Abstract
In this paper, we study the maximum number of limit cycles that can exhibit a planar discontinuous piecewise differential system separated by two parallel straight lines and formed by two arbitrary linear differential systems with isolated singularity in the lines of discontinuity and a linear Hamiltonian saddle. More precisely, we prove that when the piecewise differential systems are of type boundary focus-Hamiltonian linear saddle-boundary focus, then this class of systems has at most four crossing limit cycles. But when the piecewise differential system is of type boundary focus-Hamiltonian linear saddle-boundary center, we show that it can have at most three limit cycles, and when the piecewise differential system is of type boundary center-Hamiltonian linear saddle-boundary center, we show that it can have at most one limit cycle.
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Berbache, A., Tababouchet, I. Four crossing limit cycles of a family of discontinuous piecewise linear systems with three zones separated by two parallel straight lines. Bol. Soc. Mat. Mex. 30, 50 (2024). https://doi.org/10.1007/s40590-024-00623-6
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DOI: https://doi.org/10.1007/s40590-024-00623-6
Keywords
- Discontinuous planar piecewise linear system
- Crossing limit cycles
- First integral
- Linear Hamiltonian saddles