Renormalization group approach to a class of singularly perturbed delay differential equations

Highlights

• We present a systematic RG approach to deal with the singularly perturbed DDEs.

• We complete the entire task by unifying the piecewise RG equations into a delay one.

• By the numerical simulations, the uniformly valid of our result be proved.

Abstract

In this paper, we present a systematic renormalization group method to investigate a class of singularly perturbed delay differential equations. The uniformly valid approximate solution can be obtained, and we give a rigorous proof of the error estimate of the approximate solution. In addition, some numerical comparisons between the exact solution, the result by the averaging method and our result are given for two examples.

Introduction

Recently, a considerable number of researches have focused on singularly perturbed delay differential equations (DDEs). This is due to the versatility of such types of differential equations in the mathematical modeling of processes in various application fields, such as the first exit time problem in the modeling of the activation of neuronal variability [1], the optically bistable devices [2], a variety of models for physiological processes or diseases [3], the human pupil-light reflex [4].

It is known that most of DDEs cannot be solved exactly, hence, constructing the approximate solutions has gained substantial attention from researchers in different fields. This is especially true for singularly perturbed DDEs, and many methods have been proposed. Utilizing the WKB method and matched asymptotic expansion method, Lange and Miura [1], [5], [6] obtained the approximate solutions of boundary value problems of the second-order singularly perturbed DDEs in which the highest order derivative is multiplied by a small parameter. By the method of boundary functions and fractional steps, Wang et al. constructed the asymptotic expansion solution of the second-order nonlinear singularly perturbed DDEs with the interior layer in [7], and obtained the impulsive solution of the semi-linear singularly perturbed DDEs in [8]. Recently, Ni et al. [9] investigated the asymptotic approximation of the singularly perturbed second-order DDEs with boundary value condition by the Vasil’eva’s method. Zhao et al. used the variational iteration method to solve singular perturbation initial value problems (IVPs) with delays in [10]. Hu and Wang [11] not only found an approximation of the bifurcated periodic solution, but also determined the stability of the periodic solution of the time delay systems by the multiple scales method.

In [12], Lehman and Weibel extended the averaging method to singularly perturbed DDEs as follows

$$\dot{x}\left(t\right)=\varepsilon f(t,x_t),x_{t_0}=\varphi ,$$

where $f:\mathbb{R}\times ℂ\to {\mathbb{R}}^n$ is a KBM functional, $ℂ=ℂ([-r,0],{\mathbb{R}}^n),r\ge 0$, denoting the space of continuous functions that map $[-r,0]$ into ${\mathbb{R}}^n$, $\varphi$ is a continuous function on $[t_0-r,t_0]$, and $x_t\left(\theta \right)=x(t+\theta ),\theta \in [-r,0]$ for $t\in [t_0,L]$, $L>t_0$. By applying the averaging method to (1), they obtained two averaged models. One is the averaged model without delay,

$$\dot{\stackrel{̃}{x}}\left(t\right)=\varepsilon F_{av}\left(\tilde{x}\left(t\right)\right),\tilde{x}\left(t_0\right)=\varphi \left(t_0\right),$$

where

$$F_{av}\left(\psi \right)=\underset{T\to \infty }{lim}\frac{1}{T}{\int }_0^{t}f(s,\psi )ds,T>0,$$

and the other is the averaged model with delay,

$$\dot{\stackrel{ˇ}{x}}\left(t\right)=\varepsilon F_{av}\left({\check{x}}_t\right),{\check{x}}_{t_0}=\varphi ,$$

where $F_{av}\left({\check{x}}_t\right)$ is given by (3). Their results indicated that the averaged model (4) has a better approximation to the solution of Eq. (1) than the averaged model (2).

It is worthy pointing out that the renormalization group (RG) method can also be applied to deal with the singularly perturbed DDEs. The RG method proposed by Chen et al. [13], [14] is one of the perturbation methods for obtaining the approximate solutions of the singularly perturbed ordinary differential equations, and there has been a variety of effective applications, such as envelope theory [15], invariant manifolds [16], [17], center manifolds [18], normal forms [19], limit cycle [20] and so on. Shin-itiro Goto [21] proposed the reformulated renormalization method to weakly nonlinear systems with delay, and obtained the reduced equations with delay from the systems with large or order-one delay. In [22], Wu and Xu constructed the first order approximate solution and utilized the RG equation to analyze the stability regions of the delayed Mathieu equation. In [23], the authors investigated the approximate solution of a system of the singularly perturbed DDE with boundary layer condition by the RG method. However, there are few new and systematic results despite of the above cases. The goal of this paper is to present a systematic RG method to solve singularly perturbed DDEs.

In this paper, we mainly consider the following DDEs

$$\left\{\begin{array}{cc}\dot{x}\left(t\right)=\varepsilon g(x\left(t\right),x(t-r),t),\hfill & t\ge 0,\hfill \\ x\left(t\right)=\phi \left(t\right),\hfill & -r\le t\le 0,\hfill \end{array}\right.$$

where $r>0$. By the obtained RG procedure, we obtain the approximate solution on a time scale $\mathcal{O}(1/\varepsilon )$, moreover, we give the error estimate of the approximate solution. The highlight of our strategy is to break down delay problem into piecewise singularly perturbed DDEs, and ultimately, complete the entire task by unifying the piecewise RG equations into a delay one. By the error estimate and the numerical tests, the uniform validity of the approximate solution can be obtained.

This paper is organized as follows. In Section 2, the RG equation and the approximate solution of the singularly perturbed DDEs are obtained, then the error estimate between the approximate solution and the exact one can be given in Section 3. In the last section, we apply our result to two examples and give the numerical simulation diagram of the approximate solution.