\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2002(2002), No. 79, pp. 1-14.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2002 Southwest Texas State University.} \vspace{1cm}} \begin{document} \title [\hfilneg EJDE--2002/79\hfil RD systems with 1-homogeneous non-linearity] {Reaction-diffusion systems with 1-homogeneous non-linearity} \author[M. B\"uger\hfil EJDE--2001/79\hfilneg] {Matthias B\"uger} \address{Matthias B\"uger \hfill\break Mathematisches Institut der Justus-Liebig-Universit\"at Gie\ss en \hfill\break Arndtstra\ss e 2, D-35392 Gie\ss en, Germany} \email{matthias.bueger@math.uni-giessen.de} \date{} \thanks{Submitted November 15, 2000. Published September 27, 2002.} \subjclass[2000]{35K57} \keywords{Reaction-diffusion equations, order-preserving, \hfill\break\indent reduction to one equation, oscillation number} \begin{abstract} We describe the dynamics of a system of two reaction-diffusion equations with 1-homogeneous non-linearity. We show that either an order-preserving property holds and can be used in order to determine the limiting behaviour in some (invariant) sets or the long time behaviour of all solutions can be described by looking at one scalar reaction-diffusion equation only. \end{abstract} \maketitle \def\dist{\mathop {\rm dist}\nolimits} \font\fiverm=cmr5 \input prepictex \input pictex \input postpictex \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} The dynamics of solutions of one reaction-diffusion equation \begin{equation} \frac{d}{dt} y=y_{xx}+g(t,x,y) \label{onerd} \end{equation} with Dirichlet or other boundary conditions on an interval $(0,L)$, $L>0$, has been examined by many authors \cite{f1,h1,h2,m1,z1}. Far less is known for a system of two reaction-diffusion equations \begin{equation} \frac{d}{dt} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + f\big(t,x,\begin{pmatrix} u \\ v \end{pmatrix}\big) , \quad x\in (0,L), \; t>0. \label{ard} \end{equation} In this paper we examine system (\ref{ard}) with a $1$-homogeneous non-linearity $f(u,v)=h_1(u,v)$, which means that $h_1(\mu u,\mu v)=\mu h_1(u,v)$ for all real $\mu$. Furthermore, we assume that $h_1$ is $C^1$ in $\mathbb{R}^2\setminus\{(0,0)\}$. \\ It turns out that there are two different cases, depending on whether the set $$ E:=\Big\{ \phi_0\in\mathbb{R} : \begin{pmatrix} -\sin\phi_0 \\ \cos\phi_0 \end{pmatrix} \cdot h_1\begin{pmatrix} \cos\phi_0 \\ \sin\phi_0 \end{pmatrix} =0 \Big\} $$ is empty or not. \subsection*{Case 1: $E\ne\emptyset$.} When $E=\mathbb{R}$ the dynamics has already been discussed in \cite{b1}, so we concentrate on the case $E\ne\mathbb{R}$. Since $E\ne\emptyset$ implies that $E$ has infinitely many elements, we can take $\alpha,\beta\in E$, $\alpha<\beta$, such that $\gamma\not\in E$ for all $\alpha<\gamma<\beta$. We write the initial data $(u_0,v_0)$ in polar coordinates \begin{equation} \begin{pmatrix} u_0 \\ v_0 \end{pmatrix} = r_0 \begin{pmatrix} \cos\varphi_0 \\ \sin\varphi_0 \end{pmatrix} \, . \label{pl} \end{equation} If $r_0$ is non-negative and $\varphi_0$ has its values in the interval $(\alpha,\beta)$, then we show that the solution $(u,v)$ starting at $(u_0,v_0)$ is flattened out and tends to a planar solution \begin{equation} \bar r_0 \begin{pmatrix} \cos\bar\varphi_0 \\ \sin\bar\varphi_0 \end{pmatrix} \label{r12} \end{equation} where $\bar\varphi_0$ is a constant and equals either $\alpha$ or $\beta$. We call a state $(\bar u_0,\bar v_0)$ {\it planar}, if it can be written in the form (\ref{r12}) with a constant, i.e.\ $x$-independent angle $\bar \varphi_0$. We note that $(\bar u_0,\bar v_0)$ is planar if and only if $\bar u_0$ and $\bar v_0$ are linearly dependent. A solution $(\bar u,\bar v)$ is called planar if $(\bar u,\bar v)(t)$ is planar for all $t$. This means that the angle function $\bar\varphi$ might depend on $t$ but not on the space variable $x$. It is easy to observe that $(\bar u,\bar v)(t)$ is a planar solution if and only if the initial value is planar. This notation is motivated by the fact that for a planar solution $(\bar u,\bar v)$ the curve $$ \gamma_t:[0,L]\ni x\mapsto (x,\bar u(t,x),\bar v(t,x))\in [0,L]\times\mathbb{R}^2 $$ lies in a plane $P_t$ (which depends on $t$), as one can see in Figure 1. \begin{figure}[ht] $$ \beginpicture \setcoordinatesystem units <4cm,4cm> point at 0 0 \setplotarea x from 0 to 1.15, y from 0 to 0.5 \axis bottom ticks numbered from 0 to 1 by 2 unlabeled from 0.2 to 1 by 0.2 / \axis left label {$\bar u,\bar v$-axis} / \put{\vector(0,1){1}} [Bl] at 0 0.5 \put{\vector(1,1){1}} [Bl] at 0.36 0.36 \put{$P_t$} [Bl] at 1.2 0.28 \put{$L$} [Bl] at 0.97 -0.17 \plot 0 0 0.36 0.36 / \plot 0.3 0.2 1.3 0.2 0.7 -0.2 -0.3 -0.2 0.3 0.2 / \setquadratic \plot 0 0 0.55 0.15 0.5 0 0.45 -0.15 1 0 / \endpicture $$ \caption{Curve $\gamma_t$ for a planar solution $(\bar