\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 141, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/141\hfil Periodic solutions] {Periodic solutions for functional differential equations with periodic delay close to zero} \author[M. L. Hbid, R. Qesmi\hfil EJDE-2006/141\hfilneg] {My Lhassan Hbid, Redouane Qesmi} % in alphabetical order \address{My Lhassan Hbid \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences Semlalia, Universit\'e Cadi Ayyad, B.P. S15, Marrakech, Morocco} \email{hassan.hbid@gmail.com} \address{Redouane Qesmi \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences Semlalia, Universit\'e Cadi Ayyad, B.P. S15, Marrakech, Morocco} \email{qesmir@gmail.com} \date{} \thanks{Submitted August 25, 2006. Published November 9, 2006.} \subjclass[2000]{34K13} \keywords{Differential equation; periodic delay; bifurcation; \hfill\break\indent h-asymptotic stability; periodic solution} \begin{abstract} This paper studies the existence of periodic solutions to the delay differential equation $$ \dot{x}(t)=f(x(t-\mu\tau(t)),\epsilon)\,. $$ The analysis is based on a perturbation method previously used for retarded differential equations with constant delay. By transforming the studied equation into a perturbed non-autonomous ordinary equation and using a bifurcation result and the Poincar\'e procedure for this last equation, we prove the existence of a branch of periodic solutions, for the periodic delay equation, bifurcating from $\mu=0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Let us consider the periodic delay differential equations of the form \begin{equation} \dot{u}(t)=f(u(t-\mu\tau(t)),\epsilon),\label{5} \end{equation} under the following assumptions: \begin{itemize} \item [(H1)] $f\in C^{\infty}(\mathbb{R}^{2}\times \mathbb{R},\mathbb{R}^{2})$, $f(0,\epsilon)=0$, and $f_{u}'(0,\epsilon)=\begin{pmatrix} 0 & -\beta_{1}(\epsilon)\\ \beta_{1}(\epsilon) & 0\end{pmatrix}$ where $\beta_{1}(\epsilon)>0$ and satisfies $\beta_{1}(1)=1$ and $\beta_{1}'(1)\neq0$. Moreover, $f$ and its first and second derivatives are bounded so that there is a number $A>0$, such that $\max(\| f\|_{\infty},\| f'_{u}\|_{\infty},\| f''_{u}\|_{\infty})0$ and $\epsilon$ are parameters having values in a neighborhood of $0$ and $1$, respectively. When the function $\tau$ is independent of $t$ (i.e: $\tau(t)=\tau>0$), system \eqref{5} is an autonomous equation which is extensively studied in \cite{key1,key-03,key-04,key-01,key-02,key-11}. The aim of this paper is to prove the existence of a branch of a bifurcated periodic solutions for the differential equation \eqref{5} with periodic delay in the case where $\mu$ is small enough. The problem studied here is local in nature. The work is in line with a previous work by Arino and Hbid \cite{key1} in which the delay was assumed to be constant and small. Smallness of the delay was an essential feature which made possible the study of the equation as a perturbation of an ordinary differential equation (ODE). It was also possible to go a little further than the Hopf bifurcation and extend to this situation results obtained previously by Bernfeld and Salvadori \cite{key-20} on generalized Hopf bifurcation for ODEs. The authors have used a perturbation method to transform the functional differential equation into an ODE. A Poincar\'e map was constructed in a neighborhood of the bifurcating periodic solutions of the ordinary differential system. The fixed points of this map correspond to periodic solutions of the functional differential equations. In this paper we proceed in the same general spirit as in \cite{key1}. Though our approach could be viewed as a simple adaptation of the one described in \cite{key1}, some specific features related to the dependence of the delay on the time $t$ are to be mentioned: the main one is that the perturbed method applied here transforms the time dependent delay equation into a non-autonomous ordinary equation. Under an additional hypothesis on $h$-asymptotic stability (see