PapersNo Access

Time-Reversible Chaotic System with Conditional Symmetry

Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China

Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China

E-mail Address: goontry@126.com

E-mail Address: chunbiaolee@nuist.edu.cn

Corresponding author.

Search for more papers by this author

Julien Clinton Sprott

Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA

E-mail Address: sprott@physics.wisc.edu

Search for more papers by this author

, and

Yongjian Liu

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, P. R. China

E-mail Address: liuyongjianmaths@126.com

Search for more papers by this author

https://doi.org/10.1142/S0218127420500674Cited by:5 (Source: Crossref)

Abstract

When the polarity reversal induced by offset boosting is considered, a new regime of a time-reversible chaotic system with conditional symmetry is found, and some new time-reversible systems are revealed based on multiple dimensional offset boosting. Numerical analysis shows that the system attractor and repellor have their own dynamics in respective time domains which constitutes the fundamental property in a time-reversible system. More remarkably, when the conditional symmetry is destroyed by a slightly mismatched offset controller, the system undergoes different bifurcations to chaos, and the corresponding coexisting attractors and repellors shape their own phase trajectories.

Keywords:

References

Bao, B., Li, Q., Wang, N. & Xu, Q. [2016] “ Multistability in Chua’s circuit with two stable node-foci,” Chaos 26, 043111. Crossref, Web of Science, Google Scholar
Bao, H., Wang, N., Bao, B., Chen, M., Jin, P. & Wang, G. [2018] “ Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria,” Commun. Nonlin. Sci. Numer. Simulat. 57, 264–275. Crossref, Web of Science, Google Scholar
Chen, M., Xu, Q., Lin, Y. & Bao, B. [2017] “ Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit,” Nonlin. Dyn. 87, 789–802. Crossref, Web of Science, Google Scholar
Hoover, W. G. [1995] “ Remark on ‘some simple chaotic flows,” Phys. Rev. E 51, 759–760. Crossref, Web of Science, Google Scholar
Hoover, W. G., Kum, O. & Posch, H. A. [1996] “ Time-reversible dissipative ergodic maps,” Phys. Rev. E 53, 2123–2129. Crossref, Web of Science, Google Scholar
Jafari, S., Sprott, J. C. & Nazarimehr, F. [2015] “ Recent new examples of hidden attractors,” Eur. Phys. J. Spec. Top. 224, 1469–1476. Crossref, Web of Science, Google Scholar
Kuznetsov, N. V., Leonov, G. A., Mokaev, T. N., Prasad, A. & Shrimali, M. D. [2018a] “ Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system,” Nonlin. Dyn. 92, 267–285. Crossref, Web of Science, Google Scholar
Kuznetsov, N. V., Leonov, G. A., Mokaev, T. N., Prasad, A. & Shrimali, M. D. [2018b] “ Numerical analysis of dynamical systems: Unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension,” J. Phys.: Conf. Ser. 1205, 012034. Crossref, Google Scholar
Lai, Q. & Chen, S. [2016a] “ Generating multiple chaotic attractors from Sprott B system,” Int. J. Bifurcation and Chaos 26, 1650177. Link, Web of Science, Google Scholar
Lai, Q. & Chen, S. [2016b] “ Research on a new 3D autonomous chaotic system with coexisting attractors,” Optik 127, 3000–3004. Crossref, Web of Science, Google Scholar
Lai, Q., Akif, A., Zhao, X. & Pei, H. [2017] “ Various types of coexisting attractors in a new 4D autonomous chaotic system,” Int. J. Bifurcation and Chaos 27, 1750142. Link, Web of Science, Google Scholar
