scholarly journals Attractiveness and Hopf bifurcation for retarded differential equations

2003 ◽  
Vol 2 (2) ◽  
pp. 147-158 ◽  
Author(s):  
R. Ouifki ◽  
◽  
M. L. Hbid ◽  
O. Arino ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Retarded Differential Equations

scholarly journals Pseudospectral Approximation of Hopf Bifurcation for Delay Differential Equations

2021 ◽  
Vol 20 (1) ◽  
pp. 333-370
Author(s):  
B. A. J. de Wolff ◽  
F. Scarabel ◽  
S. M. Verduyn Lunel ◽  
O. Diekmann
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Delay Differential

Hopf Bifurcation for Implicit Neutral Functional Differential Equations

1993 ◽  
Vol 36 (3) ◽  
pp. 286-295 ◽  
Author(s):  
Tomasz Kaczynski ◽  
Huaxing Xia
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equations ◽  
Bifurcation Theorem ◽  
Functional Differential

AbstractAn analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

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