Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional maps

Abstract

Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map becomes continuous. Depending on the properties of the map near the codimension-two bifurcation point, there exist different scenarios regarding how the infinite number of periodic orbits are born, mainly the so-called period adding and period incrementing. In our work we prove that, in order to undergo a big bang bifurcation of the period incrementing type, it is sufficient for a piecewisedefined one-dimensional map that the colliding fixed points are attractive and with associated eigenvalues of different signs.

Breaking the continuity of a piecewise linear map

“…The stable orbits existing in the absorbing interval are proved to be [12] organized by the period incrementing scenario. This means that after the existence region of a periodic orbit with the period p n follows the existence region of a periodic orbit with the period p n+1 = p n + 1.…”

“…The term period adding refers to the self-similar bifurcation scenario in which the periodicity regions (sometimes referred to as mode-locking tongues or Arnold's tongues) are located in the parameter space according to the Farey addition rule for the rotation numbers of the corresponding periodic orbits. More precisely, between two periodicity region associated with the rotation numbers u 1 v 1 and u 2 v 2 increasing/decreasing case in was proved in [12] that if the map is contractive on both sides of the boundary, then the point at which the function is continuous represents an organizing center of the period incrementing type, formed by the in general pairwise overlapping existence regions of the basic orbits O RL n . This result was extended in [13], where a different proof is given, showing that if the map is in the increasing/decreasing configuration, but not necessarily contractive, then the existence regions of the same basic orbits are originating from the point at which the function is continuous.…”

Organizing centers in parameter space of discontinuous 1D maps. The case of increasing/decreasing branches

“…As we shall see below, this dynamic problem is also associated with the continuity breaking in a fixed point of a piecewise smooth map, and all the possible cases can be classified depending on the shape of the two functions involved on the right/left sides of the fixed point. The partial results are given in [4], where it is shown that assuming two contracting functions, one increasing and one decreasing on the two sides of the fixed point, the BBB occurs, leading to stable cycles of any period, with periodicity regions which are overlapping in pair.…”

“…This problem is also associated with the continuity breaking in a fixed point of a piecewise smooth map. We will relax the hypothesis needed in [4] where it was proved that in the case of an increasing/decreasing contracting functions on the left/right side of a border point, at such a crossing point, we have a big-bang bifurcation, from which infinitely many border collision bifurcation curves are issuing.AMS (2000) subject classification. 37E05, 37G10, 37G35.…”

“…Ce problème est aussi associé avec la rupture de continuité en un point fixe d'une application régulière par morceaux. Nous allons relacher l'hypothèse requise dans [4], où il aété montré que dans le cas de fonctions contractantes croissantes/décroissantes strictementà gauche/droite d'un point du bord, en un tel point de franchissement, nous avons une bifurcation big-bang, de laquelle est issue une infinité de courbes de bifurcation de collision au bord.Mots clefs. applications régulières par morceaux, bifurcations de collision au bord, centres organisateurs.…”