On the "Crossroad Area–saddle Area" and "Crossroad Area–spring Area" Transitions

Abstract

Let T be a one-dimensional or two-dimensional map. The three considered areas are related to three different configurations of fold and flip bifurcation curves, centred at a cusp point of a fold curve in the T parameter plane (b, c). The two transitions studied here occur via a codimension-three bifurcation defined in each case, when varying a third parameter a. The transition "mechanism," from an area type to another one, is given with a three-dimensional representation describing the sheet configuration of the parameter plane.

Dynamical Behavior in a DPCM System with an Order 2 Predictor. Case of a Constant Input

“…These curves are numerically obtained using equation (5). The existence of cuspidal points C 1 3 and E 3 , which are common to several bifurcation curves and the existence of point P 3 corresponds to specific bifurcation structures, have been studied in [27], [28]. Point P 3 corresponds to a tangential contact between fold and flip bifurcation curves.…”

Homoclinic saddle to saddle-focus transitions in 4D systems

“…These points are absent in a saddle area. Mira et al [17] discuss in detail the spring and saddle areas, including transitions from one case to the other and their genericity.…”

“…Each LP curve is a 'horn' composed of two branches. Close to the horn's tipping point LP and PD curves interact via spring and saddle areas [17]. Transitions between saddle and spring areas are observed.…”

Different types of critical behavior in conservatively coupled Hénon maps

“…First we recall the important property of chaotic dynamics, that it is interspersed with periodic windows of different periods [7][8][9]. The skeletons of these regions of periodic dynamics within the chaotic region in the two-dimensional parameter space are the spring area and crossroad area structures [19][20][21]. When changing the dissipation parameter b, these periodic regions "move" through parameter space, i.e., they change their location as well as their size.…”

“…In the two-dimensional parameter space such windows have a typical form which depends on the special type of organization of bifurcation lines for its main period. The two mostly common types are spring area and crossroad area structures [19][20][21] (periodic windows based on the crossroad area are often called shrimps [17,22]). Such structures in the parameter plane have been found in driven, parametrically excited, and impact oscillators [23][24][25][26], electrochemical oscillators [27], two-gene systems [28], lasers [29,30], population dynamical systems in ecology, as well as in paradigmatic models such * Corresponding author: savin.dmitry.v@gmail.com as the Rössler system [31].…”