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doi: 10.3934/dcdsb.2022045
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Coexisting singular cycles in a class of three-dimensional three-zone piecewise affine systems

School of Information and Mathematics, Yangtze University-Wuhan Campus, Wuhan, 430100, China

* Corresponding author: Wenjing Xu

Received  April 2021 Revised  December 2021 Early access March 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC) grant No. 12101078

Detecting an isolated homoclinic or heteroclinic cycle is a great challenge in a concrete system, letting alone the case of coexisting scenarios and more complicated chaotic behaviors. This paper systematically investigates the dynamics for a class of three-dimensional (3D) three-zone piecewise affine systems (PWASs) consisting of three sub-systems. Interestingly, under different conditions the considered system can display three types of coexisting singular cycles including: homoclinic and homoclinic cycles, heteroclinic and heteroclinic cycles, homoclinic and heteroclinic cycles. Furthermore, it establishes sufficient conditions for the presence of chaotic invariant sets emerged from such coexisting cycles. Finally, three numerical examples are provided to verify the proposed theoretical results.

Citation: Kai Lu, Wenjing Xu. Coexisting singular cycles in a class of three-dimensional three-zone piecewise affine systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022045
References:
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J. H. Bao and Q. G. Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011), 6526-6540.  doi: 10.1016/j.amc.2011.01.032.

[2]

J. G. Barajas-Ramírez, A. Franco-López and H. G. González-Hernández, Generating Shilnikov chaos in 3D piecewise linear systems, Appl. Math. Comput., 395 (2021), 125877, 11pp. doi: 10.1016/j.amc.2020.125877.

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V. N. Belykh, Bifurcation of separatrices of a saddle point of the Lorenz system, Differ. Equ., 20 (1984), 1184-1191. 

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V. N. Belykh, N. V. Barabash and I. V. Belykh, A Lorenz-type attractor in a piecewise-smooth system: rigorous results, Chaos, 29 (2019), 103108, 17pp. doi: 10.1063/1.5115789.

[5]

V. N. Belykh, N. V. Barabash and I. V. Belykh, Sliding homoclinic bifurcations in a Lorenz-type system: analytic proofs, Chaos, 31 (2021), 043117, 17pp. doi: 10.1063/5.0044731.

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M. Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer-Verlag London, Ltd., London, 2008.

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V. CarmonaF. Fernández-Sánchez and E. García-Medina, Including homoclinic connections and T-point heteroclinic cycles in the same global problem for a reversible family of piecewise linear systems, Appl. Math. Comput., 296 (2017), 33-41.  doi: 10.1016/j.amc.2016.10.008.

[8]

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.

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V. CarmonaF. 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.

[10]

Y. Chen, The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system, Nonlin. Dyn., 87 (2017), 1445-1452.  doi: 10.1007/s11071-016-3126-1.

[11]

Y. M. Chen and Q. G. Yang, A new Lorenz-type hyperchaotic system with a curve of equilibria, Math. Comput. Simulat., 112 (2015), 40-55.  doi: 10.1016/j.matcom.2014.11.006.

[12]

U. Chialva and W. Reartes, Heteroclinic cycles in a competitive network, Int. J. Bifurcation and Chaos, 27 (2017), 1730044, 16pp. doi: 10.1142/S0218127417300440.

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H. KokubuD. Wilczak and P. Zgliczyński, Rigorous verification of cocoon bifurcations in the Michelson system, Nonlinearity, 20 (2007), 2147-2174. 

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G. A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lü and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.

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G. A. LeonovN. V. Kuznetsov and T. N. Mokaev, Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity, Commun. Nonlin. Sci. Numer. Simulat., 28 (2015), 166-174.  doi: 10.1016/j.cnsns.2015.04.007.

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G. A. Leonov, R. N. Mokaev, N. V. Kuznetsov and T. N. Mokaev, Homoclinic bifurcations and chaos in the fishing principle for the Lorenz-like systems, Int. J. Bifurcation and Chaos, 30 (2020), 2050124, 20pp. doi: 10.1142/S0218127420501242.

[20]

Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.

[21]

Z. L. LiuH. B. Fang and J. Xu, Identification of piecewise linear dynamical systems using physically-interpretable neural-fuzzy networks: Methods and applications to origami structures, Neural Networks, 116 (2019), 74-87.  doi: 10.1016/j.neunet.2019.04.007.