for instance \cite{key-24}) the non-autonomous ODE has an attractive bifurcating branch of periodic solutions. A closed bounded convex subset of the space of Lipschitz continuous functions is constructed in the neighborhood of this branch. Finally, we set up a Poincar\'e map which transforms the convex set into itself. The Poincar\'e map being eventually compact in the space of Lipschitz continuous functions, has fixed points which yield periodic solutions of the retarded differential equation with periodic delay \eqref{5}. \section{Background} In this section we recall some aspects of bifurcation given in \cite{key-24} for the periodic ordinary system \begin{equation} \dot{z}=g(t,z,\mu,\epsilon)\label{0} \end{equation} where $g\in C^{\infty}(\mathbb{R}\times \mathbb{R}^{2}\times \mathbb{R}\times \mathbb{R},\mathbb{R})$, $g(t,0,\mu,\epsilon)=0$ and $2\pi$-periodic in $t$. $\mu$ and $\epsilon$ are parameters and have values respectively in a neighborhood of $0$ and $1$. Because of Floquet theory the Jacobian matrix $f_{z}'(t,0,\mu,\epsilon)$ may be assumed without loss of generality to be independent of $t$ and its eigenvalues will be denoted by $\alpha(\mu,\epsilon)\pm i\beta(\mu,\epsilon)$. We will assume that \begin{gather*} \alpha(0,\epsilon)=0, \quad \beta(0,1)=1\\ \alpha'_{\mu}(0,1)\neq0 \quad \beta_{\epsilon}'(0,1)\neq0. \end{gather*} By a linear transformation of $z$ independent of $t$, and involving $\mu,\epsilon$, Equation \eqref{0} may be written as \begin{equation} \label{1} \begin{gathered} \dot{x}=\alpha(\mu,\epsilon)x-\beta(\mu,\epsilon)y+X(t,x,y,\mu, \epsilon) \\ \dot{y}=\alpha(\mu,\epsilon)y+\beta(\mu,\epsilon)x +Y(t,x,y,\mu,\epsilon), \end{gathered} \end{equation} where $X,Y\in C^{\infty}$ in $(x,y,\mu,\epsilon)$ and $2\pi$-periodic in $t$, and $X,Y$ are $O(x^{2}+y^{2})$. We remark that there exist a neighborhood $N$ of $(x,y):=(0,0)$ and three positive numbers $a,b,\omega$ such that for any $t_{0}\in \mathbb{R}$, $(x_{0},y_{0})\in N$, $\mu\tau$. We denote by $(x(t,t_{0},c,\mu,\epsilon),y(t,t_{0},c,\mu,\epsilon))$ the solution of \eqref{1} such that $x_{0}=c>0,y_{0}=0,(c,0)\in N$. \begin{lemma}[\cite{key-24}] \label{a} There exist three positive numbers $\overline{a},\overline{b},\overline{c},(\overline{c},0)\in \mathbb{N}$, and a function $\overline{\epsilon}\in C^{\infty}(\mathbb{R}\times[0,\overline{c}]\times[-\overline{b},\overline{b} ],[1-\overline{a},1+\overline{a}])$, $\overline{\epsilon}(t_{0},0,0)=1$, such that for any $t_{0}\in \mathbb{R},c\in[0,\overline{c}],\mu\in[0,\overline{b}]$, and $|\epsilon-1|<\overline{a}$ the equation $y(t_{0}+2\pi,t_{0},c,\mu,\epsilon)=0$ is satisfied if and only if $\epsilon=\overline{\epsilon}(t_{0},c,\mu)$. \end{lemma} Consider now the function $V\in C^{\infty}(\mathbb{R} \times[0,\overline{c}]\times[-\overline{b},\overline{b}],\mathbb{R})$ defined by \begin{equation} V(t_{0},c,\mu)=x(t_{0}+2\pi,t_{0},c,\mu,\overline{\epsilon}(t_{0},c,\mu)) -c.\label{4} \end{equation} Clearly the $2\pi$-periodic solutions of \eqref{1} relative to any triplet $(c,\mu,\epsilon)$ for which $c\in[0,\overline{c}],|\mu|\in[0,\overline{b}]$, and $|\epsilon-1|\in[0,\overline{a}]$ correspond to the zeros of $V(t_{0},c,\mu)$. We will call $V$ the displacement function. The following theorem holds. \begin{theorem}[\cite{key-24}] \label{b} Suppose that $\overline{a},\overline{b},\overline{c}$ are sufficiently small. Assume that there exist two functions $\mu^{*}\in C^{\infty}(\mathbb{R}\times[0,\overline{c}], [-\overline{b},\overline{b}]),\epsilon^{*}\in C^{\infty} (\mathbb{R}\times[0,\overline{c}],[1-\overline{a},1+\overline{a}])$ such that if $t_{0}\in \mathbb{R},c\in[0,\overline{c}]$, $|\mu^{*}|\in[0,\overline{b}],|\epsilon-1|\in[0,\overline{a}]$. Then the solution $(x(t,t_{0},c,\mu,\epsilon),y(t,t_{0},c,\mu,\epsilon))$ of \eqref{1} is $2\pi$-periodic if and only if $\mu=\mu^{*}(t_{0},c),\epsilon=\epsilon^{*}(t_{0},c)$. Moreover $\epsilon^{*}(t_{0},c)=\overline{\epsilon}(t_{0},c,\mu^{*}(t_{0},c))$. \end{theorem} We will assume