u,\bar v)$} \end{figure} The proof of the fact that the angular function $\varphi(t,\cdot)$ associated with $(u,v)(t)$ converges either to $\alpha$ or to $\beta$ is based on the following order-preserving property: \begin{figure} $$ \beginpicture \setcoordinatesystem units <10cm,10cm> point at 0 0 \setplotarea x from 0 to 1.1, y from 0 to 0.55 \axis left / \axis bottom / \put{\vector(0,1){1}} [Bl] at 0 0.55 \put{\vector(1,0){1}} [Bl] at 1.1 0 \put{$u$} [Bl] at 1.08 -0.035 \put{$v$} [Bl] at -0.05 0.5 \plot 0 0 1.1 0.055 / \plot 0 0 1 0.1 / \plot 0 0 0.1 0.5 / \plot 0 0 0.3 0.5 / \circulararc 2.86 degrees from 0.9 0 center at 0 0 \put{$\scriptsize\alpha$} [Bl] at 0.86 0.006 \circulararc 5.7 degrees from 0.83 0 center at 0 0 \put{$\scriptsize\bar \varphi_-$} [Bl] at 0.77 0.05 \circulararc -11.3 degrees from 0 0.45 center at 0 0 \put{$\scriptsize\!{\pi\over 2}\!-\!\beta$} [Bl] at 0.004 0.415 \circulararc -30.97 degrees from 0 0.23 center at 0 0 \put{$\scriptsize\!{\pi\over 2}\!-\!\bar \varphi_+$} [Bl] at 0.005 0.18 \circulararc 15 degrees from 0.95 0.095 center at 0.6 0.06 \put{\vector(-1,4){1}} [Bl] at 0.929 0.185 \circulararc 15 degrees from 0.24 0.4 center at 0.06 0.1 \put{\vector(-4,1){1}} [Bl] at 0.15 0.438 \put{\vector(-1,0){15} \quad \,\, graph of} [Bl] at 0.7 0.45 \put{$(0,L)\ni x\mapsto \!\begin{pmatrix} u_0(x) \\ v_0(x)\end{pmatrix} \in\mathbb{R}^2$} [Bl] at 0.71 0.38 \setquadratic \plot 0 0 0.1 0.04 0.3 0.035 0.4 0.044 0.5 0.08 0.6 0.2 0.65 0.4 0.62 0.5 0.5 0.48 0.33 0.38 0.18 0.27 0.135 0.225 0.12 0.17 0.09 0.08 0 0 / \endpicture $$ \caption{A situation in which $\bar \varphi_-, \bar \varphi_+$ and $\varphi$ converge to $\beta$} \end{figure} If $(\bar u_-,\bar v_-)(t)$ is a planar solution with $\bar \varphi_-(0)\le\varphi(0,\cdot)$, then we get $\bar \varphi_-(t)\le\varphi(t,\cdot)$ for all $t\ge 0$. Similarly, we get a function $\bar \varphi_+$ satisfying $\bar \varphi_+(t)\ge\varphi(t,\cdot)$ for all $t\ge 0$. Hence, the solution $(u,v)$ lies between a lower and an upper planar solution. Then it only remains to show that the angle functions $\bar \varphi_\pm(t)$ of the planar solutions both converge either to $\alpha$ or to $\beta$. We note that in the case $E={\pi\over 2}\mathbb{Z}$ the non-linearity $h_1$ can be written in the form $$ h_1(u,v) =\begin{pmatrix} u \zeta_1(u,v) \\ v \zeta_2(u,v) \end{pmatrix} \, . $$ Then the maximum principle yields that solutions $(u,v)$ with initial value $(u_0,v_0)$, $u_0>0,v_0>0$, stay positive for all $t>0$, and our result follows from well known order-preserving properties \cite{d1,h4}. We note that in this paper we use a different ordering relation which has the advantage that it does also work if $E\ne{\pi\over 2}\mathbb{Z}$. Therefore, our result in case 1 is a kind of generalization of the well known order-preserving results for $1$-homogeneous non-linearities. \subsection*{Case 2: $E=\emptyset$.} Another concept, however, is needed in the case $E=\emptyset$. In this case, loosely speaking, the part of the non-linearity which induces a rotation on $\mathbb{R}^2$ around the origin does never vanish. Hence, we cannot expect the existence of any stationary solution besides zero, but we should be able to observe periodic (or more difficult) motion whenever zero is unstable. The description of the dynamics in this case seems to be really complicated, but we show that a major part can be done by looking at planar solutions that are obtained from a scalar (time-periodic) reaction-diffusion equation instead of a system of two equations. This method can be looked at as a generalization of the approach used in \cite{b2,b3,b4} for the model system \begin{equation} \frac{d}{dt} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + \begin{pmatrix} -v \\ u \end{pmatrix} + \begin{pmatrix} u \\ v \end{pmatrix} F\begin{pmatrix} u \\ v \end{pmatrix} \label{mrd} \end{equation} As a main result we obtain that each bounded solution of (\ref{ard}) with $1$-homogeneous non-linearity $h_1$ tends either to a stationary or periodic (planar) solution, i.e.\ we get a Poincar\'e-Bendixson result. \subsection*{Remarks on both cases.