Dichotomy results for delay differential equations with negative Schwarzian

Hopf bifurcation and period functions for Wright type delay differential equations

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.

Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $ x'(t) = -x(t) + ζf(x(t-h)), x \in \R, f'(0)= -1, ζ> 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $ζ-1 >0$ and $h(ζ-1)^{1/8}$ are less than some constant (independent on $h,ζ$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.

Almost automorphic delayed differential equations and Lasota-Wazewska model

Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.

Linearization for difference equations with infinite delay

In this article, we construct a conjugacy map for a linear difference equation with infinite delay and corresponding nonlinear perturbation. We also prove that the conjugacy map is one-one with some additional conditions. As an application of our result, we show that the cases of (uniform) exponential dichotomy follow from our result.

Non-monotone travelling waves in a single species reaction-diffusion equation with delay

We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*)$, when $g \in C^2(R_+,R_+)$ has exactly two fixed points: $x_1= 0$ and $x_2= a >0$. Recently, non-monotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay $h$ grows. For the case of $g$ with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass type equ…

Admissible wavefront speeds for a single species reaction-diffusion equation with delay

We consider equation $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*) $, when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=κ>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(κ)) = [c_*,c^*]$ such that $(*)$ has at least one (possibly, nonmonotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here $c_*>0$ is finite and $c^* \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound $c_*$ is sharp so that there are not wavefro…

On exponential stability of linear delay equations with oscillatory coefficients and kernels

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))+\int_{g(t)}^t K(t,s) x(s)ds=0, $$ where $h_k(t)\leq t$, $g(t)\leq t$, $a_k(\cdot)$ and the kernel $K(\cdot,\cdot)$ are oscillatory and, generally, discontinuous functions. The proofs are based on establishing boundedness of solutions and later using the exponential dichotomy for linear equations stating that either the homogeneous equation is exponentially stable or a non-homogeneous equation has an unbounded solution for …

关于线性与非线性延迟微分方程指数稳定的回顾与新结果

On exponential stability of linear and nonlinear delay differential equations: a review and new results

本文回顾了线性与非线性延迟微分方程的指数稳定性准则，并提出新的统一稳定性测试方法，结合全局线性化和指数分裂定理，通过数值示例验证了非线性模型的稳定性。

基于Yosida距离的线性偏泛函微分方程无界扰动

Unbounded Perturbation of Linear Partial Functional Differential Equations via Yosida Distance

该研究针对带时滞的线性偏泛函微分方程无界扰动问题，引入Yosida距离度量扰动大小，通过相空间解半群生成元的Yosida距离估计，证明原方程具指数二分性且扰动足够小时，扰动方程也具指数二分性。