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 non-local analysis is based on the equivariant degree method, which allows us to handle both generic and degenerate Hopf bifurcations. This is illustrated by examples.
We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.
We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-μf(x(t-1))$, as the parameter $μ$ passes through the critical values $μ_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.
The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a $Γ-$equivariant dynamical system, based on the group-theoretical aspects. In addition, it classifies the spatio temporal symmetries that are possible. By the contrary, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each $\mathbf{C}-$axial subgroup of $Γ\times\mathbb{S}^1.$ In this paper we identify which periodic solution types, whose existence is guaranteed by the $H~\mathrm{mod}~K$ theorem, are obtainable by Hopf bifurcation, when …
In a previous paper, the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like $λ^{1/2}$, $λ^{1/6}$, $λ^{1/18}$, etc. Such amplified Hopf branches were previously found by others in a subclass of feed-forward networks with three cells, first under a normal form assumption and later by ex…
In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting $Γ\times S^1$-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted $Γ\times S^1$-degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group $Γ=D_8$.
In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.
We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.
In the study of the periodic solutions of a $Γ$-equivariant dynamical system, the $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions, based on group-theoretical aspects. By contrast, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each $\mathbf{C}$-axial subgroup of $Γ\times\mathbb{S}^1$. In this article we compare the bifurcation of periodic solutions for generic differential equations equivariant under the full group of symmetries of the tetrahedron and the group of rotational symmetries of …
This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. H…
We study scalar delay equations $$\dot{x} (t) = λf(x(t-1)) + b^{-1} (x(t) + x(t -p/2))$$ with odd nonlinearity $f$, real nonzero parameters $λ, \, b$, and two positive time delays $1,\ p/2$. We assume supercritical Hopf~bifurcation from $x \equiv 0$ in the well-understood single-delay case $b = \infty$. Normalizing $f' (0)=1$, branches of constant minimal period $p_k = 2π/ω_k$ are known to bifurcate from eigenvalues $iω_k = i(k+\tfrac{1}{2})π$ at $λ_k = (-1)^{k+1}ω_k$, for any nonnegative integer $k$. The unstable dimension of these rapidly oscillating periodic solutions is $k$, at the local b…
Pyragas time delayed feedback control has proven itself as an effective tool to non-invasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings and applied to stabilize branches of small periodic solutions in systems of symmetrically coupled Landau oscillators near a Hopf bifurcation point. The form of the control ensures the non-invasiveness property, hence reducing the problem to finding a set of the gain matrices, which would guarantee the stabilization. In this paper, we apply this method to a system of Van der Pol oscillators coupl…
In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced m…
Nullcline geometry constrains the location of oscillatory instabilities in planar predator--prey systems
División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco（塔巴索自治朱阿雷斯大学基础科学分院）；Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）