} We note that the fact that the diffusion rates coincide in both equations is a crucial point. Numerical studies show that the dynamics can be totally different from the cases mentioned above for different diffusion rates, in general. The restrictions on the non-linearity, however, can be weakened. It turns out that all of our results hold for a non-linearity $f$ of the form $$ f(t,x,u,v)=h_1(u,v)+\begin{pmatrix} u \\ v \end{pmatrix}F(t,x,u,v) $$ where $F:[0,\infty)\times [0,L]\times\mathbb{R}^2\to\mathbb{R}$ is $C^1$. We will work with the more general system \begin{equation} \frac{d}{dt} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + h_1(u,v)+\begin{pmatrix} u \\ v \end{pmatrix} F(t,x,u,v) \label{rd} \end{equation} with either Dirichlet or Neumann boundary conditions for the rest of this paper. \section{Notation and preliminaries} Let $\mathbb{R}^2_*:=\mathbb{R}^2\setminus\{(0,0)\}$. For every $(\xi,\eta)\in\mathbb{R}^2_*$ the set $\{(\xi,\eta),(-\eta,\xi)\}$ is a base of $\mathbb{R}^2$ and, thus, $h_1(\xi,\eta)\in\mathbb{R}^2$ can be written in the form \begin{equation} h_1\begin{pmatrix} \xi \\ \eta \end{pmatrix} =\begin{pmatrix} \xi \\ \eta \end{pmatrix} f_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} +\begin{pmatrix} -\eta \\ \xi \end{pmatrix} g_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} \label{h1} \end{equation} where $f_0(\xi,\eta),g_0(\xi,\eta)\in\mathbb{R}$ are uniquely determined. Since $h_1\in C^1(\mathbb{R}^2_*,\mathbb{R}^2)$ and (\ref{h1}) implies that \begin{eqnarray*} f_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} &=& {1\over\xi^2+\eta^2} \begin{pmatrix} \xi \\ \eta \end{pmatrix} h_1\begin{pmatrix} \xi \\ \eta \end{pmatrix} \, , \\ g_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} &=& {1\over\xi^2+\eta^2} \begin{pmatrix} -\eta \\ \xi \end{pmatrix} h_1\begin{pmatrix} \xi \\ \eta \end{pmatrix} \, , \end{eqnarray*} we get $f_0,g_0\in C^1(\mathbb{R}^2_*,\mathbb{R})$. Since for all $\mu\in\mathbb{R}$ \begin{eqnarray*} \lefteqn{ \mu \begin{pmatrix} \xi \\ \eta \end{pmatrix} f_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} +\mu\begin{pmatrix} -\eta \\ \xi \end{pmatrix} g_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} }\\ &=& \mu h_1\begin{pmatrix} \xi \\ \eta \end{pmatrix} = h_1\begin{pmatrix} \mu\xi \\ \mu\eta \end{pmatrix} \\ &=& \mu \begin{pmatrix} \xi \\ \eta \end{pmatrix} f_0\begin{pmatrix} \mu\xi \\ \mu\eta \end{pmatrix} +\mu\begin{pmatrix} -\eta \\ \xi \end{pmatrix} g_0\begin{pmatrix} \mu\xi \\ \mu\eta \end{pmatrix} \, , \end{eqnarray*} we obtain that $$ f_0\begin{pmatrix} \mu\xi \\ \mu\eta \end{pmatrix} =f_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} \, , \quad g_0\begin{pmatrix} \mu\xi \\ \mu\eta \end{pmatrix} =g_0\begin{pmatrix} \xi \\ \eta \end{pmatrix} \, . $$ Therefore, it suffices to define $f_0,g_0$ on $S^1$ where $S^1$ is defined by $$ S^1:=\{ (x,y)\in\mathbb{R}^2: x^2+y^2=1\} \, . $$ In particular, this implies that $f_0,g_0$ and their derivatives are bounded (on $S^1$ and, hence, on $\mathbb{R}^2_*$). For $\phi_0\in\mathbb{R}$ we introduce the solution $\phi(\cdot;\phi_0):\mathbb{R}\to\mathbb{R}$ of the ODE \begin{equation}\begin{gathered} \frac{d}{dt} \phi = g_0(\cos\phi,\sin\phi) \, , \\ \phi(0) = \phi_0 \, . \end{gathered}\label{phi} \end{equation} Since $g_0$ is $C^1$ on $S^1$ (and has, thus, bounded derivatives), $\phi$ is well defined. \begin{lemma} \label{lm1} The function $\phi(\cdot;\phi_0)$ is either constant or strictly monotone. \end{lemma} \noindent{\bf Proof.} If $g_0(\cos\phi_0,\sin\phi_0)=0$, then $\phi(t;\phi_0)=\phi_0$ for all $t$. We assume that $g_0(\cos\phi_0,\sin\phi_0)\ne 0$, without loss of generality, we deal only with $g_0(\cos\phi_0,\sin\phi_0)>0$. If $\phi$ was not strictly increasing, then there would be some $t_0\in\mathbb{R}$ with $\frac{d}{dt}\phi(t_0;\phi_0)=0$ which would imply that $g_0(\cos\phi(t_0;\phi_0),\sin\phi(t_0;\phi_0))=0$ and, thus, $\phi$ would be constant, contradicting the assumption. \begin{lemma} \label{lm2} If $g_0(\cos\phi_*,\sin\phi_*)=0$ for some $\phi_*\in [0,2\pi)$, i.e.\ $\phi_*\in E$, then $\phi(\cdot;\phi_0)$ is bounded for all $\phi_0$; if there is no such $\phi_*$, i.e.\ $E=\emptyset$, then $\phi(\cdot;\phi_0)$ is unbounded. \end{lemma} \noindent{\bf Proof.} 1. If there is $\phi_*$ as above, then every non-constant function $\phi(\cdot;\phi_0)$ satisfies $$ \phi(t;\phi_0)\ne\phi_* \quad \mbox{mod $2\pi$} \quad \mbox{for all $t$.} $$ Since $\phi(\cdot;\phi_0)$ is continuous, $\phi(\mathbb{R};\phi_0)$ is contained in an interval of length $2\pi$; in particular, $\phi(\cdot;\phi_0)$ is bounded. \\ 2. If $\phi(\cdot;\phi_0)$ is bounded, then the limit $$\phi_*:=\lim_{t\to\infty} \phi(t;\phi_0)\quad \in \mathbb{R} $$ exists by Lemma \ref{lm1}. Thus, there is a sequence $t_n\nearrow\infty$ such that $\frac{d}{dt}\phi(t_n;\phi_0)\to 0 \quad (n\to\infty)$. This implies that $ g_0(\cos\phi_*,\sin\phi_*)=0$ which completes the proof. \section{Main results} \subsection*{The case $E=\emptyset$} An element $(u_0,v_0)$ of $C^1([0,L],\mathbb{R}^2)$ is called {\it planar}, if $u_0$ and $v_0$ are linearly dependent over $\mathbb{R}$, i.e.