2020/06/02 by Sishu Shankar Muni, Robert I. McLachlan, Muni, S. S. +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Quantum chaos and dynamical systems #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.2006.01405
We consider a homoclinic orbit to a saddle fixed point of an arbitrary C^∞ map f on ℝ2 and study the phenomenon that f has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires f to have a homoclinic tangency. We show it also necessary for f to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, λ and σ, to satisfy |λσ| = 1. The phenomenon is codimension-three in the case λσ= -1, but codimension-four in the case λσ= 1 because here the coefficients of the leading-order resonance terms associated with f at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.