2025/12/02 by David J. W. Simpson, Simpson, D. J. W., Виктор Аврутин +1
Computer Science · Physics and Astronomy · #37G25 #37G35 #39A28 #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2512.02291
openalex publication_date 2025/12/02 · openalex created_date 2025/12/04 · openalex updated_date 2026/07/28
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near certain codimension-two homoclinic bifurcations are well approximated by a three-parameter family of one-dimensional maps. Each member of the one-dimensional family is discontinuous, because the family is constructed from the first return of iterates to a subset of phase space, and comprised of infinitely many linear pieces, where each piece corresponds to a fixed number of iterations near the saddle associated with the homoclinic bifurcation. The one-dimensional family exhibits period-incrementing, period-adding, bandcount-incrementing, and bandcount-adding structures (all typical for two-piece maps), as well as unique features caused by orbits repeatedly visiting more than two pieces of the map. These structures carry through to the two-dimensional maps with only minor differences in the arrangement of the bifurcations developing with the distance from the codimension-two bifurcations. This leads to a novel and vivid elucidation of the dynamics of the two-dimensional border-collision normal form.