Hopf Bifurcation for Retarded Functional Differential Equations and for Semiflows in Banach Spaces

2013 ◽  
Vol 25 (4) ◽  
pp. 1159-1199 ◽  
Author(s):  
Bernhard Lani-Wayda
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Banach Spaces ◽  
Functional Differential Equations ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

Normal Form of Saddle-Node-Hopf Bifurcation in Retarded Functional Differential Equations and Applications

2016 ◽  
Vol 26 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Heping Jiang ◽  
Jiao Jiang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Predator Prey Model ◽  
Saddle Node ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhães to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie–Gower predator–prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.

On Semigroups in Rn X Lp Corresponding to Differential Equations with Delays

1978 ◽  
Vol 30 (5) ◽  
pp. 897-914 ◽  
Author(s):  
C. Bernier ◽  
A. Manitius
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Banach Spaces ◽  
Functional Differential Equations ◽  
Retarded Functional Differential Equations ◽  
Functional Differential

In this paper we study some properties of the semigroups associated with the linear retarded functional differential equations (FDE) in the setting of Banach spaces Rn X Lp( — h, 0, Rn), 1 < p < ∞. Earlier investigations of these equations via semigroups defined on the customary space C([ — h, 0], Rn) played an important role in problems of stability, oscillations, bifurcation, asymptotic behavior etc. [15].

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