, J.
Chen
, Z.
Yuan
, Z.
Existence of a New Three-Dimensional Chaotic Attractor
,” Chaos, Solitons Fractals
0960-0779, 42
, pp. 3053
–3057
.23.
Watada
, K.
Tetsuro
, E.
Seishi
, H.
Shil’nikov Orbits in an Autonomous Third-Order Chaotic Phase-Locked Loop
,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl.
1057-7122, 45
, pp. 979
–983
.24.
Zhou
, T.
Tang
, Y.
Chen
, G.
Chen’s Attractor Exists
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 9
, pp. 3167
–3177
.25.
Zhou
, T.
Chen
, G.
Classification of Chaos in 3-D Autonomous Quadratic Systems—I. Basic Framework and Methods
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 16
, pp. 2459
–2479
.26.
Tigan
, G.
Dumitru
, O.
Analysis of a 3D Chaotic System
,” Chaos, Solitons Fractals
0960-0779, 36
, pp. 1315
–1319
.27.
Tigan
, G.
Constantinescu
, D.
Heteroclinic Orbits in the T and the Lü Systems
,” Chaos, Solitons Fractals
0960-0779, 42
, pp. 20
–23
.28.
Jiang
, B.
Han
, X.
Bi
, Q.
Hopf Bifurcation Analysis in the T System
,” Nonlinear Anal.: Real World Appl.
1468-1218, 11
, pp. 522
–527
.29.
Van Gorder
, R. A.
Choudhury
, S. R.
Extended Hopf Bifurcation Analysis of the T System
,” unpublished.30.
Van Gorder
, R. A.
Choudhury
, S. R.
Classification of Chaotic Regimes in the T System by Use of Competitive Modes
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, accepted.31.
Wu
, Y.
Zhou
, X.
Chen
, J.
Hui
, B.
Chaos Synchronization of a New 3D Chaotic System
,” Chaos, Solitons Fractals
0960-0779, 42
, pp. 1812
–1819
.32.
Li
, X. -F.
Chu
, Y. -D.
Zhang
, J. -G.
Chang
, Y. -X.
Nonlinear Dynamics and Circuit Implementation for a New Lorenz-Like Attractor
,” Chaos, Solitons Fractals
0960-0779, 41
, pp. 2360
–2370
.33.
Yong
, C.
Zhen-Ya
, Y.
Chaos Control in a New Three-Dimensional Chaotic T System
,” Commun. Theor. Phys.
0253-6102, 49
, pp. 951
–954
.34.
Nayfeh
, A. H.
Balachandran
, B.
Applied Nonlinear Dynamics
, Wiley
, New York
.35.
Choudhury
, S. R.
On Bifurcations and Chaos in Predator-Prey Models With Delay
,” Chaos, Solitons Fractals
0960-0779, 2
, pp. 393
–409
.36.
Doedel
, E. J.
AUTO-Software for Continuation and Bifurcation Problems in ODEs
,” CALTECH Report.37.
Doedel
, E. J.
Keller
, H. B.
Kernevez
, J. P.
Numerical Analysis and Control of Bifurcation Problems I and II
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 1
, pp. 493
–520
and 745–772.Copyright © 2011
by American Society of Mechanical Engineers
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