also that the functions $X$ and $Y$ are independent of $t$ when $\mu=0$. Then system \eqref{1} may be written as \begin{equation} \label{2} \begin{gathered} \dot{x} = \alpha(\mu,\epsilon)x-\beta(\mu,\epsilon)y+\tilde{X} (x,y,\epsilon)+\mu X^{*}(t,x,y,\mu,\epsilon) \\ \dot{y} = \alpha(\mu,\epsilon)y+\beta(\mu,\epsilon)x +\tilde{Y}(x,y,\epsilon)+\mu Y^{*}(t,x,y,\mu,\epsilon). \end{gathered} \end{equation} which for $\mu=0$ has the form \begin{equation} \label{3a} \begin{gathered} \dot{x}=-\beta(0,\epsilon)y+\tilde{X}(x,y,\epsilon)\\ \dot{y}=\beta(0,\epsilon)x+\tilde{Y}(x,y,\epsilon). \end{gathered} \end{equation} \begin{definition}[\cite{key-24}] \rm Let $h\in \mathbb{N}$. The solution $\xi=0$ of system \eqref{3a} is said to be $h$-asymptotically stable (resp. $h$-completely unstable) if the following conditions are satisfied: \begin{enumerate} \item For all $\tau_{1},\tau_{2}\in C(\mathbb{R}^{2},\mathbb{R})$ of order $h$, the solution $0$ of system \begin{gather*} \dot{x} = -\beta(0,\epsilon)y+\tilde{X}(x,y,\epsilon)+\tau_{1}(x,y)\\ \dot{y} = \beta(0,\epsilon)x+\tilde{Y}(x,y,\epsilon)+\tau_{2}(x,y). \end{gather*} is asymptotically stable (resp.unstable). \item $h$ is the smallest integer such that the property (1) above is satisfied. \end{enumerate} \end{definition} We have the following equivalence between $h$-asymptotic stability and the existence of an appropriate polynomial in $(x,y)$. This polynomial may be determined by an algebraic procedure due to Poincar\'e. \begin{proposition}[\cite{key-24}] \label{pro:1} The origin of \eqref{3a} is $h$-asymptotically stable if and only if $h$ is odd and there exists a polynomial in $(x,y),F(x,y,\epsilon)$, of degree $h+1$ having the form \[ F(x,y,\epsilon)=x^{2}+y^{2}+f_{3}(x,y,\epsilon)+\dots +f_{h+1}(x,y,\epsilon) \] ($f_{i}$ is homogeneous of degree $i$ in $(x,y)$) such that the derivative along the solutions of \eqref{3a} is given by \[ \dot{F}(x,y,\epsilon)=G_{h+1}(\epsilon)(x^{2}+y^{2})^{(h+1)/2} +o((x^{2}+y^{2})^{(h+1)/2}). \] Here $G_{h+1}(\epsilon)<0$ is a constant. \end{proposition} The explicit constant $G_{h+1}(\epsilon)$, called Lyapunov constant, can be obtained for each odd integer $h$ by the following proposition \begin{proposition}[\cite{key-11}] \label{pro:2} Given an even integer $k>2$, the Poincar\'e constant $G_{k}(\epsilon)$ is given in a unique way by \[ G_{k}(\epsilon)=\frac{p_{k}+p_{k-2}/(k-1)\beta(0,\epsilon) +\sum_{s=1}^{k/2-1}c_{s}d_{s}}{\frac{k}{2(k-1)\beta(0,\epsilon)} +\sum_{s=1}^{k/2-1}c_{s}\frac{C_{s+1}^{k/2}}{(k-2s+1)\beta(0,\epsilon)}+1}, \] where $c_{s}=\frac{3\times5\times7\dots \times(2s+1)}{(k-1)\times(k-3)\dots \times(k-2s+1)}$, $d_{s}=\frac{p_{k-2s-2}}{(k-2s-1)}$ for all $s\in\{1,\dots ,\frac{k}{2}-1\}$. and the terms $p_{j},j=0..k$ are given by \[ \rho_{k}(\xi_{1},\xi_{2})=\sum_{j=0}^{k}p_{j}\xi_{1}^{k-j}\xi_{2}^{j} \] with $\rho_{j}(\xi_{1},\xi_{2})$ is the homogeneous part of degree $j$ of the function $\rho(\xi_{1},\xi_{2})$ given by \[ \rho(\xi_{1},\xi_{2})=\tilde{X}(\xi_{1},\xi_{2},\epsilon) \frac{\partial}{\partial\xi_{1}}(\sum_{l=3}^{j-1}f_{l}(\xi_{1},\xi_{2}, \epsilon))+\tilde{Y}(\xi_{1},\xi_{2},\epsilon) \frac{\partial}{\partial\xi_{2}}(\sum_{l=3}^{j-1}f_{l}(\xi_{1},\xi_{2}, \epsilon)). \] \end{proposition} We have the following results. \begin{theorem}[\cite{key-24}] \label{c} Suppose there exists an odd integer $h\geq3$ such that the origin of \eqref{3a} is $h$-asymptotically stable for every $\epsilon\in[1-\overline{a},1+\overline{a}]$. Then if $\alpha'(0,1)<0$ (resp $\alpha'(0,1)>0$) the bifurcating $2\pi$-periodic solutions of \eqref{2} occur for $\mu>0$ (resp $\mu<0$). Moreover the positive numbers $\overline{a}$, $\overline{b}$, $\overline{c}$ of Theorem \ref{b} can be chosen such that for any $t_{0}\in \mathbb{R}$ and $\mu\in[0,\overline{b}]$ (resp $\mu\in[-\overline{b},0]$) there exists one and only one $c\in[0,\overline{c}]$ such that $\mu=\mu^{*}(t_{0},c)$. \end{theorem} \section{Main result} In the sequel, we transform equation \eqref{5} into a periodic