Search

We consider a scalar delay differential equation x ˙ (t) = - d x (t) + f ((1 - α) ρ x (t - τ) + α ρ x (t - 2 τ)) with an instant mortality rate d > 0 , the nonlinear Rick reproductive function f, a survival rate during all development stages ρ , and a proportion constant α ∈ [ 0 , 1 ] with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within (3 τ , 6 τ) for a wide range of parameter values. We show this existence of periodic solutions not only for the delay τ near the first critical value τ ∗ when a local Hopf bifurcation takes place near the positive equilibrium, but for all τ > τ ∗ . We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period 3 τ . We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium. [ABSTRACT FROM AUTHOR]

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the S 1 -equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a time-delayed model of microorganism flocculation with saturated functional responses is presented. We first analyse the local dynamics of this model with bifurcations in parameter fields, and then prove the collection of microorganisms is sustainable as well as obtain an explicit eventual lower bound of microorganism concentration when threshold parameter R0>1. This model has a backward bifurcation if w under an additional condition, which implies that the microorganism-free equilibrium coexists with a microorganism equilibrium. In these cases, we establish some sufficient conditions for the global stability by using a variant of the Lyapunov-LaSalle theorem. [ABSTRACT FROM AUTHOR]

We embark on a case study for the scalar delay equation with odd nonlinearity f, real nonzero parameters $$\lambda , \, b$$ , and three positive time delays $$1,\, \vartheta ,\, p/2$$ . 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 $$p_k = 2\pi /\omega _k$$ are known to bifurcate from eigenvalues $$i\omega _k = i(k+\tfrac{1}{2})\pi $$ at $$\lambda _k = (-1)^{k+1}\omega _k$$ , for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For $$p:= p_k$$ the branch k of constant period $$p_k$$ persists as a solution, for any $$b\ne 0$$ and $$\vartheta \ge 0$$ . Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions $$\mathcal {P}$$ of controls $$b \ne 0$$ and delays $$\vartheta \ge 0$$ such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for $$\mathcal {P}$$ in the limit of large k. The only two regions $$\mathcal {P} = \mathcal {P}^\pm $$ which we were able to detect, in this setting, required delays $$\vartheta $$ near 1, controls b near $$(-1)^k \cdot 2/\omega _k$$ , and were of very small area of order $$k^{-4}$$ . Our analysis is based on a 2-scale covering lift for the frequencies involved. [ABSTRACT FROM AUTHOR]

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent DDE is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete characterisation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches. [ABSTRACT FROM AUTHOR]

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of 'classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold. [ABSTRACT FROM AUTHOR]

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions. [ABSTRACT FROM AUTHOR]