New Elements of Analysis of a Degenerate Chenciner Bifurcation

Abstract

A new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters (α1,α2)→(β1,β2) is not longer regular at (0,0); therefore, implicit function theorem (IFT) cannot be applied around the origin, and a new transformation is necessary. The approach in this article to a case of Chenciner bifurcation is theoretical, but it can provide an answer for a number of applications of dynamical systems. We studied the bifurcation scenario and found out that, by this transformation, four different bifurcation diagrams are obtained, and the non-degenerate Chenciner bifurcation can be described by two bifurcation diagrams.

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…The non-degenerate Chenciner bifurcation was firstly studied in the papers [6,13,14]. More recently this bifurcation appears in many papers from different areas of research, in "biology, physics, economy, informatics" [15] as well as multidisciplinary and applied sciences [12,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. For example, in [31], the Chenciner bifurcation was observed when a potential mechanism from bifurcation analyses was used for studying the occurrence of modulated oscillations in synchronous machine nonlinear dynamics, being reported for the first time in power engineering for this bifurcation.…”

“…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.…”

“…A parallel approach to that of [4] is studied in [35] by using another regular transformation of parameters, where the product a 10 a 01 b 10 b 01 = 0. In the article [15], the functions β 1 (α) and β 2 (α) have a 10 = 0; a 01 = 0; b 10 = 0; b 01 = 0, obtaining four bifurcation diagrams. 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].…”

“…A possible application of the degenerated Chenciner bifurcation was presented in [15], but one could analyze in all previous mentioned Chenciner papers what happens when degeneration occurs. For example, the volatility of the economics systems based on discrete Chenciner bifurcation may be interpreted as a variant of input data implying the degeneration of the bifurcation.…”