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Topological horseshoe theory is fundamental for studying chaos rigorously, which, however, has rarely applied to hyperchaos. The reason is that it is too hard to find a topological horseshoe in a hyperchaotic system, due to the high dimension of the system and the multiple expansion directions in the state space. Therefore, in this paper a practical algorithm for three-dimensional (3D) hyperchaotic maps is proposed. Usually, a hyperchaotic system has a large negative Lyapunov exponent, its attractor is often contracted closely to a certain surface. Based on this feature, the algorithm first deducts the dimension along the direction of contraction to obtain a 2D projective system; then it detects a projective horseshoe with 2D expansion; finally, it constructs a 3D horseshoe for the original system. In order to verify the validity of the algorithm, it is applied to the classic hyperchaotic Lorenz system and the famous Saito hyperchaotic circuit, and their horseshoes with 2D expansion are successfully found from the Poincaré mapping.
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Keywords:
- hyperchaos /
- topological horseshoes /
- Saito hyperchaotic system /
- Lorenz hyperchaotic system
[1] Rossler O 1979 Physics Letters A 71 155
[2] Zheng J 2011 Computers & Mathematics with Applications 61 2000
[3] Yu H, Cai G, Li Y 2012 Nonlinear Dynamics 67 2171
[4] Sheikhan M, Shahnazi R, Garoucy S 2011 Neural Computing & Applications 20 1
[5] Vaidyanathan S, Sampath S 2012 Advances in Computer Science and Information Technology. Computer Science and Engineering 85 257
[6] Uchida A, Amano K, Inoue M 2008 Nature Photonics 2 728
[7] Sun L, Jiang D P 2006 Acta Phys. Sin. 55 3288 (in Chinese) [孙琳, 姜德平 2006 物理学报 55 3283]
[8] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 60503 (in Chinese) [王晶, 蒋国平 2011 物理学报 60 60503]
[9] Kennedy J, Kocak S, Yorke J A 2001 The American Mathematical Monthly 108 411
[10] Kennedy J, Yorke J A 2001 Transactions of the American Mathematical Society 353 2513
[11] Yang X S 2004 Chaos, Solitons & Fractals 20 1149
[12] Szymczak A 1996 Topology 35 287
[13] Plumecoq J, Lefranc M 2000 Physica D: Nonlinear Phenomena 144 231
[14] Zgliczyński P, Gidea M 2004 Journal of Differential Equations 202 32
[15] Li Q, Yang X S 2006 Journal of Physics a-Mathematical and General 39 9139
[16] Yang F, Li Q, Zhou P 2007 International Journal of Bifurcation and Chaos 17 4205
[17] Li Q, Yang X S 2007 Discrete Dynamics in Nature and Society 2007 16239
[18] Li Q 2008 Physics Letters A 372 2989
[19] Li Q, Yang X S 2008 International Journal of Circuit Theory and Applications 36 19
[20] Li Q, Yang X S, Chen S 2011 International Journal of Bifurcation and Chaos 21 1719
[21] Yang X S 2009 International Journal of Bifurcation and Chaos 19 1127
[22] Yang X S, Li H, Huang Y 2005 Journal of Physics A: Mathematical and General 38 4175
[23] Li Q, Yang X S 2010 International Journal of Bifurcation and Chaos 20 467
[24] Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 5136 (in Chinese) [王兴元, 王明军 2007 物理学报 56 5136]
[25] Saito T 1990 Circuits and Systems, IEEE Transactions on 37 399
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[1] Rossler O 1979 Physics Letters A 71 155
[2] Zheng J 2011 Computers & Mathematics with Applications 61 2000
[3] Yu H, Cai G, Li Y 2012 Nonlinear Dynamics 67 2171
[4] Sheikhan M, Shahnazi R, Garoucy S 2011 Neural Computing & Applications 20 1
[5] Vaidyanathan S, Sampath S 2012 Advances in Computer Science and Information Technology. Computer Science and Engineering 85 257
[6] Uchida A, Amano K, Inoue M 2008 Nature Photonics 2 728
[7] Sun L, Jiang D P 2006 Acta Phys. Sin. 55 3288 (in Chinese) [孙琳, 姜德平 2006 物理学报 55 3283]
[8] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 60503 (in Chinese) [王晶, 蒋国平 2011 物理学报 60 60503]
[9] Kennedy J, Kocak S, Yorke J A 2001 The American Mathematical Monthly 108 411
[10] Kennedy J, Yorke J A 2001 Transactions of the American Mathematical Society 353 2513
[11] Yang X S 2004 Chaos, Solitons & Fractals 20 1149
[12] Szymczak A 1996 Topology 35 287
[13] Plumecoq J, Lefranc M 2000 Physica D: Nonlinear Phenomena 144 231
[14] Zgliczyński P, Gidea M 2004 Journal of Differential Equations 202 32
[15] Li Q, Yang X S 2006 Journal of Physics a-Mathematical and General 39 9139
[16] Yang F, Li Q, Zhou P 2007 International Journal of Bifurcation and Chaos 17 4205
[17] Li Q, Yang X S 2007 Discrete Dynamics in Nature and Society 2007 16239
[18] Li Q 2008 Physics Letters A 372 2989
[19] Li Q, Yang X S 2008 International Journal of Circuit Theory and Applications 36 19
[20] Li Q, Yang X S, Chen S 2011 International Journal of Bifurcation and Chaos 21 1719
[21] Yang X S 2009 International Journal of Bifurcation and Chaos 19 1127
[22] Yang X S, Li H, Huang Y 2005 Journal of Physics A: Mathematical and General 38 4175
[23] Li Q, Yang X S 2010 International Journal of Bifurcation and Chaos 20 467
[24] Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 5136 (in Chinese) [王兴元, 王明军 2007 物理学报 56 5136]
[25] Saito T 1990 Circuits and Systems, IEEE Transactions on 37 399
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