Related papers: Hopf bifurcation and period functions for Wright t…
We investigate the scalar autonomous equation with two discrete delays $$ \dot{x}(t)=f(x(t),x(t-r),x(t-\sigma)), $$ where $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is a continuously differentiable non-linear function such that $f(0,0,0)=0$. It…
We discuss the occurrence of Poincar\'e-Andronov-Hopf bifurcations in parameter dependent ordinary differential equations, with no a priori assumptions on special coordinates. The first problem is to determine critical parameter values from…
We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one…
We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x…
Wright's conjecture states that the origin is the global attractor for the delay differential equation $y'(t) = - \alpha y(t-1) [ 1 + y(t) ] $ for all $\alpha \in (0,\tfrac{\pi}{2}]$. This has been proven to be true for a subset of…
We apply the synergetic elimination procedure for the stable modes in nonlinear delay systems close to a dynamical instability and derive the normal form for the delay-induced Hopf bifurcation in the Wright equation. The resulting periodic…
We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that…
It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and…
Circular domains frequently appear in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a…
The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation…
This paper continues the work contained in two previous papers, devoted to the study of the dynamical system generated by a delay differential equation that models leukemia. Here our aim is to identify degenerate Hopf bifurcation points. By…
The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased.…
We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto-Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive…
We study scalar delay equations $$\dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t) + x(t -p/2))$$ with odd nonlinearity $f$, real nonzero parameters $\lambda, \, b$, and two positive time delays $1,\ p/2$. We assume supercritical…
This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady…
The bifurcation diagram of a model nonlinear Langevin equation with delayed feedback is obtained numerically. We observe both direct and oscillatory bifurcations in different ranges of model parameters. Below threshold, the stationary…
In this paper we investigate the impact of delayed tax revenues on the fiscal policy out-comes. Choosing the delay as a bifurcation parameter we study the direction and the stability of the bifurcating periodic solutions. With respect to…
In a previous work we investigated the existence of Hopf degenerate bifurcation points for a differential delay equation modeling leukemia and we actually found Hopf points of codimension two for the considered problem. If around the…
The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure…
The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form…