-
Previous Article
Periodic solutions of the planar N-center problem with topological constraints
- DCDS Home
- This Issue
-
Next Article
Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement
A new class of 3-dimensional piecewise affine systems with homoclinic orbits
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China, China |
References:
[1] |
G. F. V. Amaral, C. Letellier and L. A. Aguirre, Piecewise affine models of chaotic attractors: The Rossler and Lorenz systems, Chaos, 16 (2006), 013115, 14pp.
doi: 10.1063/1.2149527. |
[2] |
M. L. Barakat, A. S. Mansingka, A. G. Radwan and K. N. Salama, Hardware stream cipher with controllable chaos generator for colour image encryption, IET Image Process., 8 (2014), 33-43.
doi: 10.1049/iet-ipr.2012.0586. |
[3] |
S. Chakraborty and S. K. Dana, Shil'nikov chaos and mixed-mode oscillation in Chua's circuit, Chaos, 20 (2010), 23107, 7pp. |
[4] |
T. Chien and T. Liao, Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization, Chaos Solitons Fract., 24 (2005), 241-255.
doi: 10.1016/S0960-0779(04)00542-9. |
[5] |
V. Carmona, F. Fernández-Sánchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048.
doi: 10.1137/070709542. |
[6] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), 013124, 8pp.
doi: 10.1063/1.3339819. |
[7] |
M. di Bernardo and C. K. Tse, Chaos in Power Electronics: An Overview, Chaos in Circuits and Systems, World Scientific, 2002. |
[8] |
S. M. Huan, Q. D. Li and X.-S. Yang, Chaos in three-dimensional hybrid systems and design of chaos generators, Nonlinear Dyn., 69 (2012), 1915-1927.
doi: 10.1007/s11071-012-0396-0. |
[9] |
S. M. Huan and X.-S. Yang, Existence of chaotic invariant set in a class of 4-dimensional piecewise linear dynamical systems, Int. J. Bifurc. Chaos, 24 (2014), 1450158, 16pp.
doi: 10.1142/S0218127414501582. |
[10] |
T. Kousaka, T. Ueta and H. Kawakami, Chaos in a simple hybrid system and its control, Electron Lett., 37 (2001), p1.
doi: 10.1049/el:20010033. |
[11] |
J. Lü, T. Zhou, G. Chen and X.-S. Yang, Generating chaos with a switching piecewise-linear controller, Chaos, 12 (2002), 344-349. |
[12] |
R. O. Medrano-T., M. S. Baptista and I. L. Caldas, Homoclinic orbits in a piecewise system and their relation with invariant sets, Physica D, 186 (2003), 133-147.
doi: 10.1016/j.physd.2003.08.002. |
[13] |
V. Nair and R. I. Sujith, Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification, Chaos, 23 (2013), 033136, 6pp.
doi: 10.1063/1.4821475. |
[14] |
I. Pehlivan and Y. Uyaroglu, Simplified chaotic diffusionless Lorenz attractor and its application to secure communication systems, IET Commun., 1 (2007), 1015-1022. |
[15] |
L. P. Shil'nikov, A case of the existence of a countable number of periodic motions, Sov. Math.Dokl., 6 (1965), 163-166. |
[16] |
L. P. Shil'nikov, A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. |
[17] |
L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, 1998.
doi: 10.1142/9789812798596. |
[18] |
L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001.
doi: 10.1142/9789812798558_0001. |
[19] |
C. Tresser, About some theorems by L. P. Shil'nikov, Inst. H. Poincare Phys. Thoré., 40 (1984), 441-461. |
[20] |
K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. Circuits Syst., 45 (1998), 979-983. |
[21] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, $2^{nd}$ edition, Springer-Verlag, New York, 2003. |
[22] |
D. Wilczak, The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof, Found Comput. Math., 6 (2006), 495-535.
doi: 10.1007/s10208-005-0201-2. |
[23] |
X.-S. Yang and Q. D. Li, Chaos generator via Wien-bridge oscillator, Electron. Lett., 38 (2002), 623-625.
doi: 10.1049/el:20020456. |
[24] |
X.-S. Yang and Q. D. Li, Generate n-scroll attractor in linear system by scalar output feedback, Chaos Solitons Fract., 18 (2003), 25-29.
doi: 10.1016/S0960-0779(02)00638-0. |
show all references
References:
[1] |
G. F. V. Amaral, C. Letellier and L. A. Aguirre, Piecewise affine models of chaotic attractors: The Rossler and Lorenz systems, Chaos, 16 (2006), 013115, 14pp.
doi: 10.1063/1.2149527. |
[2] |
M. L. Barakat, A. S. Mansingka, A. G. Radwan and K. N. Salama, Hardware stream cipher with controllable chaos generator for colour image encryption, IET Image Process., 8 (2014), 33-43.