Leonov, G. A. & Kuznetsov, N. V. [2015] “ On differences and similarities in the analysis of Lorenz, Chen, and Lü systems,” Appl. Math. Comput. 256, 334–343. Crossref, Web of Science, Google Scholar
Leonov, G. A., Kuznetsov, N. V. & Mokaev, T. N. [2015] “ Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” Eur. Phys. J. Spec. Top. 224, 1421–1458. Crossref, Web of Science, Google Scholar
Li, C., Sprott, J. C. & Xing, H. [2016] “ Hypogenetic chaotic jerk flows,” Phys. Lett. A 380, 1172–1177. Crossref, Web of Science, Google Scholar
Li, C. & Sprott, J. C. [2016a] “ Variable-boostable chaotic flows,” Optik 127, 10389–10398. Crossref, Web of Science, Google Scholar
Li, C. & Sprott, J. C. [2016b] “ How to bridge attractors and repellors,” Int. J. Bifurcation and Chaos 27, 1750149-1–11. Web of Science, Google Scholar
Li, C., Sprott, J. C. & Xing, H. [2017] “ Constructing chaotic systems with conditional symmetry,” Nonlin. Dyn. 87, 1351–1358. Crossref, Web of Science, Google Scholar
Li, C., Sprott, J. C., Liu, Y., Gu, Z. & Zhang, J. [2018] “ Offset boosting for breeding conditional symmetry,” Int. J. Bifurcation and Chaos 28, 1850163-1–13. Link, Web of Science, Google Scholar
Li, C., Xu, Y., Chen, G., Liu, Y. & Zheng, J. [2019] “ Conditional symmetry: Bond for attractor growing,” Nonlin. Dyn. 95, 1245–1256. Crossref, Web of Science, Google Scholar
Liu, Y. & Pang, G. [2011] “ The basin of attraction of the Liu system,” Commun. Nonlin. Sci. Numer. Simulat. 16, 2065–2071. Crossref, Web of Science, Google Scholar
Sprott, J. C. [2014] “ Simplest chaotic flows with involutional symmetries,” Int. J. Bifurcation and Chaos 24, 1450009. Link, Web of Science, Google Scholar
Sprott, J. C. [2015] “ Symmetric time-reversible flows with a strange attractor,” Int. J. Bifurcation and Chaos 25, 1550078. Link, Web of Science, Google Scholar
Wang, C., Liu, X. & Xia, H. [2017] “ Multi-piecewise quadratic nonlinearity memristor and its $2 N$ $2 N$ -scroll and $2 N + 1$ -scroll chaotic attractors system,” Chaos 27, 033114. Crossref, Web of Science, Google Scholar
Wei, Z., Wang, R. & Liu, A. [2014] “ A new finding of the existence of hidden hyperchaotic attractors with no equilibria,” Math. Comput. Simulat. 100, 13–23. Crossref, Web of Science, Google Scholar
Zhang, X. & Wang, C. [2019a] “ Multiscroll hyperchaotic system with hidden attractors and its circuit implementation,” Int. J. Bifurcation and Chaos 29, 1950117. Link, Web of Science, Google Scholar
Zhang, X. & Wang, C. [2019b] “ A novel multi-attractor period multi-scroll chaotic integrated circuit based on CMOS wide adjustable CCCII,” IEEE Access 7, 16336–16350. Crossref, Web of Science, Google Scholar
Zhou, L., Wang, C. & Zhou, L. [2016] “ Generating hyperchaotic multi-wing attractor in a 4D memristive circuit,” Nonlin. Dyn. 85, 2653–2663. Crossref, Web of Science, Google Scholar
Zhou, L., Wang, C. & Zhou, L. [2017] “ Generating four-wing hyperchaotic attractor and two-wing, three-wing, and four-wing chaotic attractors in 4D memristive system,” Int. J. Bifurcation and Chaos 27, 1750027. Link, Web of Science, Google Scholar
Zhou, L., Wang, C., Zhang, X. & Yao, W. [2018a] “ Various attractors, coexisting attractors and antimonotonicity in a simple fourth-order memristive twin-T oscillator,” Int. J. Bifurcation and Chaos 28, 1850050. Link, Web of Science, Google Scholar
Zhou, L., Wang, C. & Zhou, L. [2018b] “ A novel no-equilibrium hyperchaotic multi-wing system via introducing memristor,” Int. J. Circ. Th. Appl. 46, 84–98. Crossref, Web of Science, Google Scholar