\ if there are $r_0\in C^1([0,L],\mathbb{R})$ and $\phi_0\in\mathbb{R}$ such that $$ \begin{pmatrix} u_0 \\ v_0 \end{pmatrix} = \begin{pmatrix} \cos\phi_0 \\ \sin\phi_0 \end{pmatrix} r_0 \, . $$ The set of all planar elements of $C^1([0,L],\mathbb{R}^2)$ is denoted by $\mathcal{P}$. \begin{theorem} \label{thm1} Let $(u,v)\in C^1([0,\infty)\times [0,L],\mathbb{R}^2)$ be a (classical) solution of system (\ref{rd}) with Dirichlet or Neumann boundary conditions. If $E=\emptyset$, then we get $$ \dist_{C^1}\{ (u,v)(t,\cdot),\mathcal{P}\} \to 0 \quad (t\to\infty) \, . $$ Since the $\omega$-limit set $\omega=\omega(u,v)$ associated with $(u,v)$ consists of all states at which the orbit $\{ (u,v)(t):t\ge 0\}$ accumulates, this result means that in the case $E=\emptyset$ the $\omega$-limit set of any (bounded) solution $(u,v)$ of (\ref{rd}) is contained in $\mathcal{P}$. \end{theorem} By Theorem \ref{thm1}, the crucial part of the dynamics of (\ref{rd}) takes place in the set $\mathcal{P}$ of planar elements. The following theorems deal with the dynamics on $\mathcal{P}$. \begin{theorem} \label{thm2} The set $\mathcal{P}$ is positively invariant, i.e.\ every solution $(u,v)$ of (\ref{rd}) which starts in $\mathcal{P}$ is a planar solution, i.e.\ $(u,v)(t,\cdot)\in\mathcal{P}$ for all $t\ge 0$. If $(u_0,v_0)=(\cos\phi_0,\sin\phi_0)r_0$ with $r_0\in C^1([0,L],\mathbb{R})$, $\phi_0\in\mathbb{R}$, then $(u,v)$ has the form \begin{equation} \begin{pmatrix} u \\ v \end{pmatrix}(t,\cdot) = \begin{pmatrix} \cos\phi(t;\phi_0) \\ \sin\phi(t;\phi_0) \end{pmatrix} r(t,\cdot) \label{uvr} \end{equation} where $r:[0,\infty)\times [0,L]\to\mathbb{R}$ satisfies the scalar equation \begin{equation} \frac{d}{dt} r = r_{xx}+r\left[ f_0\begin{pmatrix} \cos\phi(t;\phi_0) \\ \sin\phi(t;\phi_0) \end{pmatrix} +F\big( t,x,\begin{pmatrix} \cos\phi(t;\phi_0) \\ \sin\phi(t;\phi_0) \end{pmatrix} r\big) \right] \label{r} \end{equation} with initial value $r_0$. \end{theorem} We note that the assertion of Theorem \ref{thm2} is also true in the case $E\ne\emptyset$, but the dynamics on $\mathcal{P}$ is important only if we have the result of Theorem \ref{thm1} which ensures that all solutions finally tend to planar ones. \begin{theorem} \label{thm3} Let $(u,v)$ be a (bounded) planar solution and choose $\phi_0,r_0,r$ as in Theorem \ref{thm2}. If $E=\emptyset$, then $\phi(\cdot;\phi_0)\mod 2\pi$ is periodic and the period $p_\phi$ is given by $$ p_\phi=\inf\{\tau>0: \vert\phi(\tau;0)\vert=2\pi\} \, . $$ In particular, $p_\phi$ depends only on $g_0$. If, in addition, (\ref{rd}) is autonomous or $F$ is periodic in $t$ with a period $p_F$ such that $p_F/p_\phi$ is rational, then the $\omega$-limit set associated with $(u,v)$ consists of periodic planar elements only, i.e.\ if we take an initial value $(\bar u_0,\bar v_0)\in\omega(u,v)$ in the $\omega$-limit set associated with $(u,v)$, then $(\bar u,\bar v)$ is a periodic planar solution of (\ref{rd}). \end{theorem} \subsection*{The case $E\ne\emptyset$} Given $\alpha,\beta\in\mathbb{R}$, $\alpha<\beta$, we call \begin{eqnarray*} I_{\alpha,\beta}:&=&\Big\{ (u_0,v_0)\in C^1([0,L],\mathbb{R}^2) : \mbox{$(u_0,v_0)=r_0(\cos\varphi_0,\sin\varphi_0)$ with} \\ && r_0,\varphi_0\in C([0,L],\mathbb{R}) \mbox{such that $r_0>0$ and $\alpha<\varphi_0<\beta$ in $(0,L)$} \Big\} \end{eqnarray*} the angular space between $\alpha$ and $\beta$. If, in addition, \begin{itemize} \item $\alpha,\beta\in E$ and \item $\gamma\not\in E$ for all $\alpha<\gamma<\beta$, \end{itemize} then we call $I_{\alpha,\beta}$ an angular space associated with $E$. We note that $E\ne\emptyset$ implies that $E$ contains infinitely many elements. We show that an angular space associated with $E$ is positively invariant under the semiflow generated by (\ref{rd}) and all solutions with initial value in this angular space tend to planar solutions with (constant) angle $\alpha$ or $\beta$. \begin{theorem} \label{thm4} Let $I_{\alpha,\beta}$ be an angular space associated with $E$ and $(u,v)$ a (bounded) solution with initial value $(u,v)(0,\cdot)\in I_{\alpha,\beta}$. Then we get: \begin{itemize} \item[(a)] $(u,v)(t,\cdot)\in I_{\alpha,\beta}$ for all $t\ge 0$. \item[(b)] There are continuous functions $\varphi:(0,\infty)\to C([0,L],\mathbb{R})$ and $r:(0,\infty)\to C^1([0,L],\mathbb{R})$ such that $$ \begin{pmatrix} u \cr v \end{pmatrix} = r \begin{pmatrix} \cos\varphi \cr \sin\varphi \end{pmatrix} \, . $$ \item[(c)] If $g_0(\cos\gamma,\sin\gamma)>0$ for $\alpha<\gamma<\beta$, then $\varphi(t,\cdot)$ converges to $\beta$ in $C([0,L],\mathbb{R})$, in case $g_0(\cos\gamma,\sin\gamma)<0$ to $\alpha$. \end{itemize} \end{theorem} \section{Interpretation of the results} If we have $E=\emptyset$, then the rotation on the values of $(u,v)$ driven by $g_0$ does never stop --- formally this means that $\phi(\cdot;\phi_0)$ is strictly monotone and unbounded. In this case, our main result, Theorem \ref{thm1}, ensures that a major part of the dynamics of (\ref{rd}) can be described by looking only at the dynamics on $\mathcal{P}$, because we know that all solutions of (\ref{rd}) tend to $\mathcal{P}$ for $t\to\infty$. Then Theorems 2 and 3 show that the