doi: 10.1049/iet-ipr.2012.0586. |
[3] |
S. Chakraborty and S. K. Dana, Shil'nikov chaos and mixed-mode oscillation in Chua's circuit, Chaos, 20 (2010), 23107, 7pp. |
[4] |
T. Chien and T. Liao, Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization, Chaos Solitons Fract., 24 (2005), 241-255.
doi: 10.1016/S0960-0779(04)00542-9. |
[5] |
V. Carmona, F. Fernández-Sánchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048.
doi: 10.1137/070709542. |
[6] |
V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), 013124, 8pp.
doi: 10.1063/1.3339819. |
[7] |
M. di Bernardo and C. K. Tse, Chaos in Power Electronics: An Overview, Chaos in Circuits and Systems, World Scientific, 2002. |
[8] |
S. M. Huan, Q. D. Li and X.-S. Yang, Chaos in three-dimensional hybrid systems and design of chaos generators, Nonlinear Dyn., 69 (2012), 1915-1927.
doi: 10.1007/s11071-012-0396-0. |
[9] |
S. M. Huan and X.-S. Yang, Existence of chaotic invariant set in a class of 4-dimensional piecewise linear dynamical systems, Int. J. Bifurc. Chaos, 24 (2014), 1450158, 16pp.
doi: 10.1142/S0218127414501582. |
[10] |
T. Kousaka, T. Ueta and H. Kawakami, Chaos in a simple hybrid system and its control, Electron Lett., 37 (2001), p1.
doi: 10.1049/el:20010033. |
[11] |
J. Lü, T. Zhou, G. Chen and X.-S. Yang, Generating chaos with a switching piecewise-linear controller, Chaos, 12 (2002), 344-349. |
[12] |
R. O. Medrano-T., M. S. Baptista and I. L. Caldas, Homoclinic orbits in a piecewise system and their relation with invariant sets, Physica D, 186 (2003), 133-147.
doi: 10.1016/j.physd.2003.08.002. |
[13] |
V. Nair and R. I. Sujith, Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification, Chaos, 23 (2013), 033136, 6pp.
doi: 10.1063/1.4821475. |
[14] |
I. Pehlivan and Y. Uyaroglu, Simplified chaotic diffusionless Lorenz attractor and its application to secure communication systems, IET Commun., 1 (2007), 1015-1022. |
[15] |
L. P. Shil'nikov, A case of the existence of a countable number of periodic motions, Sov. Math.Dokl., 6 (1965), 163-166. |
[16] |
L. P. Shil'nikov, A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. |
[17] |
L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, 1998.
doi: 10.1142/9789812798596. |
[18] |
L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001.
doi: 10.1142/9789812798558_0001. |
[19] |
C. Tresser, About some theorems by L. P. Shil'nikov, Inst. H. Poincare Phys. Thoré., 40 (1984), 441-461. |
[20] |
K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. Circuits Syst., 45 (1998), 979-983. |
[21] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, $2^{nd}$ edition, Springer-Verlag, New York, 2003. |
[22] |
D. Wilczak, The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof, Found Comput. Math., 6 (2006), 495-535.
doi: 10.1007/s10208-005-0201-2. |
[23] |
X.-S. Yang and Q. D. Li, Chaos generator via Wien-bridge oscillator, Electron. Lett., 38 (2002), 623-625.
doi: 10.1049/el:20020456. |
[24] |
X.-S. Yang and Q. D. Li, Generate n-scroll attractor in linear system by scalar output feedback, Chaos Solitons Fract., 18 (2003), 25-29.
doi: 10.1016/S0960-0779(02)00638-0. |
[1] |
W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 |
[2] |
Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943 |
[3] |
Nina Lebedeva, Vladimir Matveev, Anton Petrunin, Vsevolod Shevchishin. Smoothing 3-dimensional polyhedral spaces. Electronic Research Announcements, 2015, 22: 12-19. doi: 10.3934/era.2015.22.12 |
[4] |
Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641 |
[5] |
Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485 |
[6] |
Rafel Prohens, Antonio E. Teruel. Canard trajectories in 3D piecewise linear systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595 |
[7] |
Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181 |
[8] |
Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529 |
[9] |
Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma. Moduli of 3-dimensional diffeomorphisms with saddle-foci. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5021-5037. doi: 10.3934/dcds.2018220 |
[10] |
Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control and Related Fields, 2022, 12 (2) : 303-326. doi: 10.3934/mcrf.2021023 |
[11] |
Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803 |
[12] |
Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59 |
[13] |
P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 |
[14] |
Wade Hindes. Orbit counting in polarized dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 189-210. doi: 10.3934/dcds.2021112 |
[15] |
Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1 |
[16] |
Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems and Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053 |
[17] |
M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343 |
[18] |
Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020 |
[19] |
Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497 |
[20] |
Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]