Multiple Time Scale Analysis for Bifurcation From a Double-Zero Eigenvalue

1999 ◽  
Author(s):  
Angelo Luongo ◽  
Achille Paolone ◽  
Angelo Di Egidio
Keyword(s):  
Normal Form ◽  
Center Manifold ◽  
Jacobian Matrix ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Multiple Time ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Unknown Amplitude ◽  
Normal Form Method

Abstract The multiple scale method is applied to analyze bifurcations from a double zero eigenvalue of general multiparameter dynamical systems. Due to the coalescence of the eigenvalues, the Jacobian matrix at the bifurcation is nilpotent. This entails using time scales with fractional powers of the perturbation parameter. The reconstitution method is employed lo obtain a second-order o.d.e. in the unique unknown amplitude. It coincides with Bogdanova-Arnold’s normal form for the bifurcation equation. Referring to an example, the present approach and the classical center manifold plus normal form method are compared. Finally, the mechanical behavior of a non-conservative two d.o.f. system is studied.

scholarly journals Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Stable Equilibrium ◽  
Feedback Loops ◽  
Asymptotically Stable ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
Coupling Parameters ◽  
Normal Form Method

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

PRACTICAL COMPUTATION OF NORMAL FORMS ON CENTER MANIFOLDS AT DEGENERATE BOGDANOV–TAKENS BIFURCATIONS

2005 ◽  
Vol 15 (11) ◽  
pp. 3535-3546 ◽  
Author(s):  
YU. A. KUZNETSOV
Keyword(s):  
Normal Form ◽  
Numerical Evaluation ◽  
Center Manifold ◽  
Normal Forms ◽  
Center Manifolds ◽  
Linear Transformations ◽  
Double Zero ◽  
Practical Computation ◽  
Generalized Eigenvectors

Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

CODIMENSION ONE BIFURCATION OF EQUIVARIANT NEURAL NETWORK MODEL WITH DELAY

2010 ◽  
Vol 20 (04) ◽  
pp. 1255-1259
Author(s):  
CHUNRUI ZHANG ◽  
BAODONG ZHENG
Keyword(s):  
Neural Network ◽  
Normal Form ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Double Zero ◽  
Codimension One ◽  
Bam Neural Network ◽  
Manifold Theory ◽  
Form Approach

In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.

scholarly journals Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

2021 ◽  
Vol 26 (3) ◽  
pp. 375-395
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Distribution Of Eigenvalues ◽  
Normal Form Method ◽  
The Stability

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.

COMPUTATION OF SIMPLEST NORMAL FORMS OF DIFFERENTIAL EQUATIONS ASSOCIATED WITH A DOUBLE-ZERO EIGENVALUE

2001 ◽  
Vol 11 (05) ◽  
pp. 1307-1330 ◽  
Author(s):  
Y. YUAN ◽  
P. YU
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Computational Efficiency ◽  
Normal Forms ◽  
Computer Systems ◽  
Computer Programs ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Explicit Formulae ◽  
User Friendly

In this paper a method is presented for computing the simplest normal form of differential equations associated with the singularity of a double zero eigenvalue. Based on a conventional normal form of the system, explicit formulae for both generic and nongeneric cases are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The recursive algebraic formulae have been implemented on computer systems using Maple. The user-friendly programs can be executed without any interaction. Examples are given to demonstrate the computational efficiency of the method and computer programs.

ANALYSIS OF THE TAKENS–BOGDANOV BIFURCATION ON m-PARAMETERIZED VECTOR FIELDS

2010 ◽  
Vol 20 (04) ◽  
pp. 995-1005 ◽  
Author(s):  
FRANCISCO A. CARRILLO ◽  
FERNANDO VERDUZCO ◽  
JOAQUÍN DELGADO
Keyword(s):  
Imaginary Axis ◽  
Vector Fields ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Versal Deformation ◽  
Two Dimensional ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
The Family ◽  
Parameterized Family

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

scholarly journals Synchronized Hopf Bifurcation Analysis in a Delay-Coupled Semiconductor Lasers System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Semiconductor Lasers ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Stability Switches ◽  
Center Manifold Theorem ◽  
Coupling Parameters ◽  
Normal Form Method ◽  
The Stability

The dynamics of a system of two semiconductor lasers, which are delay coupled via a passive relay within the synchronization manifold, are investigated. Depending on the coupling parameters, the system exhibits synchronized Hopf bifurcation and the stability switches as the delay varies. Employing the center manifold theorem and normal form method, an algorithm is derived for determining the Hopf bifurcation properties. Some numerical simulations are carried out to illustrate the analysis results.

Mathematical analysis of a model for thymus infection with discrete and distributed delays

2014 ◽  
Vol 07 (06) ◽  
pp. 1450070 ◽  
Author(s):  
M. Prakash ◽  
P. Balasubramaniam
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Different Types ◽  
Normal Form Method ◽  
Bifurcation And Stability ◽  
Theoretical Results ◽  
Hiv 1

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

scholarly journals BIFURCATION ANALYSIS OF DAMPED VISCO-ELASTIC PLANAR BEAMS UNDER SIMULTANEOUS GRAVITATIONAL AND FOLLOWER FORCES

2012 ◽  
Vol 26 (25) ◽  
pp. 1246015 ◽  
Author(s):  
ANGELO LUONGO ◽  
FRANCESCO D'ANNIBALE
Keyword(s):  
A Priori ◽  
Elastic Beam ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Stability Diagram ◽  
System Behavior ◽  
Dead Load ◽  
Double Zero ◽  
Scale Method ◽  
Transversal Displacement

The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.

Bogdanov–Takens Bifurcation in a Neutral Delayed Hopfield Neural Network with Bidirectional Connection

2020 ◽  
Vol 13 (06) ◽  
pp. 2050049
Author(s):  
Houssem Achouri ◽  
Chaouki Aouiti ◽  
Bassem Ben Hamed
Keyword(s):  
Neural Network ◽  
Normal Form ◽  
Characteristic Equation ◽  
Center Manifold ◽  
Hopfield Neural Network ◽  
Second Step ◽  
Double Zero ◽  
The Third ◽  
Dynamical Behaviors ◽  
Linearized System

In this paper, a neutral Hopfield neural network with bidirectional connection is considered. In the first step, by choosing the connection weights as parameters bifurcation, the critical point at which a zero root of multiplicity two occurs in the characteristic equation associated with the linearized system. In the second step, we studied the zeros of a third degree exponential polynomial in order to make sure that except the double zero root, all the other roots of the characteristic equation have real parts that are negative. Moreover, we find the critical values to guarantee the existence of the Bogdanov–Takens bifurcation. In the third step, the normal form is obtained and its dynamical behaviors are studied after the use of the reduction on the center manifold and the theory of the normal form. Furthermore, for the demonstration of our results, we have given a numerical example.

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