ODE perturbed by a small time dependent-delay term. We define $\tau_{\infty}:=\sup\{ |\tau(t)|:t\in[0,2\pi]\} $ and $C$ the space of continuous functions from $[-\mu\tau_{\infty},0]$ to $\mathbb{R}^{2}$, then we have the following result \begin{proposition}\label{f} Under hypothesis (H2), the periodic delay system \eqref{5} can be written in the form \[ \dot{u}(t)=g(t,u(t),\mu,\epsilon)+H(t,u_{t},\mu,\epsilon), \] where \[ g(t,u,\mu,\epsilon):=\big[I+\mu\tau(t)f_{u}'(u,\epsilon)\big]^{-1} f(u,\epsilon) \] and $H$ a function defined on $\mathbb{R}\times C^{1}\times \mathbb{R}\times \mathbb{R}$ and satisfies \begin{align*} &H(t,u_{t},\mu,\epsilon)\\ &=\big[I+\mu\tau(t)f_{u}'(u(t),\epsilon)\big]^{-1} \int_{-\mu\tau(t)}^{0}[f_{u}'(u(t),\epsilon)\dot{u}(t) -f_{u}'(\dot{u}(t+\sigma),\epsilon)\dot{u}(t+\sigma)]d\sigma \end{align*} for all solutions $u$ of system \eqref{5}. \end{proposition} \begin{proof} Let $u$ be a solution of equation \eqref{5}. We have \[ f(u(t-\mu\tau(t)),\epsilon)=f(u(t),\epsilon)-\int_{-\mu\tau(t)}^{0} f_{u}'(u(t+\sigma),\epsilon)\dot{u}(t+\sigma)d\sigma \] Then \begin{align*} f(u(t-\mu\tau(t)),\epsilon) & = f(u(t),\epsilon)-\mu\tau(t)f_{u}'(u(t),\epsilon)\dot{u}(t)\\ &\quad +\int_{-\mu\tau(t)}^{0}[f_{u}'(u(t),\epsilon)\dot{u}(t) -f_{u}'(u(t+\sigma),\epsilon)\dot{u}(t+\sigma)]d\sigma. \end{align*} Since $u$ is a solution of equation \eqref{5}, we obtain \begin{align*} &\big(I+\mu\tau(t)f_{u}'(u(t),\epsilon)\big)f(u(t-\mu\tau(t), \epsilon)\\ &=f(u(t),\epsilon) +\int_{-\mu\tau(t)}^{0}[f_{u}'(u(t),\epsilon)\dot{u}(t) -f_{u}'(u(t+\sigma),\epsilon)\dot{u}(t+\sigma)]d\sigma. \end{align*} For $\mu$ small enough, the matrix $\big(I+\mu\tau(t)f_{u}'(u(t),\epsilon)\big)$ is invertible and we can write \[ f(u(t-\mu\tau(t),\epsilon)=g(t,u(t),\mu,\epsilon) +H(t,u_{t},\mu,\epsilon) \] with $g$ and $H$ as defined above. \end{proof} In the sequel, the equation under study is \begin{equation} \dot{u}(t)=g(t,u(t),\mu,\epsilon)+H(t,u_{t},\mu,\epsilon). \label{eq:56} \end{equation} In what follows we look for periodic solutions of the following $2$-dimensional system \begin{equation} \dot{w}(t)=g(t,w(t),\mu,\epsilon).\label{6} \end{equation} \begin{theorem} \label{d} Suppose (H1)--(H3) hold. Then there exists a sufficiently small positive numbers $\overline{a},\overline{b},\overline{c}$, and there exist two functions $\mu^{*}$ in $C^{\infty}(\mathbb{R}\times[0,\overline{c}],[0,\overline{b}])$, and $\epsilon^{*}$ in $C^{\infty}(\mathbb{R}\times[0,\overline{c}], [1-\overline{a},1+\overline{a}])$ such that if $t_{0}\in \mathbb{R}$, $\mu\in[0,\overline{b}],|\epsilon-1|\in[0,\overline{a}]$, then there exists one and only one $c\in[0,\overline{c}]$, such that the solution \[ (w_{1}(t,t_{0},c,\mu,\epsilon),w_{2}(t,t_{0},c,\mu,\epsilon)) \] of \eqref{6} is $2\pi$-periodic if and only if $\mu=\mu^{*}(t_{0},c),\epsilon=\epsilon^{*}(t_{0},c)$. Moreover the family of the bifurcating solutions are of amplitude of order $\sqrt{\mu}$. \end{theorem} \begin{proof} We first show the conditions imposed in \cite{key-24}: We have $g(t,u(t),0,\epsilon):=f(u(t),\epsilon)$, and the eigenvalues of the Jacobian matrix $g'_{w}(t,0,\mu,\epsilon)$ have the form $\alpha(\mu,\epsilon)\pm i\beta(\mu,\epsilon)$, where $\alpha(0,\epsilon)=0\textrm{ and }\beta(0,\epsilon)=\beta_{1}(\epsilon)$, then from hypothesis (H1) we have $\alpha(0,\epsilon)=0,\beta(0,1)=1$ and $\beta'_{\epsilon}(0,1)\neq0$, it remains to prove that $\alpha'_{\mu}(0,1)\neq0$, however, $\lambda(\mu):=\alpha(\mu,1)+i\beta(\mu,1)$ is the characteristic exponent of the Jacobian matrix $g'_{w}(t,0,\mu,1)=[I+\mu\tau(t)f'_{u}(0,1)]^{-1}f'_{u}(0,1)$, and with a few computations, we obtain that \[ \frac{\partial}{\partial\mu}g'_{w}(t,0,\mu,1) =-\tau(t)[[I+\mu\tau(t)f'_{u}(0,1)]^{-1}f'_{u}(0,1)]^{2}, \] then \[ \big[\frac{\partial}{\partial\mu}g'_{w}(t,0,\mu,1)\big]_{\mu=0} =-\tau(t)\beta_{1}^{2}(1)I, \] and because of the regularity of $\lambda(.)