[22]

J. LlibreE. Ponce and A. E. Teruel, Horseshoes near homoclinic orbits for piecewise linear differential systems in $\mathbb{R}^3$, Int. J. Bifurcation and Chaos, 17 (2007), 1171-1184.  doi: 10.1142/S0218127407017756.

[23]

E. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[24]

K. LuW. J. Xu and Q. M. Xiang, Coexistence of singular cycles in a new kind of 3D non-smooth systems with two discontinuous boundaries, Nonlin. Dyn., 104 (2021), 149-164. 

[25]

K. Lu, W. J. Xu and Q. G. Yang, Chaos generated by a class of 3D three-zone piecewise affine systems with coexisting singular cycles, Int. J. Bifurcation and Chaos, 30 (2020), 2050209, 17pp. doi: 10.1142/S0218127420502090.

[26]

K. Lu, Q. G. Yang and G. R. Chen, Singular cycles and chaos in a new class of 3D three-zone piecewise affine systems, Chaos, 29 (2019), 043124, 12pp. doi: 10.1063/1.5089662.

[27]

K. LuQ. G. Yang and W. J. Xu, Heteroclinic cycles and chaos in a class of 3D three-zone piecewise affine systems, J. Math. Anal. Appl., 478 (2019), 58-81.  doi: 10.1016/j.jmaa.2019.04.070.

[28]

A. A. P. Rodrigues, Strange attractors and wandering domains near a homoclinic cycle to a bifocus, J. Differ. Equ., 269 (2020), 3221-3258.  doi: 10.1016/j.jde.2020.02.027.

[29]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics (Part II), World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558_0001.

[30]

N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcation and Chaos, 27 (2017), 1730038, 18 pp. doi: 10.1142/S0218127417300385.

[31]

L. Wang and X. S. Yang, Heteroclinic cycles in a class of 3-dimensional piecewise affine systems, Nonlinear Anal. Hybrid Syst., 23 (2017), 44-60.  doi: 10.1016/j.nahs.2016.07.001.

[32]

L. Wang and X. S. Yang, Existence of homoclinic cycles and periodic orbits in a class of three-dimensional piecewise affine systems, Int. J. Bifurcation and Chaos, 28 (2018), 1850024, 15pp. doi: 10.1142/S0218127418500244.

[33]

D. WilczakS. Serrano and R. Barrio, Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler System: a computer-assisted proof, SIAM J. Appl. Dyn. Syst., 15 (2016), 356-390.  doi: 10.1137/15M1039201.

[34]

T. T. Wu and X. S. Yang, On the existence of bifocal heteroclinic cycles in a class of four-dimensional piecewise affine systems, Chaos, 26 (2016), 053104, 8pp. doi: 10.1063/1.4949474.

[35]

T. T. Wu and X. S. Yang, A new class of 3-dimensional piecewise affine systems with homoclinic orbits, Discr. Contin. Dyn. Syst. A, 36 (2016), 5119-5129.  doi: 10.3934/dcds.2016022.

[36]

Q. G. Yang and Y. M. Chen, Complex dynamics in the unified Lorenz-type system, Int. J. Bifurcation and Chaos, 24 (2014), 1450055, 30pp. doi: 10.1142/S0218127414500552.

[37]

Q. G. Yang and K. Lu, Homoclinic orbits and an invariant chaotic set in a new 4D piecewise affine systems, Nonlin. Dyn., 93 (2018), 2445-2459.  doi: 10.1007/s11071-018-4335-6.

[38]

Q. G. Yang and T. Yang, Complex dynamics in a generalized Langford system, Nonlin. Dyn., 91 (2018), 2241-2270.  doi: 10.1007/s11071-017-4012-1.

[39]

T. Yang, Homoclinic orbits and chaos in the generalized Lorenz system, Discr. Contin. Dyn. Syst. B, 25 (2020), 1097-1108.  doi: 10.3934/dcdsb.2019210.

show all references

References:
[1]

J. H. Bao and Q. G. Yang, A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011), 6526-6540.  doi: 10.1016/j.amc.2011.01.032.

[2]

J. G. Barajas-Ramírez, A. Franco-López and H. G. González-Hernández, Generating Shilnikov chaos in 3D piecewise linear systems, Appl. Math. Comput., 395 (2021), 125877, 11pp. doi: 10.1016/j.amc.2020.125877.

[3]

V. N. Belykh, Bifurcation of separatrices of a saddle point of the Lorenz system, Differ. Equ., 20 (1984), 1184-1191. 

[4]

V. N. Belykh, N. V. Barabash and I. V. Belykh, A Lorenz-type attractor in a piecewise-smooth system: rigorous results, Chaos, 29 (2019), 103108, 17pp. doi: 10.1063/1.5115789.

