Equivariant Hopf bifurcation with general pressure laws

2015 ◽  
Vol 310 ◽  
pp. 79-94 ◽  
Author(s):  
Tong Li ◽  
Jinghua Yao
Keyword(s):  
Hopf Bifurcation ◽  
Equivariant Hopf Bifurcation

Reversible Equivariant Hopf Bifurcation

2004 ◽  
Vol 175 (1) ◽  
pp. 39-84 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Jeroen S.W. Lamb
Keyword(s):  
Hopf Bifurcation ◽  
Equivariant Hopf Bifurcation

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

2020 ◽  
Vol 30 (09) ◽  
pp. 2050130 ◽  
Author(s):  
Shangzhi Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Banach Spaces ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delay Effect ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Equivariant Hopf Bifurcation ◽  
Manifold Reduction

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

An algebraic criterion for symmetric Hopf bifurcation

1993 ◽  
Vol 440 (1910) ◽  
pp. 727-732 ◽  
Keyword(s):  
Fixed Point ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Spherical Harmonics ◽  
Irreducible Representations ◽  
Two Dimensional ◽  
Bifurcation Theorem ◽  
Algebraic Criterion ◽  
Equivariant Hopf Bifurcation

The equivariant Hopf bifurcation theorem states that bifurcating branches of periodic solutions with certain symmetries exist when the fixed-point subspace of that subgroup of symmetries is two dimensional. We show that there is a group-theoretic restriction on the subgroup of symmetries in order for that subgroup to have a two-dimensional fixed-point subspace in any representation. We illustrate this technique for all irreducible representations of SO(3) on the space V l of spherical harmonics for l even.

EFFECTS OF SHORT-CUT IN A DELAYED RING NETWORK

2012 ◽  
Vol 22 (03) ◽  
pp. 1250054 ◽  
Author(s):  
JUANJUAN MAN ◽  
SHANGJIANG GUO ◽  
YIGANG HE
Keyword(s):  
Hopf Bifurcation ◽  
Ring Network ◽  
Distribution Of Zeros ◽  
Bifurcation Theorem ◽  
Trivial Equilibrium ◽  
Secondary Bifurcation ◽  
Steady State Bifurcation ◽  
Spatio Temporal ◽  
Equivariant Hopf Bifurcation ◽  
Absolute Synchronization

This paper presents a detailed analysis on the dynamics of a ring network with short-cut. We first investigate the absolute synchronization on the basis of Lyapunov stability approach, and then discuss the linear stability of the trivial solution by analyzing the distribution of zeros of the characteristic equation. Based on the equivariant branching lemma, we not only obtain the existence of primary steady state bifurcation but also analyze the patterns and stability of the bifurcated nontrivial equilibria. Moreover, by means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of connection strength on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. In particular, we further consider the secondary bifurcation of the nontrivial equilibria. These studies show that short-cut may be used as a simple but efficient switch to control the dynamics of a system.

Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling

Nonlinearity ◽  
2005 ◽  
Vol 18 (6) ◽  
pp. 2827-2846 ◽  
Author(s):  
Sue Ann Campbell ◽  
Yuan Yuan ◽  
Sharene D Bungay
Keyword(s):  
Hopf Bifurcation ◽  
Delayed Coupling ◽  
Equivariant Hopf Bifurcation

scholarly journals Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120472 ◽  
Author(s):  
Isabelle Schneider
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Non Invasive ◽  
Selective Stabilization ◽  
Spatio Temporal ◽  
The Individual ◽  
Equivariant Hopf Bifurcation

The modest aim of this case study is the non-invasive and pattern-selective stabilization of discrete rotating waves (‘ponies on a merry-go-round’) in a triangle of diffusively coupled Stuart–Landau oscillators. We work in a setting of symmetry-breaking equivariant Hopf bifurcation. Stabilization is achieved by delayed feedback control of Pyragas type, adapted to the selected spatio-temporal symmetry pattern. Pyragas controllability depends on the parameters for the diffusion coupling, the complex control amplitude and phase, the uncontrolled super-/sub-criticality of the individual oscillators and their soft/hard spring characteristics. We mathematically derive explicit conditions for Pyragas control to succeed.

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