$, we deduce that $\lambda'_{\mu}(0)$ is the characteristic exponent of the matrix $-\tau(t)\beta_{1}^{2}(1)I$, that's $\alpha'_{\mu}(0,1)=\frac{\beta_{1}^{2}(1)}{2\pi}\int_{0}^{2\pi}\tau(s)ds$, and by hypothesis (H2) we have that $\alpha'_{\mu}(0,1)>0$. Then the first part of theorem is a consequence of Theorems \ref{b} and \ref{c}. If the hypothesis (H3) is satisfied then the origin of \eqref{6} is $3-$asymptotically stable, consequently for any $t_{0}\in \mathbb{R}$ we have (see the proof of Theorem \ref{c} in \cite{key-24}): \[ \frac{\partial V}{\partial c}(t_{0},0,0) =\frac{\partial^{2}V}{\partial c^{2}}(t_{0},0,0)=0 \quad\text{and}\quad \frac{\partial^{3}V}{\partial c^{3}}(t_{0},0,0)<0, \] Moreover, we have \[ \frac{\partial^{2}\mu^{*}}{\partial c^{2}}(t_{0},0) =-\frac{1}{6\pi\alpha'_{\mu}(0,1)} \frac{\partial^{3}V}{\partial c^{3}}(t_{0},0,0)>0, \] so by developing the function $c\mapsto\mu^{*}(t_{0},c)$ in a neighborhood of zero, we obtain \[ \mu^{*}(t_{0},c)=\frac{1}{2}\frac{\partial^{2}\mu^{*}}{\partial c^{2}} (t_{0},0)c^{2}+o(c^{2}). \] Then $\mu^{*}(t_{0},c)$ is of order $c^{2}$, however, the first part of the theorem tells us that the map $c\mapsto\mu^{*}(t_{0},c)$ is injective, consequently the inverse function $c(t_{0},\mu)$ of $\mu^{*}(t_{0},.)$ is of order $\sqrt{\mu}$. This shows the second part of the theorem. \end{proof} \begin{remark} \label{e} \rm According to the above theorem, for a given $\mu$ and $t_{0}$ there is one and only one periodic solution of \eqref{6}. Precisely, this periodic solution is obtained by assuming $\epsilon=\epsilon_{1}(t_{0},\mu)$, where $\epsilon_{1}(t_{0},\mu)=\epsilon^{*}(t_{0},c(t_{0},\mu))$, in \eqref{6}. \end{remark} In the sequel, we let $t_{0}=0$ and we assume that $\epsilon=\epsilon_{1}(0,\mu)$ for any $\mu$ in equation \eqref{5}. Denote by $y(\mu):=(y_{1}(\mu),0)$ the initial data of the bifurcating periodic solutions of \eqref{6}. From theorem \ref{d} we see that there exists a constant $C>0$ such that $\| y(\mu)\|\leq C\mu^{1/2}$. Let $u(\phi)$ be the solution of \eqref{5} with initial data $u_{0}=\phi$. From lemma \ref{a} and remark \ref{e} one can find a solution $w^{*}$of \eqref{6} such that $w^{*}(0)=\phi(0)$, $w_{2}^{*}(2\pi)=0$ and $w_{1}^{*}(2\pi)>0$. To state the nest proposition, we introduce the subset \[ \mathcal{B}(\mu):=\{ \phi\in C^{1}:\|\phi(s)-y(\mu)\|\leq C\mu^{3/2}\} . \] \begin{proposition} \label{m} Under the hypothesis (H1)--(H3), there exists a constant $C_{1}>0$, such that for a given $T>0$ and $\mu$ close to zero, we have \[ \| u(\phi)(t)\|\leq C_{1}\mu^{1/2} \] for all $\phi\in\mathcal{B}(\mu)$ and $t\in[0,T]$. \end{proposition} \begin{proof} let $\tau_{0}:=\inf\{ \tau(t):t\in[0,2\pi]\} $, $t\in[0,\mu\tau_{0}]$, then $t-\mu\tau(t)\leq0$, and \[ u(\phi)(t)=\phi(0)+\int_{0}^{t}f(\phi(s-\mu\tau(s)))ds \] it follows that \[ \| u(\phi)(t)\|\leq\|\phi\|_{\infty}+\mu\tau_{0}A\|\phi\|_{\infty} \leq C(1+\mu\tau_{0}A)\mu^{1/2}. \] In a similar manner, we show by iteration that for $t\in[0,k\mu\tau_{0}]$, we have \[ \| u(\phi)(t)\|\leq C(1+\mu\tau_{0}A)^{k}\mu^{1/2}. \] Let $k$ the unique natural integer such that $k\mu\tau_{0}\leq T<(k+1)\mu\tau_{0}$, then for $t\in[0,T]$ we have \[ \| u(\phi)(t)\| \leq C(1+\mu\tau_{0}A)^{k}\mu^{1/2} \leq Ce^{\mu\tau_{0}A(k+1)}\mu^{1/2} \leq Ce^{(\mu\tau_{0}A+AT)}\mu^{1/2}. \] Finally, for $\mu$ close to zero, one obtain a constant $C_{1}$ independent of $\mu$ such that \[ \| u(\phi)(t)\|\leq C_{1}\mu^{1/2}. \] \end{proof} \begin{proposition} \label{g} Under the hypothesis (H1)--(H3), there exist positive constant $C_{2}$ such that for $\mu$ close to zero and $\phi\in\mathcal{B}(\mu)$, we have \[ \| H(t,u_{t},\mu,\epsilon)\|\leq C_{2}\mu^{5/2} \quad\text{for }t\in[3\mu\tau_{\infty},T]. \] Moreover \[ \| H(t,u_{t},\mu,\epsilon)\|\leq C_{2}\mu^{3/2}\quad \text{for }t\in[0,3\mu\tau_{\infty}]. \] \end{proposition} \begin{proof} Note that from theorem \ref{f}, for $t\in[3\mu\tau_{\infty},T]$, we have \begin{align*} &H(t,u_{t},\mu,\epsilon)\\ &=[I+\mu\tau(t)f_{u}'(u(t),\epsilon)]^{-1} \int_{-\mu\tau(t)}^{0}[f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma), \epsilon)\dot{u}(t+\sigma)]d\sigma. \end{align*} Using