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Investigations of bifurcations of limit cycles in \(Z_ 2\)-equivariant planar vector fields of degree 5. (English) Zbl 1047.34043

Summary: Some distributions of limit cycles of \(Z_2\)-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 are investigated. These include examples of specific \(Z_2\)-equivariant fields and \(Z_4\)-equivariant fields having up to 23 limit cycles. The configurations of compound eyes are also obtained by using the bifurcation theory of planar dynamical systems and the method of detection functions.

MSC:

34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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