dynamics on $\mathcal{P}$ can be described by solving first the ODE (\ref{phi}) and then looking at the scalar reaction-diffusion equation (\ref{r}) which has been studied in \cite{b5,c1,c2}. This is obviously much easier than dealing with the full system (\ref{rd}); in particular we can use oscillation number results in order to attack equation (\ref{r}) which are, in general, not available for systems. Nevertheless, we note that the global attractor of (\ref{rd}) (if it exists) contains, in general, more elements than just the union of $\omega(u_0,v_0)$ for all points $(u_0,v_0)$. We can even show that in some cases the global attractor will contain non-planar elements. The results \cite{b4} on unstable directions for periodic solutions of the model system (\ref{mrd}) could, for example, be used in order to prove the existence of non-planar heteroclinic solutions. If we have $E\ne\emptyset$ and $\alpha\in E$, then $\phi(t;\alpha)=\alpha$ for all $t$ and the set $\{ (u_0,v_0):=r_0(\cos\alpha,\sin\alpha) : r_0\in C^1([0,L],\mathbb{R})\}$ is invariant under the semiflow induced by (\ref{rd}). In this case Theorem \ref{thm4} describes the dynamics of all solutions starting in an angular space $I_{\alpha,\beta}$ between two consecutive elements of $E$, i.e.\ in a space which is bounded by invariant sets. Theorem \ref{thm4} says that all solutions which start in the angular space $I_{\alpha,\beta}$ will be flattened out and converge to the boundary of this angular space (see Figure 2). In the proof of Theorem \ref{thm4} we will see that the crucial point is that $\alpha\le\phi_-\le\varphi(t_0,\cdot)\le\phi_+\le\beta$ for some $t_0$ yields $$ \phi(t-t_0;\phi_-)\le\varphi(t,\cdot)\le\phi(t-t_0;\phi_+) \quad \mbox{for all $t\ge t_0$.} $$ This (weak) order-preserving property together with the fact that $\phi(\cdot;\phi_0)$ converges for any $\phi_0$ is the main idea of the proof of Theorem \ref{thm4}. We note that outside the angular spaces even non-planar equilibria may exist. We can, for example, take $L=\pi$, $h_1(u,v)=(u,4v)$, and $F(t,x,u,v)=0$ which has the non-planar equilibrium $(\sin x,\sin(2x))$ (and $E={\pi\over 2}\mathbb{Z}$). \section{Construction of an appropriate transformation} Let $(u,v)\in C^1([0,\infty)\times [0,L],\mathbb{R}^2)$ be a solution of (\ref{rd}) with Dirichlet or Neumann boundary conditions. For $\varphi\in\mathbb{R}$ we consider $$ M(\varphi):=\begin{pmatrix} \cos\varphi & -\sin\varphi \\ \sin\varphi & \cos\varphi \end{pmatrix} \, . $$ Given $\phi_0\in\mathbb{R}$ we introduce $$ \begin{pmatrix} w \\ z \end{pmatrix}(t,\cdot) := M(-\phi(t;\phi_0)) \begin{pmatrix} u \\ v \end{pmatrix}(t,\cdot) \, . $$ An elementary computation shows that $(w,z)$ satisfies \begin{eqnarray} \lefteqn{ \frac{d}{dt} \begin{pmatrix} w \\ z \end{pmatrix} }\nonumber\\ &=& \begin{pmatrix} w_{xx} \\ z_{xx} \end{pmatrix} + \begin{pmatrix} -z \\ w \end{pmatrix} \Big[ g_0\big( M(\phi(t;\phi_0)) \begin{pmatrix} w \\ z \end{pmatrix}\big) -\frac{d}{dt}\phi(t;\phi_0) \Big] \label{rrd} \\ && + \begin{pmatrix} w \\ z \end{pmatrix} \Big[ F\big(t,x,M(\phi(t;\phi_0)) \begin{pmatrix} w \\ z \end{pmatrix} \big) + f_0 \big( M(\phi(t;\phi_0)) \begin{pmatrix} w \\ z \end{pmatrix}\big) \Big] \, . \nonumber \end{eqnarray} \section{Oscillation number results} The concept of oscillation numbers \cite{a1,h3,m2} (sometimes also called zero number, lap number or Matano number) deals with solutions $y:[0,\infty)\times [0,L]\to\mathbb{R}$ of a linear equation \begin{equation} \frac{d}{dt} y = y_{xx}+q(t,x)y \label{rdlin} \end{equation} where $q:[0,\infty)\times [0,L]\to\mathbb{R}$ is bounded on $[0,T]\times [0,L]$ for all $T>0$. We define the number of sign changes (the oscillation number) by \begin{eqnarray*} Z(t)&:=&\sup\big\{ k\in\mathbb{N} : \exists\,\, 00$, no matter how many zeros the initial state $y(0,\cdot)$ has, as Angenent \cite{a1} proved. The most important result about oscillation numbers is that $Z:(0,\infty)\to\mathbb{N}$ is a non-increasing function. Precisely, the value of $Z(t)$ decreases whenever $y(t,\cdot)$ has a multiple zero, i.e.\ if $y(t_0,\cdot)$ has a multiple zero, then $Z(t_1)>Z(t_2)$ for all $t_10$. If $q_1$ is bounded, then $z$ will solve equation (\ref{rdlin}) with $$ q:=q_1+F\big(t,x,M(\phi(t;\phi_0)) \begin{pmatrix} w \\ z \end{pmatrix} \big) + f_0 \big( M(\phi(t;\phi_0)) \begin{pmatrix} w \\ z \end{pmatrix}\big) $$ and we will be able to apply oscillation number results to $z$. We assume that $q_1$ is not bounded. Then there is $T>0$ and a sequence $(t_n,x_n)$ in $[0,T]\times [0,L]$ such that $\vert q_1(t_n,x_n)\vert\to\infty\quad (n\to\infty)$ and $z(t_n,x_n)\ne 0$ for all $n\in\mathbb{N}$. We may assume that $(t_n,x_n)$ converges to $(t_0,x_0)\in [0,T]\times [0,L]$. Since $g_0$ and, by definition of $\phi$, also $\frac{d}{dt}\phi$ are bounded, this