scholarly journals New Elements of Analysis of a Degenerate Chenciner Bifurcation

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 77
Author(s):  
Sorin Lugojan ◽  
Loredana Ciurdariu ◽  
Eugenia Grecu
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Bifurcation Diagrams ◽  
Classical Transformation ◽  
Independent Parameters ◽  
Discrete Time Dynamical Systems

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.

Analysis of Degenerate Chenciner Bifurcation

2020 ◽  
Vol 30 (16) ◽  
pp. 2050245
Author(s):  
G. Tigan ◽  
S. Lugojan ◽  
L. Ciurdariu
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Bifurcation Diagrams ◽  
Discrete Time Dynamical Systems

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.

Degenerate Chenciner Bifurcation Revisited

2021 ◽  
Vol 31 (10) ◽  
pp. 2150160
Author(s):  
G. Tigan ◽  
O. Brandibur ◽  
E. A. Kokovics ◽  
L. F. Vesa
Keyword(s):  
Discrete Time ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Independent Parameters

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.

CHAOTIC DYNAMICS FOR MAPS IN ONE AND TWO DIMENSIONS: A GEOMETRICAL METHOD AND APPLICATIONS TO ECONOMICS

2009 ◽  
Vol 19 (10) ◽  
pp. 3283-3309 ◽  
Author(s):  
ALFREDO MEDIO ◽  
MARINA PIREDDU ◽  
FABIO ZANOLIN
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Dynamics ◽  
Economic Models ◽  
Two Dimensions ◽  
Geometrical Method ◽  
Mathematical Ideas ◽  
Definition Of ◽  
Applications To Economics ◽  
Discrete Time Dynamical Systems

This article describes a method — called here "the method of Stretching Along the Paths" (SAP) — to prove the existence of chaotic sets in discrete-time dynamical systems. The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications. The paper provides a description of the basic mathematical ideas behind the method, as well as three applications to economic models. Incidentally, the paper also discusses some questions concerning the definition of chaos and some problems arising from economic models in which the dynamics are defined only implicitly.

Large-Scale Discrete-Time Interconnected Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Large Scale ◽  
Discrete Energy ◽  
Flow Balance ◽  
Subsystem Level ◽  
Coupling Strengths ◽  
Definition Of ◽  
Energy Dissipated ◽  
Discrete Time Dynamical Systems

This chapter develops vector dissipativity notions for large-scale nonlinear discrete-time dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear discrete-time dynamical systems in terms of a vector dissipation inequality involving a vector supply rate, a vector storage function, and a nonnegative, semistable dissipation matrix. On the subsystem level, the proposed approach provides a discrete energy flow balance in terms of the stored subsystem energy, the supplied subsystem energy, the subsystem energy gained from all other subsystems independent of the subsystem coupling strengths, and the subsystem energy dissipated. The chapter also develops extended Kalman–Yakubovich–Popov conditions, in terms of the local subsystem dynamics and the interconnection constraints, for characterizing vector dissipativeness via vector storage functions for large-scale discrete-time dynamical systems.

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