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\markboth{\hfil Behavior of forced asymmetric oscillators at
resonance \hfil EJDE--2000/74}
{EJDE--2000/74\hfil C. Fabry \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~74, pp.~1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Behavior of forced asymmetric oscillators at resonance
%
\thanks{ {\em Mathematics Subject Classifications:} 34C15, 34C25, 70K30.
\hfil\break\indent
{\em Key words:} Resonance, frequency-response curves, jumping nonlinearity,
Fu\v c\'\i k spectrum.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted September 29, 2000. Published December 14, 2000.} }
\date{}
%
\author{ C. Fabry }
\maketitle
\begin{abstract}
This article collects recent results concerning the behavior at
resonance of forced oscillators driven by an
asymmetric restoring force, with or without damping. This synthesis
emphasizes the key role played by a function denoted by
$\Phi_{\alpha,\beta,p}$, which is, up to a sign reversal of its argument,
a correlation product of the forcing term $p$ and of a function
representing a free oscillation for theundamped equation.
The theoretical results are accompanied by graphical representations
illustrating the behavior of the damped and undamped oscillators.
In particular, the damped oscillator is considered, with a forcing
term whose frequency is close to the frequency of the free oscillations.
For that problem, frequency-response curves are studied, both
theoretically and through numerical computations, revealing a
hysteresis phenomenon, when $\Phi_{\alpha,\beta,p}$ is of constant sign.
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{prop}{Proposition}
\section{Introduction}
The oscillators studied here are represented by the equation
\begin{equation}
x'' + \alpha x^+- \beta x^-=p(t), \label{eq:osc}
\end{equation}
where
$x^+=\max\{x,0\}, x^-=\max\{-x,0\},$ $\alpha,\beta$ being positive
numbers. We also consider the
equation with damping
\begin{equation}
x'' + \varepsilon x'+\alpha x^+- \beta x^-=p(t).\label{eq:dam}
\end{equation}
These equations provide a fairly natural generalization of the
classical linear oscillator, the restoring force
being here piecewise linear. The interest for such equations has been
motivated in particular by models of
suspension bridges \cite{lm1,lm2}.
We will consider only periodic forcing terms; it is convenient to
work with the period $2\pi.$ We then speak of resonance when
the period $2\pi$ is the period of the free oscillations, i.e. of the
solutions of the homogeneous equation
\begin{equation}
x'' + \alpha x^+- \beta x^-=0.\label{eq:hom}
\end{equation}
It is easy to see that this occurs when
\begin{equation}
\frac{1}{\sqrt{\alpha}}+\frac{1}{\sqrt{\beta}}=\frac{2}{n}\,,\label{eq:fuc}
\end{equation}
for some integer $n$. The curve defined by (\ref{eq:fuc}), which
passes through the point $(n^2,n^2),$ is one of
the so-called Fu\v c\'\i k
curves \cite{fu1,fu2} for this boundary value problem; we will
denote it by $C_n.$ When $(\alpha,\beta) \in C_n, $ all
solutions of (\ref{eq:hom}) are $2\pi$-periodic; we will denote by
$\varphi_{\alpha,\beta}$ the particular solution corresponding to the
initial conditions $x(0)=0,x'(0)=1;$ it
is easily computed that
%\renewcommand{arraystretch}{1.2}
\begin{eqnarray}\label{varphi}
\varphi_{\alpha,\beta}(t)=\left\{
\begin{array}{lll}
\displaystyle \frac{1}{\sqrt {\alpha}}\sin\left(\sqrt{\alpha}\,
t\right) & \mbox{ for } & \displaystyle t\in
\left[0,\frac{\pi}{\sqrt{\alpha}}\right],\\
\displaystyle- \frac{1}{\sqrt{\beta}}\sin\left(\sqrt{\beta}
\left(t-\frac{\pi}{\sqrt{\alpha}})\right)\right)
&\mbox{ for } & \displaystyle t\in \left[\frac{\pi}{\sqrt{\alpha}},
\frac{2\pi}{n}\right],
\end{array}\right.
\end{eqnarray}
%\renewcommand{arraystretch}{1}
$\varphi_{\alpha,\beta}$ being of minimal period $2\pi/n.$
Despite the simplicity of equations (\ref{eq:osc}), (\ref{eq:dam}),
the behavior of the solutions turns out to be more
complicated that could be expected at first sight. As will appear
from the results recalled below, the global picture depends
crucially on the fact that the function $\Phi_{\alpha,\beta,p}, $ defined by
\begin{eqnarray}\label{phip}
\Phi_{\alpha,\beta,p}(\theta) :=
\frac{1}{2\pi}\int_0^{2\pi}p(t)\varphi_{\alpha,\beta}(t+\theta)\,dt
\quad
(\theta \in \mathbb R),
\end{eqnarray}
vanishes or not at some point; that function has been introduced by
Dancer \cite{dan1,dan2}. Notice that, if $(\alpha,\beta)
\in C_n,$ the function $\Phi_{\alpha,\beta,p}, $ as
$\varphi_{\alpha,\beta},$ is of period $2\pi/n;$ its number of zeros
in
the interval $[0,2\pi/n)$ will be determining for the behavior of the
oscillator.