implies that $$ {z(t_n,x_n)\over w(t_n,x_n)} \to 0 \quad (n\to\infty) \, . $$ Hence, $(\omega_n,\zeta_n)$ defined by $$ \begin{pmatrix} \omega_n \\ \zeta_n \end{pmatrix} := {1\over\sqrt{w^2(t_n,x_n)+z^2(t_n,z_n)}} \begin{pmatrix} w(t_n,x_n) \\ z(t_n,x_n) \end{pmatrix} $$ fulfill $\zeta_n\to 0 \quad (n\to\infty)$. This means that $\vert\omega_n\vert\to 1\quad (n\to\infty)$ and without loss of generality, we may assume that $\omega_n\to 1\quad (n\to\infty)$. Since the map $$ [0,2\pi]\times S^1\ni (\varphi,(\omega,\zeta))\mapsto g_0\big( M(\varphi) \begin{pmatrix} \omega \\ \zeta \end{pmatrix}\big) \in \mathbb{R} $$ is (uniformly) continuously differentiable, it is Lipschitz-continuous in $(\omega,\zeta)\in S^1$ uniformly for all $\varphi\in [0,2\pi]$ with Lipschitz-constant $L_g$. Hence, we get \begin{eqnarray*} \lefteqn{ \Big| g_0\big( M(\phi(t_n;\phi_0)) \begin{pmatrix} w(t_n,x_n) \\ z(t_n,x_n) \end{pmatrix}\big) - \frac{d}{dt} \phi(t_n;\phi_0) \Big| } \\ &=& \Big|g_0\big( M(\phi(t_n;\phi_0)) \begin{pmatrix} \omega_n \\ \zeta_n \end{pmatrix}\big) - g_0\big( M(\phi(t_n;\phi_0)) \begin{pmatrix} 1 \\ 0 \end{pmatrix}\big) \Big| \\ &\le & L_g \Big| \begin{pmatrix} \omega_n \\ \zeta_n \end{pmatrix} - \begin{pmatrix} 1 \\ 0 \end{pmatrix} \Big| \\ &=& L_g\sqrt{(\omega_n-1)^2+\zeta_n^2} \, . \end{eqnarray*} We define $\alpha_n\in\mathbb{R}$ by $(\omega_n,\zeta_n)=(\cos\alpha_n,\sin\alpha_n)$. Then $\tan\alpha_n=z(t_n,x_n)/w(t_n,x_n)$ and $\alpha_n\to 0$ ($\mod\pi$) ($n\to\infty$). Furthermore, it follows that $$ \sqrt{(\omega_n-1)^2+\zeta_n^2} =2\big| sin({\alpha_n\over 2})\big| =O(\tan\alpha_n) =O\big({z(t_n,x_n)\over w(t_n,x_n)}\big) \quad (n\to\infty) $$ and, thus, $q_1(t_n,x_n)$ is bounded for $n\to\infty$ contradicting our assumption. \section{Proof of Theorem \ref{thm1}} \noindent\textbf{Step 1: Assume that the assertion is false and construct a non-planar element in the $\omega$-limit set under this assumption.} By standard arguments \cite{h2,p1} we know that the solutions of (\ref{rd}) build a (local) semiflow on some Sobolev space; in particular, the orbit $$ \Gamma:=\{ (u,v)(t): t\ge 0\} $$ of the bounded solution $(u,v)$ is a precompact subset of the space $X^\alpha:=D(\Delta^\alpha)$, $\alpha\in (0,1)$ (in the notation of Henry \cite{h2}. Taking $\alpha<1$ sufficiently large and using the Sobolev embedding theorem, we obtain that $\Gamma$ lies precompact in $C^1([0,L],\mathbb{R}^2)$. This implies that, if the assertion of our theorem does not hold, then there is $(u_0',v_0')\in C^1([0,L],\mathbb{R}^2)\setminus\mathcal{P}$ and a sequence $(t_n)$, $t_n\nearrow\infty$, such that $$ \| (u,v)(t_n,\cdot)-(u_0',v_0')\|_{C^1} \to 0 \quad (n\to\infty)\, . $$ This means that $(u_0',v_0')$ is contained in the $\omega$-limit set associated with $(u,v)$. Let $(u',v')$ be the solution of (\ref{rd}) with initial value $(u_0',v_0')$. We note that $(u',v')$ is defined for all $t\in\mathbb{R}$ since $(u_0',v_0')$ is an element of the $\omega$-limit set. \smallskip \noindent\textbf{Step 2: Definition of $\phi_0$.} We want to take $\phi_0\in\mathbb{R}$ such that $z'(0,\cdot)$ defined using the formula $$ \begin{pmatrix} w' \\ z' \end{pmatrix}(t,\cdot) := M(-\phi(t;\phi_0)) \begin{pmatrix} u' \\ v' \end{pmatrix}(t,\cdot) $$ has a multiple zero. If $u_0'$ or $v_0'$ have a multiple zero, then we can take $\phi_0=\pi/2$ or $\phi_0=0$. If neither $u_0'$ nor $v_0'$ have multiple zeros, then we proceed as follows: Using the result of the last section, we can apply oscillation number results to $z'$ --- for every $\phi_0$. Since $(u',v')$ are also defined for negative $t$, $z'$ solves an equation of the form (\ref{rdlin}) for all $t\in\mathbb{R}$. Then Angenent's result \cite{a1} ensures that the number of zeros of $z'(t,\cdot)$ is finite for all $t$; in particular for $t=0$. Applying this result for $\phi_0=\pi/2$ and $\phi_0=0$, we obtain that $u_0'$ as well as $v_0'$ have only finitely many zeros (which are all simple by assumption). Hence, there is a continuous function $\varphi_0':[0,L]\to\mathbb{R}$ which satisfies $$ \tan\varphi_0'(x)={v_0'(x)\over u_0'(x)} \quad \mbox{for all $x\in [0,L]$ with $u_0'(x)\ne 0$.} $$ This result is an easy consequence of de l'Hospital's formula, and it can also be found in \cite[p.696,Lemma 2]{b1}. We set $$ \phi_0:=\max\big\{ \varphi_0'(x):x\in [0,L] \big\} $$ and take $x_0\in [0,L]$ such that $\varphi_0'(x_0)=\phi_0$. Then we get $z'(0,x_0)=0$ using the definition of $z'$. It remains to show that ${\partial\over\partial x} z'(0,x_0)=0$. \noindent{\bf Case 1: $w'(0,x_0)=0$.} Then $z'(0,x_0)=0$ implies that $u_0'(x_0)=v_0'(x_0)=0$ and, thus, $$ \tan\phi_0={{\partial\over\partial x} v_0'(x_0)\over {\partial\over\partial x} u_0'(x_0)} \, . $$ Hence, it follows that $$ {{\partial\over\partial x} z'(0,x_0)\over {\partial\over\partial x} w'(0,x_0)} =\tan( \varphi_0'(x_0)-\phi_0) =0 \, . $$ In particular, we have ${\partial\over\partial x} z'(0,x_0)=0$. \noindent{\bf Case 2: $w'(0,x_0)\ne 0$.