Downloads provided by UsageCountsLimit cycles and Lie symmetries
Copyright policy ) Freire, Emilio; Gasull Embid, Armengol; Guillamon Grabolosa, Antoni;
Limit cycles and Lie symmetries
Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.
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Differential equations, Hyperbolicity, Geometric methods in ordinary differential equations, Mathematics(all), hyperbolicity, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Limit cycles, Isochrons, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], :34 Ordinary differential equations::34A General theory [Classificació AMS], Periodic orbits of vector fields and flows, Differentiable dynamical systems, Lie symmetries, Equacions diferencials ordinàries, Classificació AMS::34 Ordinary differential equations::34A General theory, Symmetries, invariants of ordinary differential equations, isochrons
Differential equations, Hyperbolicity, Geometric methods in ordinary differential equations, Mathematics(all), hyperbolicity, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Sistemes dinàmics diferenciables, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Limit cycles, Isochrons, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], :34 Ordinary differential equations::34A General theory [Classificació AMS], Periodic orbits of vector fields and flows, Differentiable dynamical systems, Lie symmetries, Equacions diferencials ordinàries, Classificació AMS::34 Ordinary differential equations::34A General theory, Symmetries, invariants of ordinary differential equations, isochrons
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This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
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This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
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This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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