New Elements of Analysis of a Degenerate Chenciner Bifurcation

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.

Normal Form of Bifurcation for Caputo-Hadamard Fractional Differential System With a Parameter

This paper focuses on the normal form computation of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter. By using Taylor’s expansion and Implicit Function Theorem, we derive the normal form of the pitchfork bifurcation for the Caputo-Hadamard fractional differential system with a parameter.

Existence of Impact Period-1 Orbit After Grazing Bifurcation for a Soft Impacting System

In the paper, the theoretical study on some experimental and numerical results of grazing bifurcation for a soft impacting system are analyzed. After the conditions under which nonimpact period-1 orbit and grazing bifurcation exist are given, we prove that an impact period-1 orbit exists by the implicit function theorem. The details of these periodic orbits are also investigated, which also helps us numerically find them. The method in this paper is still efficient for the multiperiodic orbits with single impact.