the inequality \begin{align*} &\| f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma), \epsilon)\dot{u}(t+\sigma)\|\\ &\leq \| f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma),\epsilon)\dot{u}(t)\|\\ &\quad +\| f_{u}'(u(t+\sigma),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma), \epsilon)\dot{u}(t+\sigma)\|, \end{align*} we obtain \begin{align*} &\| f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma),\epsilon) \dot{u}(t+\sigma)\| \\ & \leq A\| u(t+\sigma)-u(t)\|\|\dot{u}(t)\| +A\|\dot{u}(t+\sigma)-\dot{u}(t)\|. \end{align*} On the other hand, for $t\in[3\mu\tau_{\infty},T]$ and $\sigma\in[-\mu\tau_{\infty},0]$, we have \begin{align*} \| u(t+\sigma)-u(t)\| & \leq -\sigma\sup_{s\in[t,t+\sigma]}\|\dot{u}(s)\|\\ & = -\sigma\sup_{s\in[t,t+\sigma]}\| f(u(s-\mu\tau(s))\|\\ & \leq -\sigma A\sup_{s\in[t,t+\sigma]}\| u(s)\|\\ & \leq -\sigma AC\mu^{1/2}:=-\sigma A_{1}\mu^{1/2}, \end{align*} and \begin{align*} \|\dot{u}(t+\sigma)-\dot{u}(t)\| & \leq -\sigma\sup_{s\in[t,t+\sigma]}\|\ddot{u}(s)\|\\ & = -\sigma\sup_{s\in[t,t+\sigma]}\| f_{u}'(u(s-\mu\tau(s)),\epsilon) \dot{u}(s-\mu\tau(s))(1-\mu\dot{\tau}(s))\|\\ & \leq -\sigma A^{2}C\mu^{1/2}(1+\mu\sup_{s\in[0,2\pi]}\|\dot{\tau}(s) \|\leq-\sigma A_{2}\mu^{1/2}, \end{align*} for some constant $A_{2}>0$. This implies \[ \| f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma),\epsilon) \dot{u}(t+\sigma)\|\leq-\sigma AA_{1}^{2}\mu-\sigma AA_{2}\mu^{1/2}, \] it follows that there exists a constant $C_{2}^{*}>0$ such that for $\mu$ close to zero we have \[ \|\int_{-\mu\tau(t)}^{0}[f_{u}'(u(t),\epsilon)\dot{u}(t)-f_{u}'(u(t+\sigma), \epsilon)\dot{u}(t+\sigma)]d\sigma\|\leq\frac{C_{2}^{*}}{2}\mu^{5/2}. \] On the other hand, for $\mu$ close to zero such that $\mu A\tau^{\infty}<\frac{1}{2}$ we obtain \[ \|[I+\mu\tau(t)f_{u}'(u(t),\epsilon)]^{-1}\|\leq2. \] Which prove the first inequality of the proposition. Now let $t$ be in the interval $[0,3\mu\tau_{\infty}]$ and $u(t)=u(\phi)(t)$ for some $\phi\in\mathcal{B}(\mu)$. We have \[ g(t,u(t),\mu,\epsilon) = [I+\mu\tau(t)f_{u}'(u(t),\epsilon)]^{-1}f(u(t),\epsilon) = f(u(t),\epsilon)+\mu O(u(t)). \] Using proposition \ref{m} we obtain \[ g(t,u(t),\mu,\epsilon)=f(u(t),\epsilon)+O(\mu^{3/2}). \] Then \[ H(t,u_{t},\mu,\epsilon) = f(u(t-\mu\tau(t)),\epsilon)-f(u(t),\epsilon) +O(\mu^{3/2}) \] and \[ \| H(t,u_{t},\mu,\epsilon)\|\leq\| f(u(t-\mu\tau(t)),\epsilon) -f(u(t),\epsilon)\|+O(\mu^{3/2}). \] Since $f$is a smooth function, we deduce that \begin{align*} \| H(t,u_{t},\mu,\epsilon)\| & \leq A\| u(t-\mu\tau(t))-u(t)\|+O(\mu^{3/2})\\ & \leq A\int_{t-\mu\tau_{\infty}}^{t}\|\dot{u}(s)\| ds+O(\mu^{3/2})\\ & \leq A\int_{t-\mu\tau_{\infty}}^{t}\| f((u(s-\mu\tau(s)),\epsilon)\| ds +O(\mu^{3/2})\\ & \leq A^{2}\int_{t-\mu\tau_{\infty}}^{t}\| u(s-\mu\tau(s))\| ds +O(\mu^{3/2}) \end{align*} Using once more proposition \ref{m} we deduce that there exist $C_{2}^{**}>0$ such that \[ \| H(t,u_{t},\mu,\epsilon)\|\leq C_{2}^{**}\mu^{3/2} \] which completes the proof with $C_{2}:=\min(C_{2}^{*},C_{2}^{**})$. \end{proof} As a result of proposition \ref{f} and the above theorem, the equation \eqref{5} can be written as a perturbation of an ordinary differential equation by a small term. We are now in position to give an estimation of the difference between $u(\phi)$ and $w^{*}$. \begin{lemma} \label{h} There exist a positive constant $C_{3}$ such that for all $\phi\in\mathcal{B}(\mu)$ and $t\in[0,2\pi]$ we have \[ \| u(\phi)(t)-w^{*}(t)\|\leq C_{3}\mu^{5/2}. \] \end{lemma} \begin{proof} Let $\phi\in\mathcal{B}(\mu)$, we have \[ \frac{d}{dt}[\dot{u}(\phi)(t)-\dot{w}^{*}(t)]=g(t,u(\phi)(t),\mu,\epsilon) +H(t,u_{t}(\phi),\mu,\epsilon)-g(t,w^{*}(t),\mu,\epsilon), \] then from hypothesis (H1) and (H2) and using the inner product in $\mathbb{R}^{2}$, we obtain \begin{align*} &\frac{1}{2}\frac{d}{dt}\| u(\phi)(t)-w^{*}(t)\|^{2}\\ &\leq 2A^{2}\|u(\phi)(t)-w^{*}(t)\|^{2}+\| u(\phi)(t)-w^{*}(t)\| \| H(t,u_{t}(\phi),\mu, \epsilon)\|, \end{align*} form