scholarly journals Analytic Normal forms and inverse problems for unfoldings of 2-dimensional saddle-nodes with analytic center manifold

2021 ◽  
Vol 54 (1) ◽  
pp. 133-233
Author(s):  
Christiane Rousseau ◽  
Loïc Teyssier
Keyword(s):  
Inverse Problems ◽  
Center Manifold ◽  
Normal Forms ◽  
Analytic Center

scholarly journals On the Computation of Degenerate Hopf Bifurcations forn-Dimensional Multiparameter Vector Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Michail P. Markakis ◽  
Panagiotis S. Douris
Keyword(s):  
Vector Fields ◽  
Center Manifold ◽  
Normal Forms ◽  
Computer Assisted ◽  
Hopf Bifurcations ◽  
Symbolic Form ◽  
Parametric System ◽  
Analytical Expressions

The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

Bifurcation Analysis of a Diffusive Activator–Inhibitor Model in Vascular Mesenchymal Cells

2015 ◽  
Vol 25 (10) ◽  
pp. 1530026 ◽  
Author(s):  
Rui Yang ◽  
Yongli Song
Keyword(s):  
Steady State ◽  
Center Manifold ◽  
Normal Forms ◽  
Mesenchymal Cells ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Form Theory ◽  
Spatially Homogeneous ◽  
The Stability ◽  
Activator Inhibitor

In this paper, a diffusive activator–inhibitor model in vascular mesenchymal cells is considered. On one hand, we investigate the stability of the equilibria of the system without diffusion. On the other hand, for the unique positive equilibrium of the system with diffusion the conditions ensuring stability, existence of Hopf and steady state bifurcations are given. By applying the center manifold and normal form theory, the normal forms corresponding to Hopf bifurcation and steady state bifurcation are derived explicitly. Numerical simulations are employed to illustrate where the spatially homogeneous and nonhomogeneous periodic solutions and the steady states can emerge. The numerical results verify the obtained theoretical conclusions.

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.

CONDITIONS FOR APPEARANCE AND DISAPPEARANCE OF LIMIT CYCLES IN THE SHIMIZU–MORIOKA SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2489-2503
Author(s):  
LINGLING LIU ◽  
BO GAO
Keyword(s):  
Numerical Simulations ◽  
Limit Cycles ◽  
Center Manifold ◽  
Normal Forms ◽  
Hopf Bifurcations ◽  
Algebraic Varieties ◽  
System A ◽  
Parameter Dependent ◽  
Generalized Lorenz Canonical Form ◽  
Two Equilibria

This paper deals with the Shimizu–Morioka system, a special generalized Lorenz canonical form. Using techniques of elimination in the computation of algebraic varieties we obtain parameter-dependent normal forms on a center manifold. Our computation shows that the maximal number of limit cycles produced from Hopf bifurcations is four and only even number of limit cycles can be bifurcated near the two equilibria because of [Formula: see text]-symmetry. Our parameter-dependent normal forms enable us to give parameter conditions for the cases of none, two and four limit cycles separately. Furthermore, considering exterior perturbations, we give conditions under which one or three limit cycles can be produced from Hopf bifurcations. Moreover, we also give conditions for fold bifurcations, under which limit cycles coincide or disappear. Finally, our results are illustrated by numerical simulations.

Computation of Normal Forms of Differential Equations Associated with Non-Semisimple Zero Eigenvalues

1998 ◽  
Vol 08 (12) ◽  
pp. 2279-2319 ◽  
Author(s):  
Q. Bi ◽  
P. Yu
Keyword(s):  
Differential Equations ◽  
Center Manifold ◽  
Normal Forms ◽  
Specific Problem ◽  
Dimensional System ◽  
Center Manifold Theory ◽  
Zero Eigenvalue ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.

scholarly journals Birkhoff normal forms in semi-classical inverse problems

2002 ◽  
Vol 9 (3) ◽  
pp. 337-362 ◽  
Author(s):  
A. Iantchenko ◽  
J. Sjöstrand ◽  
M. Zworski
Keyword(s):  
Inverse Problems ◽  
Normal Forms

scholarly journals Simple-Zero and Double-Zero Singularities of a Kaldor-Kalecki Model of Business Cycles with Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-29 ◽  
Author(s):  
Xiaoqin P. Wu
Keyword(s):  
Business Cycles ◽  
Center Manifold ◽  
Normal Forms ◽  
Simple Zero ◽  
Bifurcation Diagrams ◽  
Double Zero ◽  
Center Manifold Reduction ◽  
Manifold Reduction ◽  
Theoretical Results ◽  
Double Limit

We study the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. Simple-zero and double-zero singularities are investigated when bifurcation parameters change near certain critical values. By performing center manifold reduction, the normal forms on the center manifold are derived to obtain the bifurcation diagrams of the model such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.

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