Limit Cycles for Continuous Piecewise Linear Systems with Three Zones Having Two Degenerate Subsystems

Li Xiong; Kuilin Wu

doi:10.62762/JMIA.2025.453419

Volume 1, Issue 1, Journal of Mathematics and Interdisciplinary Applications

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Journal of Mathematics and Interdisciplinary Applications, Volume 1, Issue 1, 2025: 3-19

Open Access | Research Article | 05 November 2025

Limit Cycles for Continuous Piecewise Linear Systems with Three Zones Having Two Degenerate Subsystems

Li Xiong 1

Kuilin Wu 1 *

1 School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

* Corresponding Author: Kuilin Wu, [email protected]

DOI: 10.62762/JMIA.2025.453419

ARK: ark:/57805/jmia.2025.453419

Received: 24 August 2025, Accepted: 22 September 2025, Published: 05 November 2025

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Abstract

This paper deals with continuous piecewise linear differential systems with three zones separated by two parallel straight lines (for short, CPWL3). The number of limit cycles of CPWL3 systems with two degenerate subsystems is not clear. In the paper, we provide a complete study on the maximum number of limit cycles by geometric techniques when the continuous piecewise linear systems with three zones have two degenerate subsystems. During our analysis, we also detect some bifurcation phenomena, such as boundary equilibrium bifurcation, scabbard bifurcation, grazing bifurcation, heteroclinic bifurcation and Hopf bifurcation.

Graphical Abstract

Keywords

limit cycle

piecewise linear systems

bifurcation

Data Availability Statement

Data will be made available on request.

Funding

This work was supported by the Science and Technology Plan Project of Guizhou Province under Grant ZK[2022]G118.

Conflicts of Interest

The authors declare no conflicts of interest.

Ethical Approval and Consent to Participate

Not applicable.

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Xiong, L., & Wu, K. (2025). Limit Cycles for Continuous Piecewise Linear Systems with Three Zones Having Two Degenerate Subsystems. Journal of Mathematics and Interdisciplinary Applications, 1(1), 3–19. https://doi.org/10.62762/JMIA.2025.453419

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TY  - JOUR
AU  - Xiong, Li
AU  - Wu, Kuilin
PY  - 2025
DA  - 2025/11/05
TI  - Limit Cycles for Continuous Piecewise Linear Systems with Three Zones Having Two Degenerate Subsystems
JO  - Journal of Mathematics and Interdisciplinary Applications
T2  - Journal of Mathematics and Interdisciplinary Applications
JF  - Journal of Mathematics and Interdisciplinary Applications
VL  - 1
IS  - 1
SP  - 3
EP  - 19
DO  - 10.62762/JMIA.2025.453419
UR  - https://www.icck.org/article/abs/JMIA.2025.453419
KW  - limit cycle
KW  - piecewise linear systems
KW  - bifurcation
AB  - This paper deals with continuous piecewise linear differential systems with three zones separated by two parallel straight lines (for short, CPWL3). The number of limit cycles of CPWL3 systems with two degenerate subsystems is not clear. In the paper, we provide a complete study on the maximum number of limit cycles by geometric techniques when the continuous piecewise linear systems with three zones have two degenerate subsystems. During our analysis, we also detect some bifurcation phenomena, such as boundary equilibrium bifurcation, scabbard bifurcation, grazing bifurcation, heteroclinic bifurcation and Hopf bifurcation.
SN  - 3070-393X
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  -

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@article{Xiong2025Limit,
  author = {Li Xiong and Kuilin Wu},
  title = {Limit Cycles for Continuous Piecewise Linear Systems with Three Zones Having Two Degenerate Subsystems},
  journal = {Journal of Mathematics and Interdisciplinary Applications},
  year = {2025},
  volume = {1},
  number = {1},
  pages = {3-19},
  doi = {10.62762/JMIA.2025.453419},
  url = {https://www.icck.org/article/abs/JMIA.2025.453419},
  abstract = {This paper deals with continuous piecewise linear differential systems with three zones separated by two parallel straight lines (for short, CPWL3). The number of limit cycles of CPWL3 systems with two degenerate subsystems is not clear. In the paper, we provide a complete study on the maximum number of limit cycles by geometric techniques when the continuous piecewise linear systems with three zones have two degenerate subsystems. During our analysis, we also detect some bifurcation phenomena, such as boundary equilibrium bifurcation, scabbard bifurcation, grazing bifurcation, heteroclinic bifurcation and Hopf bifurcation.},
  keywords = {limit cycle, piecewise linear systems, bifurcation},
  issn = {3070-393X},
  publisher = {Institute of Central Computation and Knowledge}
}

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