} Let $U_0$ be a neighborhood of $x_0$ in $[0,L]$ in which $w'(0,\cdot)$ has no zeros. Since $\varphi_0'(x)\le \phi_0$ for all $x\in [0,L]$, we get $$ 0\ge \tan(\varphi_0'(x)-\phi_0) = {z'(0,x)\over w'(0,x)} \quad \mbox{for all $x\in U_0$.} $$ Since $w'(0,\cdot)$ has no sign change in $U_0$, it follows that $z'(0,\cdot)$ must not have a sign change in $U_0$, too. Hence $x_0$ must be a multiple zero of $z'(0,\cdot)$. \noindent\textbf{Step 3: Applying oscillation number results to $z'$.} As shown in Section 6, we can apply oscillation number results to $z'$. Let $Z':\mathbb{R}\to \mathbb{N}$ be the corresponding oscillation number. We note that we can define $Z'$ on the whole real axis because $z'$ is defined for all $t\in\mathbb{R}$ (and solves a linear equation of the form (\ref{rdlin}) on each interval $[-\tau,\infty)$, $\tau\ge 0$). Since $z'(0,\cdot)$ has a multiple zero by Step 2, we get $$ Z'(t_-)>Z'(t_+) \quad \mbox{for all $t_-<0Z'(t_+)$, the number $Z(t)$ of zeros of $z(t,\cdot)$ accumulates at two different values $Z'(t_\pm)$. Since oscillation number results hold for $z$ by Section 6, $Z$ has to be non-increasing which is a contradiction. \smallskip \noindent\textbf{Remark.} We note that the assumption $E=\emptyset$ is used in Step 4 only. It ensures that $\phi(\cdot;\phi_0)$ is unbounded and, thus, periodic. Otherwise (\ref{st4}) does not hold and we cannot conclude that $(w',z')(0)$ is a limit point of some function $(w,z)$ defined as above. \section{Proof of Theorems \ref{thm2} and \ref{thm3}} Suppose that $(u_0,v_0)$, $\phi_0$ and $r_0$ are given as in Theorem \ref{thm2}. First define $r$ by (\ref{r}) and then $(u,v)$ by (\ref{uvr}). An elementary computation shows that $(u,v)$ is a solution of (\ref{rd}) with initial value $(u_0,v_0)$. Using the uniqueness of the solutions of (\ref{rd}) (see, for example, \cite{h3}), the assertion of Theorem \ref{thm2} follows. By Lemma \ref{lm2}, $\phi(\cdot;\phi_0)$ is unbounded. In particular, $\phi(\cdot;\phi_0)$ is not constant and Lemma \ref{lm1} implies that it is strictly monotone. Without loss of generality assume that $\phi(\cdot;\phi_0)$ is strictly increasing. Since $\phi(t;\phi_0)$ is not bounded for $t\to\infty$, there is a uniquely determined $p_\phi>0$ such that $$ \phi(p_\phi;\phi_0)=\phi_0+2\pi \, . $$ Thus, $\phi(\cdot;\phi_0)$ is periodic with period $p_\phi$. Furthermore, there is $t_0\in [\phi_0,\phi_0+p_\phi)$ such that $\phi(t_0;\phi_0)=2\pi k$ with some integer $k$. Then it follows that $$ \phi(t_0+p_\phi j;\phi_0)=2\pi(k+j) \quad \mbox{for all integers $j$.} $$ In particular, we get $\phi(t_0-p_\phi k;\phi_0)=0$ and, thus, $$ \phi(p_\phi;0)=\phi(p_\phi;\phi(t_0-p_\phi k;\phi_0))=\phi(t_0-p_\phi k+p_\phi;\phi_0) =\phi(t_0-p_\phi(k-1);\phi_0)=\!2\pi $$ which shows that $p_\phi$ is given by the expression mentioned in Theorem \ref{thm3}. We assume that $F$ is $t$-periodic with period $p_F$ where $p_\phi/p_F$ is rational. This means that there is $p>0$ such that $p/p_\phi$ as well as $p/p_F$ are positive integers (in the autonomous case, one can just take $p=p_\phi$). In particular, equation (\ref{r}) depends periodically on $t$ (with period $p$). Let $(\bar u_0,\bar v_0)$ be an element of $\omega(u,v)$ and $(\bar u,\bar v)$ be the corresponding solution of (\ref{rd}). Then $(\bar u,\bar v)$ is planar by Theorem \ref{thm1}. By Theorem \ref{thm2}, there is $\bar\phi_0\in\mathbb{R}$ and a function $\bar r$ such that (\ref{uvr}) and (\ref{r}) hold. Since $(\bar u_0,\bar v_0)=\bar r(0)(\cos\bar\phi_0,\sin\bar\phi_0)$ is an element of $\omega(u,v)$, it is a chain-recurrent point. Then an elementary analysis like it was done in \cite[Theorem 2]{b4} shows that $\bar r(0)$ is a also chain-recurrent point for equation (\ref{r}). Then \cite{c2} ensures that $\bar r$ is a $p$-periodic solution of (\ref{r}) and, since $\phi(\cdot;\phi_0)$ is $p$-periodic, $(\bar u,\bar v)$ is a $p$-periodic solution of (\ref{rd}). \section{Proof of Theorem \ref{thm4}} \noindent\textbf{1.} We will prove the assertion only in the case $\alpha=-\pi/2$. The general case can easily be reduced to this situation setting $$ \begin{pmatrix} u' \\ v' \end{pmatrix}(t) := M(-\alpha-\pi/2) \begin{pmatrix} u \\ v \end{pmatrix}(t)\, . $$ Then $(u',v')$ solves an equation similar to (\ref{rd}) replacing $F,f_0,g_0$ by \begin{gather*} f_0'\begin{pmatrix} \xi \\ \eta \end{pmatrix} := f_0\big( M(\alpha+\pi/2)\begin{pmatrix} \xi \\ \eta \end{pmatrix} \big) \, , \\ g_0'\begin{pmatrix} \xi \\ \eta \end{pmatrix} := g_0\big( M(\alpha+\pi/2)\begin{pmatrix} \xi \\ \eta \end{pmatrix} \big) \, , \\ F'\left(t,x,\begin{pmatrix} \xi \\ \eta \end{pmatrix}\right) := F\big( t,x, M(\alpha+\pi/2)\begin{pmatrix} \xi \\ \eta \end{pmatrix} \big) \, . \end{gather*} Hence, we have $\alpha'=-\pi/2$, $\beta'=\beta-\alpha-\pi/2$ and the assertion follows using $\varphi=\varphi'+\alpha+\pi/2$. \noindent\textbf{2.