Degenerate hopf bifurcations in discontinuous planar systems

By B. Coll, Armengol Gasull, R. Prohens. 2001

158 citations recorded by the source corpus.

Abstract We study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities. This is done, for the most generic cases, by computing some kind of Lyapunov constants. Our computations are based on the so called ( R , θ, p , q )-generalized polar coordinates, introduced by Lyapunov, and they are essentially different from the ones used in the smooth case. These Lyapunov constants are also used to generate limit cycles for some concrete examples.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/603a7abbb9f2beacf682751e21704ba09c7efba4

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems (2013, background): “In [2,3, 8,22,29], Hopf bifurcations are studied for planar piecewise smooth differential systems.”
On Hopf bifurcation in non-smooth planar systems (2010, methodology): “Recently, Hopf bifurcations in general or some kinds of non-smooth planar systems were studied by [3,12,13].”
Bifurcation of limit cycles at infinity in piecewise polynomial systems (2018, methodology): “[16] developed a method for computing the Lyapunov constants to study bifurcation of small-amplitude limit cycles.”
Uniqueness of limit cycles for sewing planar piecewise linear systems (2015, methodology): “The stability conditions of the statement follow from the computation of the Lyapunov constants, see [6], because of the monodromic property.”
The discontinuous matching of two planar linear foci can have three nested crossing limit cycles (2014, background): “The Hopf bifurcation in non-smooth systems was studied in [12], [3] and [19].”
Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop (2011, methodology): “If h→ 1, Lh approaches the origin which is an elementary center of parabolic-focus type(see [4] for the definition).”
Limit cycles for discontinuous quadratic differential systems with two zones (2014, background): “As far as we know for discontinuous quadratic differential systems only the center problem and the Hopf bifurcation has been studied partially, see [8, 9, 10].”