[5]

V. N. Belykh, N. V. Barabash and I. V. Belykh, Sliding homoclinic bifurcations in a Lorenz-type system: analytic proofs, Chaos, 31 (2021), 043117, 17pp. doi: 10.1063/5.0044731.

[6]

M. Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer-Verlag London, Ltd., London, 2008.

[7]

V. CarmonaF. Fernández-Sánchez and E. García-Medina, Including homoclinic connections and T-point heteroclinic cycles in the same global problem for a reversible family of piecewise linear systems, Appl. Math. Comput., 296 (2017), 33-41.  doi: 10.1016/j.amc.2016.10.008.

[8]

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.

[9]

V. CarmonaF. 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.

[10]

Y. Chen, The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system, Nonlin. Dyn., 87 (2017), 1445-1452.  doi: 10.1007/s11071-016-3126-1.

[11]

Y. M. Chen and Q. G. Yang, A new Lorenz-type hyperchaotic system with a curve of equilibria, Math. Comput. Simulat., 112 (2015), 40-55.  doi: 10.1016/j.matcom.2014.11.006.

[12]

U. Chialva and W. Reartes, Heteroclinic cycles in a competitive network, Int. J. Bifurcation and Chaos, 27 (2017), 1730044, 16pp. doi: 10.1142/S0218127417300440.

[13]

L. O. Chua and R. Ying, Canonical piecewise-linear analysis, IEEE Trans. Circuits Syst., 30 (1983), 125-140.  doi: 10.1109/TCS.1983.1085342.

[14]

H. KokubuD. Wilczak and P. Zgliczyński, Rigorous verification of cocoon bifurcations in the Michelson system, Nonlinearity, 20 (2007), 2147-2174. 

[15]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Springer, Berlin, 2004. doi: 10.1007/978-3-540-44398-8.

[16]

G. A. Leonov, On estimates of the bifurcation values of the parameters of a Lorenz system, Russ. Math. Surveys, 43 (1988), 216-217.  doi: 10.1070/RM1988v043n03ABEH001766.

[17]

G. A. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lü and Chen systems, Phys. Lett. A, 376 (2012), 3045-3050.  doi: 10.1016/j.physleta.2012.07.003.

[18]

G. A. LeonovN. V. Kuznetsov and T. N. Mokaev, Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity, Commun. Nonlin. Sci. Numer. Simulat., 28 (2015), 166-174.  doi: 10.1016/j.cnsns.2015.04.007.

[19]

G. A. Leonov, R. N. Mokaev, N. V. Kuznetsov and T. N. Mokaev, Homoclinic bifurcations and chaos in the fishing principle for the Lorenz-like systems, Int. J. Bifurcation and Chaos, 30 (2020), 2050124, 20pp. doi: 10.1142/S0218127420501242.

[20]

Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal. Real World Appl., 11 (2010), 2563-2572.  doi: 10.1016/j.nonrwa.2009.09.001.

[21]

Z. L. LiuH. B. Fang and J. Xu, Identification of piecewise linear dynamical systems using physically-interpretable neural-fuzzy networks: Methods and applications to origami structures, Neural Networks, 116 (2019), 74-87.  doi: 10.1016/j.neunet.2019.04.007.

[22]

J. LlibreE. Ponce and A. E. Teruel, Horseshoes near homoclinic orbits for piecewise linear differential systems in $\mathbb{R}^3$, Int. J. Bifurcation and Chaos, 17 (2007), 1171-1184.  doi: 10.1142/S0218127407017756.

[23]

E. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[24]

K. LuW. J. Xu and Q. M. Xiang, Coexistence of singular cycles in a new kind of 3D non-smooth systems with two discontinuous boundaries, Nonlin. Dyn., 104 (2021), 149-164. 

[25]

K. Lu, W. J. Xu and Q. G. Yang, Chaos generated by a class of 3D three-zone piecewise affine systems with coexisting singular cycles, Int. J. Bifurcation and Chaos, 30 (2020), 2050209, 17pp. doi: 10.1142/S0218127420502090.

[26]

K. Lu, Q. G. Yang and G. R. Chen, Singular cycles and chaos in a new class of 3D three-zone piecewise affine systems, Chaos, 29 (2019), 043124, 12pp. doi: 10.1063/1.5089662.

