Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds
Zhu, B., Wei, Z., Escalante-González, R.J., & Kuznetsov, N. V. (2020). Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds. Chaos, 30(12), Article 123143. https://doi.org/10.1063/5.0032702
Published in
ChaosDate
2020Copyright
© 2020 Author(s).
In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as examples and the results are consistent with the predictions of theorems.
From the Shil’nikov theorems, it is known that the existence of a homoclinic orbit and a heteroclinic cycle play a key and important role in the chaos research of dynamic systems. Moreover, chaos and piecewise linear (PWL) systems have important applications in many fields such as electronic circuits, biology, machinery, and so on. Naturally, the research on the existence of homoclinic orbits and heteroclinic cycles of piecewise linear systems is very meaningful. However, it is not easy to find the homoclinic orbit and the heteroclinic cycle of smooth dynamic systems, especially for higher-dimensional piecewise linear systems with multiple switching manifolds, which will inevitably make the exploration of this problem more complicated. Therefore, this article studies the existence of homoclinic orbits and heteroclinic cycles in a style of the piecewise linear system with three switching manifolds.
...
![showless](/themes/JYX2//images/showless.png)
![showmore](/themes/JYX2//images/showmore.png)
Publisher
American Institute of PhysicsISSN Search the Publication Forum
1054-1500Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/47617359
Metadata
Show full item recordCollections
Additional information about funding
This work was supported by the National Natural Science Foundation of China (NNSFC) (No. 11772306), the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY20A020001), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (No. CUGGC05). The last author acknowledges support from the Russian Leading Scientific school programm (NSh-2624.2020.1).![showless](/themes/JYX2//images/showless.png)
![showmore](/themes/JYX2//images/showmore.png)
License
Related items
Showing items with similar title or keywords.
-
Numerical analysis of dynamical systems : unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension
Kuznetsov, Nikolay; Mokaev, Timur (IOP Publishing, 2019)In this article, on the example of the known low-order dynamical models, namely Lorenz, Rössler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz ... -
Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system
Danca, Marius-F.; Fečkan, Michal; Kuznetsov, Nikolay; Chen, Guanrong (Springer, 2018) -
The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension
Kuznetsov, N. V.; Mokaev, T. N.; Kuznetsova, O. A.; Kudryashova, E. V. (Springer, 2020)On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global ... -
Dynamics of the Shapovalov mid-size firm model
Alexeeva, Tatyana A.; Barnett, William A.; Kuznetsov, Nikolay V.; Mokaev, Timur N. (Elsevier, 2020)Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., ... -
Study of irregular dynamics in an economic model : attractor localization and Lyapunov exponents
Alexeeva, Tatyana A.; Kuznetsov, Nikolay V.; Mokaev, Timur N. (Elsevier, 2021)Cyclicality and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in ...