} We note that $\alpha=-\pi/2$ (Step 1) implies $\pi/2\in E$ using the fact that $g_0(0,1)=g_0(0,-1)$ ($0$-homogeneity). This gives $\beta\in (-\pi/2,\pi/2]$. Hence, $(u,v)(0)\in I_{\alpha,\beta}$ implies $u_0=r_0\cos\varphi_0$ with $r_0\ge 0$ and $-\pi/2\le\alpha\le\varphi_0\le\beta\le\pi/2$ and, thus, $u_0\ge 0$. Furthermore, we get $g_0(0,-1)=g_0(\cos\alpha,\sin\alpha)=0$. Since the restriction of $g_0$ on $S^1$ is continuously differentiable, then function $\hat g_0$ given by $$ \hat g_0:S^1\ni (\cos\phi,\sin\phi)\mapsto {g_0(\cos\phi,\sin\phi)\over\cos\phi}\sin\phi \in\mathbb{R} $$ is well defined and continuous. Setting $\hat g_0(\xi,\eta):={g_0(\xi,\eta)\over\xi}\eta$ for all $(\xi,\eta)\in\mathbb{R}^2_*$, $\hat g_0$ is $0$-homogeneous, continuous and bounded; in particular, it is contained in $L^\infty(\mathbb{R}^2)$. Since $u$ satisfies $$ \frac{d}{dt} u = u_{xx}+u(F+f_0)-vg_0 = u_{xx}+u(F+f_0-\hat g_0) $$ and $u(0,\cdot)=u_0\ge 0$, maximum principle arguments imply that, for all $t>0$, $u(t,x)>0$ for $x\in (0,L)$ and $u_x(t,0),u_x(t,L)\ne 0$ in case of Dirichlet boundary conditions and $u(t,x)>0$ for $x\in [0,L]$ in case of Neumann boundary conditions. Thus, there is a continuous function $q:(0,\infty)\times [0,L]\to\mathbb{R}$ such that $$ q(t,x)={v(t,x)\over u(t,x)} \, , \quad x\in (0,L) \, . $$ (Note that $q(t,0),q(t,L)$ can be defined using de'l Hospital's formula.) We define $$ \varphi:(0,\infty)\times [0,L]\ni (t,x)\mapsto \arctan q(t,x) \in (-\pi/2,\pi/2)\, , $$ $\varphi(t,\cdot)\in C([0,L],\mathbb{R})$, and $r$ by $r(t,\cdot):=u(t,\cdot)/\cos\varphi(t,\cdot)$. Then, (b) follows. We note that the positivity of $u$ and the fact that $\varphi(t,\cdot)$ has its values in $(-\pi/2,\pi/2)$ implies that $r(t,\cdot)$ has only positive values in $(0,L)$ for all $t>0$. \noindent\textbf{3.} We take $t_0>0$ and set $\phi_-:=\min_{[0,L]}\varphi(t_0,\cdot)$, $\phi_+:=\max_{[0,L]}\varphi(t_0,\cdot)$. Note that $\varphi$ is continuous by Step 2. Let $z=z_\pm$ be defined as in Section 5 where the constant $\phi_0$ should have the value $\phi(-t_0;\phi_\pm)$. Then we get $$ z_\pm(t_0,\cdot)=r(t_0,\cdot)\sin(\varphi(t_0,\cdot)-\phi_\pm) \, . $$ Using the fact that $r(t_0,\cdot)$ is positive in $(0,L)$ by Step 2 and $\varphi(t_0,\cdot)-\phi_-\in [0,\pi)$, it follows that $z_-(t_0,\cdot)\ge 0$. We take $x_-\in [0,L]$ such that $\varphi(t_0,x_-)=\phi_-$. Then $z_-(t_0,\cdot)$ has a multiple zero at $x=x_-$. By Section 5, we can apply oscillation number results to $z_-$. Hence, $z_-(t,\cdot)$ is either the zero function or strictly positive in $(0,L)$ for all $t>t_0$. In particular, we get $$ \min_{[0,L]}\varphi(t,\cdot)\ge\phi(t-t_0;\phi_-) \quad \forall t>t_0. $$ Analogously, we get $$ \max_{[0,L]}\varphi(t,\cdot)\le\phi(t-t_0;\phi_+) \quad \forall t>t_0. $$ \noindent\textbf{4.} The solutions of (\ref{rd}) form a (local) semiflow on the space $C^1([0,L],\mathbb{R}^2)$. Hence, $(u,v)(0)\in I_{\alpha,\beta}$ implies that there is $\delta>0$ such that $$ \alpha<\varphi(t,\cdot)<\beta \quad \forall 0\le t<\delta. $$ Thus, we have $(u,v)(t)\in I_{\alpha,\beta}$ for $t\in [0,\delta)$. Take $t_0:=\delta/2$ and define $\phi_\pm$ as in Step 3. Then we get $\alpha<\phi_-\le\phi_+<\beta$ and, by definition of the function $\phi(\cdot;\phi_\pm)$, $$ \alpha<\phi(t;\phi_-)\le\phi(t;\phi_+)<\beta \quad \forall t\ge 0. $$ Thus, Step 3 implies that $(u,v)(t)\in I_{\alpha,\beta}$ for all $t\ge\delta/2$ and, using the result mentioned above, for all $t\ge 0$. This proves (a). \noindent\textbf{5.} Take $t_0>0$ and define $\phi_\pm$ as in Step 3. Then we get $(u,v)(t_0)\in I_{\alpha,\beta}$ by Step 4 and, thus, $$ \alpha <\phi_-\le \phi_+<\beta \, . $$ If $g_0(\cos\phi_0,\sin\phi_0)>0$ for $\alpha<\phi_0<\beta$, then Lemmata \ref{lm1} and \ref{lm2} yield $\phi(t;\phi_\pm)\to\beta\quad (t\to\infty)$. Since $$ \phi(t-t_0;\phi_-)\le \min_{[0,L]}\varphi(t,\cdot) \le \max_{[0,L]}\varphi(t,\cdot)\le\phi(t-t_0;\phi_+) \quad \forall t>t_0 $$ by Step 3, it follows that $\varphi(t,\cdot)$ converges to $\beta$ in $C([0,L],\mathbb{R})$. Analogously, we get convergence to $\alpha$ in the case $g_0(\cos\phi_0,\sin\phi_0)<0$ for $\alpha<\phi_0<\beta$. This proves (c). \smallskip \noindent{\bf Acknowledgement.} The author thanks Professor B.\ Fiedler, Freie Universit\"at Berlin, for helpful suggestions that led to the examination of the problem dealt with in this paper. \begin{thebibliography}{00} \frenchspacing \bibitem{a1} S.Angenent, The zero set of a solution of a parabolic equation, {\it J. reine u. angew. 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