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.

The k-Zero Bifurcation Theorem for m-Parameterized Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1850100
Author(s):  
Martin A. Carrillo ◽  
Fernando Verduzco ◽  
Francisco A. Carrillo
Keyword(s):  
Equilibrium Point ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Geometric Multiplicity ◽  
Parameterized Systems ◽  
Multiplicity One ◽  
The Family ◽  
Parameterized Family ◽  
Multiplicity Formula ◽  
Eigenvalue Zero

Given an [Formula: see text]-parameterized family of [Formula: see text]-dimensional vector fields, with an equilibrium point with linearization of eigenvalue zero with algebraic multiplicity [Formula: see text], with [Formula: see text], and geometric multiplicity one, our goal in this paper is to find sufficient conditions for the family of vector fields such that the dynamics on the [Formula: see text]-dimensional [Formula: see text]-parameterized center manifold around the equilibrium point becomes locally topologically equivalent to a given unfolding. Finally, the result is applied to the study of the Rössler system.

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.

CONTROL OF CODIMENSION ONE STATIONARY BIFURCATIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 575-582 ◽  
Author(s):  
FERNANDO VERDUZCO
Keyword(s):  
Control Systems ◽  
Vector Fields ◽  
Center Manifold ◽  
Control Law ◽  
Center Manifold Theory ◽  
Zero Eigenvalue ◽  
Codimension One ◽  
Pitchfork Bifurcations ◽  
Manifold Theory ◽  
Dimension One

The control of the saddle-node, transcritical and pitchfork bifurcations are analyzed in nonlinear control systems with one zero eigenvalue. It is shown that, provided some conditions on the vector fields are satisfied, it is possible to design a control law such that the bifurcation properties can be modified in some desirable way. To simplify the analysis to dimension one, the center manifold theory is used.

Codimension 2 Symmetric Homoclinic Bifurcations and Application to 1:2 Resonance

1990 ◽  
Vol 42 (2) ◽  
pp. 191-212 ◽  
Author(s):  
Chengzhi Li ◽  
Christiane Rousseau
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Hopf Bifurcation ◽  
Vector Fields ◽  
Jordan Block ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Homoclinic Bifurcations ◽  
Image Position ◽  
Flow Map

In this paper we study a codimension 3 form of the 1:2 resonance. It was first noted by Arnold [3] that the study of bifurcations of symmetric vector fields under a rotation of order q yields information about Hopf bifurcation for a fixed point of a planar diffeomorphism F with eigenvalues . The map Fq can be identified to arbitrarily high order with the flow map of a symmetric vector field having a double-zero eigenvalue ([3], [4], [10], [23]). The resonance of order 2 (also called 1:2 resonance) considered here is the case of a pair of eigenvalues —1 with a Jordan block of order 2. The diffeomorphism then has normal form around the origin given by [4]:

scholarly journals On Convolution and Convex Combination of Harmonic Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ahmad Sulaiman Ahmad El-Faqeer ◽  
Zhen Chuan Ng ◽  
Shamani Supramaniam
Keyword(s):  
Unit Disk ◽  
Imaginary Axis ◽  
Convex Combination ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
The Family ◽  
Univalent Harmonic Mappings ◽  
Adequate Condition

In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

scholarly journals Liftable vector fields, unfoldings and augmentations

2021 ◽  
Author(s):  
J. J. Nuño-Ballesteros ◽  
R. Oset Sinha
Keyword(s):  
Vector Fields ◽  
The Family ◽  
Singularity Type ◽  
Smooth Map ◽  
Finite Singularity

AbstractWe study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family $$H_k$$ H k in Mond’s list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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