A criticality result for polycycles in a family of quadratic reversible\n centers

2017/05/15 by David Rojas, Rojas, David, Jordi Villadelprat +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1705.05408

openalex publication_date 2017/05/15 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider the family of dehomogenized Loud's centers\nX_\μ=y(x-1)\∂_x+(x+Dx²+Fy²)\∂_y, where\n\μ=(D,F)\∈\ℝ², and we study the number of critical periodic orbits\nthat emerge or dissapear from the polycycle at the boundary of the period\nannulus. This number is defined exactly the same way as the well-known notion\nof cyclicity of a limit periodic set and we call it criticality. The previous\nresults on the issue for the family X_\μ,\μ\∈\ℝ² \ndistinguish between parameters with criticality equal to zero (regular\nparameters) and those with criticality greater than zero (bifurcation\nparameters). A challenging problem not tackled so far is the computation of the\ncriticality of the bifurcation parameters, which form a set \Γ_B of\ncodimension 1 in \ℝ². In the present paper we succeed in proving\nthat a subset of \Γ_B has criticality equal to one.\n

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