Guaranteed Estimates for the Length of Branches of Periodic Orbits for Equivariant Hopf Bifurcation

Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the assumption that the nonlinear terms satisfy a linear estimate in a ball. If the estimate is global, then the branch is unbounded. The results are formulated in an equivariant setting where the system can have multiple branches of periodic orbits characterized by different groups of symmetries. The nonlocal analysis is based on the equivariant degree method, which allows us to handle both generic and degenerate Hopf bifurcations. This is illustrated by examples.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

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.

Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants

We consider an intrahost malaria model allowing for antigenic variation within a single species. The host’s immune response is compartmentalised into reactions to major and minor epitopes. We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium. We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay.

Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in 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.

Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

EFFECTS OF SHORT-CUT IN A DELAYED RING NETWORK

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 Delay

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.