Analysis of Degenerate Chenciner Bifurcation

Abstract

Degenerate Chenciner bifurcation in generic discrete-time dynamical systems is studied in this work. While the nondegenerate Chenciner bifurcation can be described by two bifurcation diagrams, the degeneracy we studied in this work gives rise to 32 different bifurcation diagrams.

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…Continuous and discrete-time dynamical systems can be used for modeling many applications in the surrounding world [1][2][3]. Discrete dynamical systems may appear in "practical applications when a phenomenon cannot be observed continuously in time" [4], but in certain moments of time [5]. Additionally, they can be obtained from dynamic systems with continuous time by discretizing time, that is, if we only take certain values for time [6] or as return maps that are return applications defined by the intersections of the system flows with certain "surfaces transversal to the flows" [4].…”

“…That results in a much more difficult scenario. 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].…”

“…In [4], the bifurcation diagrams were discovered in a general case, where the functions β 1 (α) and β 2 (α) both have linear terms different from zero that satisfy the degeneracy condition a 10 b 01 − a 01 b 10 = 0, or a 10 = a 01 = 0, see (1). In that case, 32 bifurcation diagrams were obtained.…”

“…In that case, 32 bifurcation diagrams were obtained. 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.…”

“…[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].…”