Chenciner Bifurcation Presenting a Further Degree of Degeneration

Abstract

Chenciner bifurcation appears for some two-dimensional systems with discrete time having two independent variables. Investigated here is a special case of degeneration where the implicit function theorem cannot be used around the origin, so a new approach is necessary. In this scenario, there are many more bifurcation diagrams than in the two non-degenerated cases. Several numerical simulations are presented.

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…In this article, all eight regions corresponding to the eight phase portraits (see Figure A1) appear in the bifurcation diagrams, unlike [15] or [10], where all of these are not present. In [15], only regions 1-4 appear in the bifurcation diagrams.…”

“…As it is not possible to choose new coordinates β 1 , β 2 , the idea is to use only the initial parameters (α 1 , α 2 ). This leads to the modifications of the structure of the sets of points B 1,2 and C, thus obtaining concurrent lines at the origin, similar to the situation analyzed in other articles [10,15], but different from the cases studied in [4,35]. We want to specify how many bifurcation diagrams are obtained, many or few.…”

“…From a computational point of view, the use of dynamical systems with discrete time is more efficient in modeling because it can capture complex behaviors that cannot be easily captured otherwise [7][8][9]. Among the most "important topics in the qualitative theory" [10] of continuous and discrete dynamic systems is the analysis of bifurcations (see [11]).…”

“…A first type of such a degenerated Chenciner discrete dynamical was solved in [4]. Two other types of possible degeneration were studied in [10,15]. Each of those cases has a quite different method of solving.…”

“…Ref. [10] studied the case when a 20 = a 11 = a 02 = 0 and b 10 = 0, b 01 = 0 or a 20 = a 11 = a 02 = 0 and b 10 = 0, b 01 = 0, obtaining 18 different bifurcation diagrams. The stability of the fixed point O for |α| that is sufficiently small and, respectively, "the existence of closed invariant curves in the" [4] truncated normal form in all the cases was treated before [10,15,35].…”