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<p>Heteroclinic cycles and homoclinic orbits play a vital role in studying global and complicated dynamics of systems. This paper studies the existence of heteroclinic cycles and homoclinic orbits in a class of 3-dimensional piecewise linear systems with three intersecting switching boundaries. Based on four intersection scenarios of stable and unstable manifolds of the three subsystems, the system in study can have (1) five heteroclinic cycles (2) three heteroclinic cycles and a homoclinic orbit (3) two heteroclinic cycles and two homoclinic orbits (4) a heteroclinic cycle and three homoclinic orbits. Specifically, each heteroclinic orbit in consideration intersects a switching manifold transversely at one point, and each homoclinic orbit in consideration intersects a switching manifold transversely at two points. Two numerical examples with simulations of heteroclinic cycles and homoclinic orbits are offered to show feasibility of the results.</p>

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Coexistence of heteroclinic cycles and homoclinic orbits in a class of 3-dimensional piecewise linear systems

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Xiaoteng Yang,
  • Yangsheng Zhang,
  • Tiantian Wu

摘要

Heteroclinic cycles and homoclinic orbits play a vital role in studying global and complicated dynamics of systems. This paper studies the existence of heteroclinic cycles and homoclinic orbits in a class of 3-dimensional piecewise linear systems with three intersecting switching boundaries. Based on four intersection scenarios of stable and unstable manifolds of the three subsystems, the system in study can have (1) five heteroclinic cycles (2) three heteroclinic cycles and a homoclinic orbit (3) two heteroclinic cycles and two homoclinic orbits (4) a heteroclinic cycle and three homoclinic orbits. Specifically, each heteroclinic orbit in consideration intersects a switching manifold transversely at one point, and each homoclinic orbit in consideration intersects a switching manifold transversely at two points. Two numerical examples with simulations of heteroclinic cycles and homoclinic orbits are offered to show feasibility of the results.