The paper is organized as follows. Section 2 is a short presentation
of formulas concerning the
function
$\Phi_{\alpha,\beta,p},$ that will be needed later. In Section 3, we
present results concerning the equation without
damping (\ref{eq:osc}), whereas Section 4 is devoted to the equation
with damping (\ref{eq:dam}). In Section 5, we also discuss
the equation with damping, when the forcing term is no longer of period
$2\pi,$ but of period close to $2\pi;$ we will study the variation of
the amplitude of
$2\pi$-periodic solutions with respect to the frequency of the
forcing term and draw
frequency-response curves.
\section{The function $\Phi_{\alpha,\beta,p}$}
We list here, for later reference, some results about the function
$\Phi_{\alpha,\beta,p}.$ Remember that we are particularly interested
in the number of zeros of $\Phi_{\alpha,\beta,p}$ in the
interval
$[0,2\pi/n).$
If (\ref{eq:fuc}) is satisfied with
$\alpha=\beta=n,$ the differential equation (\ref{eq:hom}) is linear
and we have $\varphi_{n,n}(t)= \sin (nt)/n\,;$
the function $\Phi_{n,n,p}(\theta)$ is then, up to a factor $2/n,$
made of the Fourier components of order $n$ of the function
$p.$ Hence,
$\Phi_{n,n,p}$ is of the form
$A
\cos(nt) +B\sin(nt);$ it is identically zero when the corresponding
Fourier coefficients $A,B$ are equal to $0;$ otherwise,
it has $2$ zeros in the interval $[0,2\pi/n).$ Clearly, the function
$\Phi_{n,n,p}(\theta)$ cannot be of constant sign, a
situation which, by contrast, can occur when $\alpha \neq \beta.$
In the examples considered below for $\alpha \neq \beta,$ we will
deal for instance with right-hand sides $p$ of the form
$p(t) = a + b
\cos (kt).$ For such a function
$p,$ assuming that
$(\alpha,
\beta)$ satisfy (\ref{eq:fuc}) with $n=1$ and that $\alpha \neq
\beta,$ it is computed that
\begin{eqnarray}
\Phi_{\alpha,\beta,p}(\theta)
&=&\frac{a}{\pi}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\nonumber
\\
& &\hspace{-2.8cm} +\frac{b}{2\pi}
\,\frac{\beta-\alpha}{(k^2-\beta)(k^2-\alpha)} \left\{\left[1+\cos
\left(\frac{k\pi}{\sqrt{\alpha}}\right)\right]\cos (kt) +
\sin\left(\frac{k\pi}{\sqrt{\alpha}}\right) \sin (kt)\right\} . \label{eq:for}
\end{eqnarray}
The function $\Phi_{\alpha,\beta,p}$ is of constant sign if
\begin{eqnarray}
\frac{|a|}{|b|} >
\,\frac{\alpha \beta
}{|k^2-\beta||k^2-\alpha|}\cos\left(\frac{k\pi}{2
\sqrt{\alpha}}\right)\,,\label{eq:fs}
\end{eqnarray}
the sign being that of $a (\beta-\alpha).$ It immediately results
from the expression (\ref{eq:for}) that, when the
inequality (\ref{eq:fs}) is reversed,
$\Phi_{\alpha,\beta,p}$ has
$2k$ simple zeros in $[0,2\pi).$
\section{Forced oscillator without damping}
Concerning equation (\ref{eq:osc}), with a forcing term $p$ of period
$2\pi,$ the comparison with the linear oscillator
suggests the following two questions:
\begin{itemize}
\item
For which $p$ does the problem have $2\pi$-periodic solutions?
\item
For which $p$ does the problem have unbounded solutions?
\end{itemize}
A fairly complete answer to those questions is provided by the
results recalled below. We start with sufficient conditions
for the existence of $2\pi$-periodic solutions.
\begin{prop}[\cite{fafo}] \label{theo:fafo} Assume that
$(\alpha,\beta) \in C_n,$ and that $p$ is continuous and
$2\pi$-periodic. Then, if
$\Phi_{\alpha,\beta,p} $ has $2z$ zeros in $[0,2\pi/n)\,,$ all zeros
being simple, and if $z\neq 1,$
equation (\ref{eq:osc}) has
(at least) one $2\pi$-periodic solution.