[27]

K. LuQ. G. Yang and W. J. Xu, Heteroclinic cycles and chaos in a class of 3D three-zone piecewise affine systems, J. Math. Anal. Appl., 478 (2019), 58-81.  doi: 10.1016/j.jmaa.2019.04.070.

[28]

A. A. P. Rodrigues, Strange attractors and wandering domains near a homoclinic cycle to a bifocus, J. Differ. Equ., 269 (2020), 3221-3258.  doi: 10.1016/j.jde.2020.02.027.

[29]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics (Part II), World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812798558_0001.

[30]

N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. O. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurcation and Chaos, 27 (2017), 1730038, 18 pp. doi: 10.1142/S0218127417300385.

[31]

L. Wang and X. S. Yang, Heteroclinic cycles in a class of 3-dimensional piecewise affine systems, Nonlinear Anal. Hybrid Syst., 23 (2017), 44-60.  doi: 10.1016/j.nahs.2016.07.001.

[32]

L. Wang and X. S. Yang, Existence of homoclinic cycles and periodic orbits in a class of three-dimensional piecewise affine systems, Int. J. Bifurcation and Chaos, 28 (2018), 1850024, 15pp. doi: 10.1142/S0218127418500244.

[33]

D. WilczakS. Serrano and R. Barrio, Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler System: a computer-assisted proof, SIAM J. Appl. Dyn. Syst., 15 (2016), 356-390.  doi: 10.1137/15M1039201.

[34]

T. T. Wu and X. S. Yang, On the existence of bifocal heteroclinic cycles in a class of four-dimensional piecewise affine systems, Chaos, 26 (2016), 053104, 8pp. doi: 10.1063/1.4949474.

[35]

T. T. Wu and X. S. Yang, A new class of 3-dimensional piecewise affine systems with homoclinic orbits, Discr. Contin. Dyn. Syst. A, 36 (2016), 5119-5129.  doi: 10.3934/dcds.2016022.

[36]

Q. G. Yang and Y. M. Chen, Complex dynamics in the unified Lorenz-type system, Int. J. Bifurcation and Chaos, 24 (2014), 1450055, 30pp. doi: 10.1142/S0218127414500552.

[37]

Q. G. Yang and K. Lu, Homoclinic orbits and an invariant chaotic set in a new 4D piecewise affine systems, Nonlin. Dyn., 93 (2018), 2445-2459.  doi: 10.1007/s11071-018-4335-6.

[38]

Q. G. Yang and T. Yang, Complex dynamics in a generalized Langford system, Nonlin. Dyn., 91 (2018), 2241-2270.  doi: 10.1007/s11071-017-4012-1.

[39]

T. Yang, Homoclinic orbits and chaos in the generalized Lorenz system, Discr. Contin. Dyn. Syst. B, 25 (2020), 1097-1108.  doi: 10.3934/dcdsb.2019210.

Figure 1.  Geometric structure of $ R_{k} $ and $ P_1(R_{k}) $
Figure 2.  Geometric structure of $ P(R_{k}) $ and $ R_{ki} $ for $ i = 1, 2 $
Figure 3.  Geometric structure of $ R_k $ and $ P_1(R_k) $
Figure 4.  Geometric structure of Geometry of $ R_{ni} $ and $ R_{kij} $ for $ i = 1, 2 $, $ j = 1, 2 $
Figure 5.  Geometric structure of Geometry of $ R_{ni} $ and $ R_{kij} $ for $ i = 1, 2 $, $ j = 1, 2 $
Figure 6.  Geometric structure of $ R_{lk} $ and $ R_{rk} $ under Poincaré return map $ P $
Figure 7.  System (35) in Example 1 has (a) a homoclinic $ \Upsilon_1 $, (b) a homoclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $
Figure 8.  (a) The phase portrait of a chaotic set of system (35). The projection of the chaotic set in (b) x-y plane, (c)y-z plane
Figure 9.  System (36) in Example 2 has (a) a heteroclinic cycle $ \Upsilon_1 $, (b) a heteroclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $
Figure 10.  (a) The phase portrait of a chaotic set of system (36). The projection of the chaotic set in (b) x-z plane, (c)y-z plane
Figure 11.  System (37) in Example 3 has (a) a heteroclinic cycle $ \Upsilon_1 $, (b) a homoclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $
Figure 12.  (a) The phase portrait of a chaotic set of system (37). The projection of the chaotic set in (b) x-y plane, (c)x-z plane
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Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215

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