which it follows that \[ D^{+}\| u(\phi)(t)-w^{*}(t)\|\leq2A^{2}\| u(\phi)(t)-w^{*}(t)\| +\| H(t,u_{t}(\phi),\mu,\epsilon)\|, \] where $D^{+}$ denotes the derivative from the right. Using the Gronwall's inequality and in view of $u(\phi)(0)=\phi(0)=w^{*}(0)$, we obtain \[ \| u(\phi)(t)-w^{*}(t)\|\leq\int_{0}^{t}e^{2A^{2}(t-s)}\| H(s,u_{s}(\phi),\mu,\epsilon)\|, \] then \begin{align*} &\| u(\phi)(t)-w^{*}(t)\|\\ &\leq\int_{0}^{3\mu\tau_{\infty}}e^{2A^{2}(t-s)} \| H(s,u_{s}(\phi),\mu,\epsilon)\|+\int_{3\mu\tau_{\infty}}^{t}e^{2A^{2} (t-s)}\| H(s,u_{s}(\phi),\mu,\epsilon)\|. \end{align*} From proposition \ref{g}, we have \[ \int_{0}^{3\mu\tau_{\infty}}e^{2A^{2}(t-s)}\| H(s,u_{s}(\phi),\mu,\epsilon)\| \leq\int_{0}^{3\mu\tau_{\infty}}e^{2A^{2}(t-s)}O(\mu^{3/2}) \leq\frac{C_{3}}{2}\mu^{5/2} \] and \[ \int_{3\mu\tau_{\infty}}^{t}e^{2A^{2}(t-s)}\| H(s,u_{s}(\phi),\mu,\epsilon)\| \leq\int_{3\mu\tau_{\infty}}^{t}e^{2A^{2}(t-s)}O(\mu^{5/2}) \leq\frac{C_{3}}{2}\mu^{5/2} \] for some constant $C_{3}>0$. Thus \[ \| u(\phi)(t)-w^{*}(t)\|\leq C_{3}\mu^{5/2} \] \end{proof} \begin{lemma}\label{i} For any $t\in[2\pi-\mu\tau_{\infty},2\pi]$ and any $\phi\in\mathcal{B}(\mu)$, we have \[ \| u(\phi)(t)-u(\phi)(2\pi)\|\leq C_{4}\mu^{3/2} \] where $C_{4}$ is a positive constant independent of $\mu$. \end{lemma} \begin{proof} Let $t\in[2\pi-\mu\tau_{\infty},2\pi]$, $\mu$ close to zero such that $\mu<\frac{2\pi}{\tau_{\infty}}$ and $\phi\in\mathcal{B}(\mu)$, we have \begin{align*} \| u(\phi)(t)-u(\phi)(2\pi)\| & \leq \mu\tau_{\infty}\sup_{s\in[2\pi-\mu\tau_{\infty},2\pi]} \|\frac{d}{ds}u(\phi)(s)\|\\ & \leq \mu\tau_{\infty}\sup_{s\in[2\pi-\mu\tau_{\infty},2\pi]} \| f(u(s-\mu\tau(s))\|\\ & \leq \mu\tau_{\infty}A\sup_{\sigma\in[0,2\pi]}\| u(\sigma)\| \leq\mu\tau_{\infty}AC_{1}\mu^{1/2}:=C_{4}\mu^{3/2}. \end{align*} Which completes the proof. \end{proof} \begin{proposition} \label{j} Assume (H1)--(H3) are satisfies, then there exists $K_{1}>0$ such that for $\mu$ close to zero and $\phi\in\mathcal{B}(\mu)$, we have \[ \| w^{*}(2\pi)-y(\mu)\|\leq C[1-K_{1}| y(\mu)|^{2}]\mu^{3/2}. \] \end{proposition} \begin{proof} Put $c':=w_{1}^{*}(0)$ and $c:=y_{1}(\mu)$, we have \[ \| w^{*}(2\pi)-y(\mu)\|=| V(0,c',\mu(c)))+c'-c|. \] On the other hand we have \[ V(0,c',\mu(c))=V(0,c,\mu(c))+\frac{\partial}{\partial c}V(0,\eta,\mu(c)(c'-c) \] for some $\eta\in]\min(c,c'),\max(c,c')[$ and \[ \frac{\partial}{\partial c}V(0,\eta,\mu(c)) =\frac{\partial}{\partial c}V(0,\eta,0) +\frac{\partial^{2}V(0,\eta,v_{0}\mu(c))}{\partial\mu\partial c}\mu(c) \] for some $v_{0}\in]0,1[$. However, we have $V(0,c,\mu(c))=0$ since $c:=y(\mu)$ the initial data of the bifurcating periodic solutions of \eqref{6}, then \[ V(0,c',\mu(c))=[\frac{\partial}{\partial c}V(0,\eta,0) +\frac{\partial^{2}V(0,\eta,v_{0}\mu(c))}{\partial\mu\partial c}\mu(c)](c'-c). \] According to the $3$-asymptotic stability, we have \[ \frac{\partial V}{\partial c}(0,0,0) =\frac{\partial^{2}V}{\partial c^{2}}(0,0,0)=0\quad \text{and}\quad \frac{\partial^{3}V}{\partial c^{3}}(0,0,0)<0. \] Moreover, we have \[ \mu^{*}(0,0)=0,\frac{\partial\mu^{*}}{\partial c}(0,0)=0 \quad\text{and}\quad \frac{\partial^{2}\mu^{*}}{\partial c^{2}}(0,0) =-\frac{1}{3}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)/ \frac{\partial^{2}}{\partial\mu\partial c}V(0,0,0), \] then \[ \frac{\partial V}{\partial c}(0,\eta,0) =\frac{1}{2!}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)\eta^{2}+o(\eta^{2}) \] and \begin{align*} \frac{\partial^{2}V(0,\eta,v_{0}\mu(c))}{\partial\mu\partial c}\mu(c) & = \frac{1}{2!}\frac{\partial^{2}V(0,0,0)}{\partial\mu\partial c} \frac{\partial^{2}\mu^{*}}{\partial c^{2}}(0,0)c^{2}+o(c^{2})\\ & = -\frac{1}{6}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)c^{2}+o(c^{2}), \end{align*} it follows that \begin{align*} &\frac{\partial}{\partial c}V(0,\eta,0) +\frac{\partial^{2}V(0,\eta,v_{0}\mu(c))}{\partial\mu\partial c}\mu(c)\\ &=\frac{1}{2!