CHARACTERIZATION OF DYNAMIC BIFURCATIONS IN THE FREQUENCY DOMAIN

2002 ◽  
Vol 12 (01) ◽  
pp. 87-101 ◽  
Author(s):  
GRISELDA R. ITOVICH ◽  
JORGE L. MOIOLA
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Frequency Domain ◽  
Building Blocks ◽  
Hopf Bifurcations ◽  
Frequency Domain Approach ◽  
Dynamic Bifurcations ◽  
Bifurcation Behavior

In this paper dynamical systems with certain degenerate Hopf bifurcations are considered. An analysis of the bifurcation behavior is proposed using several tools from the frequency domain approach. The analyzed bifurcations are the building blocks to understand the multiplicity of Hopf bifurcation points and to propose certain strategies in the future for controlling the bifurcation behavior in nonlinear systems.

CHARACTERIZATION OF STATIC BIFURCATIONS FOR MAPS IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (03) ◽  
pp. 975-983 ◽  
Author(s):  
LI ZENG ◽  
YI ZHAO
Keyword(s):  
Frequency Domain ◽  
Building Blocks ◽  
Complex Singularities ◽  
Smooth Maps ◽  
Continuous Time Systems ◽  
Nonlinear Maps ◽  
Frequency Domain Approach ◽  
Time Systems ◽  
Bifurcation Behavior

In this paper n-dimensional discrete-time systems with static bifurcations are considered from the viewpoint of control theory. This paper presents an adaption of available formulas for bifurcation analysis in two-dimensional continuous-time systems to the case of smooth maps using a frequency domain approach. The analyzed bifurcations are the building blocks to understand other more complex singularities and to propose certain methods for controlling the bifurcation behavior in nonlinear maps in the future.

CHARACTERIZATION OF STATIC BIFURCATIONS IN THE FREQUENCY DOMAIN

2001 ◽  
Vol 11 (03) ◽  
pp. 677-688 ◽  
Author(s):  
GRISELDA R. ITOVICH ◽  
JORGE L. MOIOLA
Keyword(s):  
Frequency Domain ◽  
Building Blocks ◽  
Two Dimensional ◽  
Complex Singularities ◽  
Frequency Domain Approach ◽  
Bifurcation Behavior

In this paper two-dimensional systems with static bifurcations are considered. An analysis of the bifurcation behavior is proposed using a frequency domain approach. The analyzed bifurcations are known as elementary since they are the building blocks to understand other more complex singularities.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

Frequency-Domain Approach to Hopf Bifurcation Analysis

10.1142/11418 ◽  
2019 ◽  
Author(s):  
Franco Sebastián Gentile ◽  
Jorge Luis Moiola ◽  
Guanrong Chen
Keyword(s):  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Bifurcation Analysis ◽  
Frequency Domain Approach

Complex Networks Approach for Dynamical Characterization of Nonlinear Systems

2019 ◽  
Vol 29 (13) ◽  
pp. 1950188 ◽  
Author(s):  
Vander L. S. Freitas ◽  
Juliana C. Lacerda ◽  
Elbert E. N. Macau
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Networks ◽  
Discrete Time ◽  
Betweenness Centrality ◽  
Bifurcation Diagrams ◽  
Computationally Expensive ◽  
Nonlinear Maps

Bifurcation diagrams and Lyapunov exponents are the main tools for dynamical systems characterization. However, they are often computationally expensive and complex to calculate. We present two approaches for dynamical characterization of nonlinear systems via the generation of an undirected complex network that is built from their time series. Periodic windows and chaos can be detected by analyzing network statistics like average degree, density and betweenness centrality. Results are assessed in two discrete time nonlinear maps.

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