Doubling bifurcation of a closed invariant curve in 3D maps

Abstract

Abstract. The object of the present paper is to give a qualitative description of the bifurcation mechanisms associated with a closed invariant curve in three-dimensional maps, leading to its doubling, not related to a standard doubling of tori. We propose an explanation on how a closed invariant attracting curve, born via Neimark-Sacker bifurcation, can be transformed into a repelling one giving birth to a new attracting closed invariant curve which has doubled loops.

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

“…Similar to quasiperiodic closed invariant curves, mode-locked periodic orbits also undergo doubling bifurcation. Such bifurcation was conjectured in [18] and can only be found in discrete maps of dimension greater than or equal to three. But such bifurcations are special in the sense that for such a bifurcation to occur, doubling of both stable and saddle periodic orbit is required.…”

“…The doubling bifurcations of closed invariant curves have important consequences on the dynamics of the system and can lead to the formation of Shilnikov attractor [15]. Recently, such doubling bifurcations were found in many dynamical systems [16][17][18][19][20], but to our knowledge such doubling bifurcations have not been discussed in neuron systems. Firstly in a neuron system, we show the phenomenon of doubling bifurcation of closed invariant curves.…”