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Weak centers and local criticality on planar Z2-symmetric cubic differential systems

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  • Liu, Yuanyuan
  • He, Dongping
  • Huang, Wentao

Abstract

There exist the necessary and sufficient conditions for planar Z2-symmetric cubic differential systems with a bi-center and an isochronous bi-center in the existing literature. Based on these bi-center conditions, we give a complete discussion on the order of weak centers and the local criticality for planar Z2-symmetric cubic systems at the origin and the symmetric equilibria (±1,0) separately. More precisely, we first give the parametric conditions for the origin and the equilibria (±1,0) being weak centers of exact order separately. Then we prove that the local criticality from the weak centers of finite order at the origin is at most 4, and at each of symmetric equilibria (±1,0) is at most 3. Finally, we also obtain that the local criticality from the isochronous center at the origin is at most 3, and at each of equilibria (±1,0) is at least 2. More importantly, these numbers are reachable.

Suggested Citation

  • Liu, Yuanyuan & He, Dongping & Huang, Wentao, 2023. "Weak centers and local criticality on planar Z2-symmetric cubic differential systems," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:chsofr:v:177:y:2023:i:c:s0960077923011591
    DOI: 10.1016/j.chaos.2023.114257
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    References listed on IDEAS

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    1. Giné, Jaume, 2007. "On some open problems in planar differential systems and Hilbert’s 16th problem," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1118-1134.
    2. Chen, Ting & Huang, Lihong & Huang, Wentao & Li, Wenjie, 2017. "Bi-center conditions and local bifurcation of critical periods in a switching Z2 equivariant cubic system," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 157-168.
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