"Crossroad Area–spring Area" Transition (Ii) Foliated Parametric Representation

Abstract

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

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.…”

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.…”

“…It is necessary to recall here briefly the structure of a typical periodic window. Crossroad area and spring area structures based on the cycle of period n are formed by two fold lines for this cycle emanating from a cusp point [19][20][21], hence one can find multistability and the parameter space becomes divided into two "multistability sheets", which mean that in some region in the parameter space two attractors with different basins of attraction and independent dynamics coexist. On each of these multistability sheets there exists a period-doubling line.…”

“…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].…”

Stable cycles in a Cournot duopoly model of Kopel

“…Bifurcation of maps have been studied intensively in the literature, cf [20,12,11,10]. A comprehensive discussion is given in [18].…”