scholarly journals 3-dimensional Hopf bifurcation via averaging theory

2007 ◽  
Vol 17 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Claudio A. Buzzi ◽  
Paulo R. da Silva ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
3 Dimensional

scholarly journals $3$ - dimensional Hopf bifurcation via averaging theory of second order

2009 ◽  
Vol 25 (4) ◽  
pp. 1287-1295 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Amar Makhlouf ◽  
Sabrina Badi ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Second Order ◽  
Averaging Theory ◽  
3 Dimensional

scholarly journals On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1137
Author(s):  
Maoan Han ◽  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Equilibrium Points ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
3 Dimensional ◽  
Volterra Systems

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

scholarly journals On the 3-dimensional Hopf bifurcation via averaging theory of third order

2017 ◽  
Vol 41 ◽  
pp. 1053-1071
Author(s):  
Elouahma BENDIB ◽  
Sabrina BADI ◽  
Amar MAKHLOUF
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
Third Order ◽  
3 Dimensional

scholarly journals Degenerate Fold–Hopf Bifurcations in a Rössler-Type System

2017 ◽  
Vol 27 (05) ◽  
pp. 1750068 ◽  
Author(s):  
G. Tigan ◽  
J. Llibre ◽  
L. Ciurdariu
Keyword(s):  
Hopf Bifurcation ◽  
Differential System ◽  
Type System ◽  
Averaging Theory ◽  
Hopf Bifurcations

We study the Hopf and the fold–Hopf bifurcations of the Rössler-type differential system [Formula: see text] with [Formula: see text]. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold–Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium [Formula: see text] when [Formula: see text]. This Hopf bifurcation becomes a fold–Hopf bifurcation when [Formula: see text].

scholarly journals PERIODIC ORBITS NEAR EQUILIBRIA VIA AVERAGING THEORY OF SECOND ORDER

2012 ◽  
Vol 17 (5) ◽  
pp. 715-731
Author(s):  
Luis Barreira ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Periodic Orbits ◽  
Differential System ◽  
Sufficient Conditions ◽  
First Integral ◽  
Second Order ◽  
Averaging Theory ◽  
First Order

Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation.

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