}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)\eta^{2} -\frac{1}{6}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)c^{2} +o(\eta^{2})+o(c^{2}). \end{align*} However, we have $| c-\eta|\leq| c-c'|$ and \[ | c-c'| = \| w^{*}(0)-y(\mu)\| = \|\phi(0)-y(\mu)\| \leq C\mu(c)^{3/2}, \] then \[ \lim_{c\longrightarrow0}\frac{1}{c^{2}}[\frac{\partial}{\partial c} V(0,\eta,0)+\frac{\partial^{2}V(0,\eta,v_{0}\mu(c))}{\partial\mu\partial c} \mu(c)]=\frac{1}{3}\frac{\partial^{3}V}{\partial c^{3}}(0,0,0)<0. \] Consequently, there exists a constant $K_{1}>0$ such that for $\mu$ close to zero we have \[ \frac{1}{c'-c}V(0,c',\mu(c))\leq-K_{1}c^{2}, \] hence \begin{align*} | V(0,c',\mu(c))+c'-c| & = | c'-c||1+\frac{1}{c'-c}V(0,c',\mu(c))|\\ & \leq | c'-c|(1-K_{1}c^{2})\\ & \leq C\mu^{3/2}(1-K_{1}c^{2}), \end{align*} which implies \[ \| w^{*}(2\pi)-y(\mu)\|\leq C[1-K_{1}| y(\mu)|^{2}]\mu^{3/2}. \] The proof is complete. \end{proof} \begin{proposition} \label{k} For each $\phi\in\mathcal{B}(\mu)$, we have $u_{2\pi}(\phi)\in\mathcal{B}(\mu)$. \end{proposition} \begin{proof} From Lemma \ref{h}, \ref{i} and Proposition \ref{j}, we have \begin{align*} &\| u(\phi)(t)-y(\mu)\| \\ & \leq \| u(\phi)(t)-u(\phi)(2\pi)\|+\| u(\phi)(2\pi)-w^{*}(2\pi)\| +\| w^{*}(2\pi)-y(\mu)\|\\ & \leq C_{4}\mu^{3/2}+C_{3}\mu^{5/2}+C[1-K_{1}| y(\mu)|^{2}]\mu^{3/2}. \end{align*} Then \[ \| u(\phi)(t)-y(\mu)\|\leq[C_{4}+C_{3}\mu+(1-K_{1}| y(\mu)|^{2})C]\mu^{3/2}, \] from which we conclude that $u_{2\pi}(\phi)\in\mathcal{B}(\mu)$ for $\mu$ close to zero. \end{proof} \begin{theorem} \label{l} Under hypotheses (H1)--(H3), equation \eqref{5} has at least one nontrivial periodic solution for $\mu$ close to zero. \end{theorem} \begin{proof} Define the Poincar\'e operator \[ \mathcal{P}:\mathcal{B}(\mu)\rightarrow C([-\mu\tau_{\infty},0], \mathbb{R}^{2}) \] such that for $\phi\in\mathcal{B}(\mu)$, $\mathcal{P}\phi:=u_{2\pi}(\phi)$. Proposition \ref{k} shows that $\mathcal{P}$ is defined from $\mathcal{B}(\mu)$, (which is a convex bounded set) into itself and that $\mathcal{P}$ as continuous and compact (see \cite{key-28}). So using the second Schauder fixed point theorem (see, for example \cite{4}) we conclude that $\mathcal{P}$ has at least one fixed point which corresponds to a periodic solution of the retarded equation \eqref{5}. Sine $\mathcal{B}(\mu)$ does not contain zero, the obtained periodic solutions are nontrivial. \end{proof} \section{Examples} Consider the system of equations \begin{equation} \label{eq:exp} \begin{aligned} \frac{d}{dt}x_{1}(t) &=\epsilon x_{2}(t-\mu\tau(t)) +a_{1}x_{1}^{2}(t-\mu\tau(t))+b_{1}x_{2}^{2}(t-\mu\tau(t))\\ &\quad +O(x_{1}^{3}(t-\mu\tau(t)),x_{2}^{3}(t-\mu\tau(t))) \\ \frac{d}{dt}x_{2}(t) &=-\epsilon x_{1}(t-\mu\tau(t))+a_{2}x_{1}^{2}(t-\mu\tau(t)) +b_{2}x_{2}^{2}(t-\mu\tau(t))\\ &\quad +O(x_{1}^{3}(t-\mu\tau(t)),x_{2}^{3}(t-\mu\tau(t))) \end{aligned} \end{equation} where $a_{1},a_{2},b_{1},b_{2},\mu,\epsilon$ are real parameters. Here $\mu>0,\epsilon$ has values respectively in a neighborhood of $0$ and $1$. $\tau\in C^{1}(\mathbb{R},\mathbb{R}^{+})$ is $2\pi$-periodic in $t$ and $\int_{0}^{2\pi}\tau(s)ds\neq0$. Thus, applying the formulas given in \cite{key-11} for the computation of the Lyapunov constant we obtain the expression of $G_{4}(\epsilon)$ (see proposition \ref{pro:2}) \[ G_{4}(\epsilon)=-\frac{1}{2}\frac{4b_{1}b_{2}+7a_{1}b_{2}+a_{2}a_{1}}{\epsilon}. \] which implies that system (\ref{eq:exp}) with $\mu=0$ is $3$-asymptotically stable if $(4b_{1}b_{2}+7a_{1}b_{2}+a_{2}a_{1})\epsilon>0$. From theorem \ref{l} we have the following proposition \begin{proposition} If $(4b_{1}b_{2}+7a_{1}b_{2}+a_{2}a_{1})\epsilon>0$, then for any $\mu$ sufficiently small there exists a periodic solution for system (\ref{eq:exp}). \end{proposition} \begin{thebibliography}{00} \bibitem{key1} O. Arino, M. L. Hbid; \emph{Periodic solutions for retarded differential equations close to ordinary one}, Non linear analysis 14, (1990) 23-24. \bibitem{key-20} S. R. Bernfeld, L. 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