\end{prop}
Notice that, since $\Phi_{\alpha,\beta,p} $ is $2\pi/n$-periodic, the
number of zeros in $[0,2\pi/n)\,,$ if they are simple,
must be even. The above result contains of course the case of a function
$\Phi_{\alpha,\beta,p}
$ of constant sign, a case already treated by Dancer
\cite{dan1,dan2}. It also follows from Proposition \ref{theo:fafo}
that
the only situations where equation (\ref{eq:osc}) can have no
$2\pi$-periodic solution, are the case where $\Phi_{\alpha,\beta,p} $
has $2$ zeros in $[0,2\pi/n)\,,$ and the case
where
$\Phi_{\alpha,\beta,p} $ has multiple zeros. For the first case, it
is possible to exhibit $2\pi$-periodic forcing $p$
such that equation (\ref{eq:osc})
has no
$2\pi$-periodic solution (see \cite{dan1,dan2,lm1,wan}). In
particular, Wang Zaihong \cite{wan}, extending results of Lazer and
McKenna
\cite{lm1}, has shown that, if $(\alpha,\beta) \in C_n,$
$$ x'' + \alpha u^+ - \beta u^-= \cos nt$$
has no $2\pi$-periodic
solution. We do not know if functions $p$ exist, for which
$\Phi_{\alpha,\beta,p} $ has multiple zeros and for which equation
(\ref{eq:osc}) has no
$2\pi$-periodic solution.
On the other hand, it can be shown that, with $(\alpha,\beta) \in
C_n,$ the set of $2\pi$-periodic solutions of equation
(\ref{eq:osc}) is bounded, unless $\Phi_{\alpha,\beta,p} $ has
multiple zeros. Examples of unbounded sets of $2\pi$-periodic
solutions are however easy to construct. For instance, if $\psi$ is a
twice differentiable function such that $\psi(t)
\varphi_{\alpha,\beta}(t)>0,$ for all
$t$ such that
$\varphi_{\alpha,\beta}(t) \neq 0,$ and if we define $p$ by $p(t)=
\psi''(t) +\alpha \psi^+(t)-\beta\psi^-(t),$ it is easy to
verify that, for any $k>0,$ $\psi +k\varphi_{\alpha,\beta}$ is a
solution of (\ref{eq:osc}).
With that choice of
$p,$ the function
$\Phi_{\alpha,\beta,p} $ can be seen to be identically zero.
\medbreak
It has been observed by Ortega \cite{ort} that, because of a result
of Massera \cite{mas}, if equation (\ref{eq:osc}) admits no
$2\pi$-periodic solutions, then all solutions must be unbounded. That
condition is of course not necessary to have unbounded
solutions. Indeed, a result of Alonso and Ortega \cite{ao} shows that,
as soon as $\Phi_{\alpha,\beta,p} $ has zeros (assumed to be
simple), equation (\ref{eq:osc}) will admit unbounded solutions.
\begin{prop} \label{theo:ao} If the function $\Phi_{\alpha,\beta,p}$
takes both positive
and negative values, and if all its zeros are simple, then there exists $R > 0$
such that every solution $x$ of equation (\ref{eq:osc}) with $$x(t_0)^2 +
x'(t_0)^2 > R$$ for some $t_0 \in \mathbb R,$ is such that $$x(t)^2 +
x'(t)^2 \to
+\infty$$ as $t \to +\infty$ or $t \to -\infty.$
\end{prop}
Combining Propositions \ref{theo:fafo} and \ref{theo:ao}, it clearly
appears that, when $\alpha \neq \beta,$ equation
(\ref{eq:osc}) can have both
$2\pi$-periodic and unbounded solutions. It is the case for instance
for the equation
$$ x'' + \alpha x^+- \beta x^- = \cos (3t),$$
with $(\alpha, \beta) \in C_1, \alpha \neq \beta.$ Indeed, it can be
shown that, with $p(t)= \cos
(3t),$ we have $\Phi_{\alpha,\beta,p}(t) = A_{\alpha,\beta}
\cos(3t) + B_{\alpha,\beta}
\sin(3t),$ with $A_{\alpha,\beta}^2+B_{\alpha,\beta}^2 \neq 0,
\forall (\alpha,\beta) \in C_1,
\alpha \neq \beta.$ Both Proposition \ref{theo:fafo} and Proposition
\ref{theo:ao} thus apply to that equation.
\medbreak
By contrast to Proposition \ref{theo:ao}, it has been proved by Liu
Bin \cite{liu} that, when $\Phi_{\alpha,\beta,p} $ is of
constant sign, and provided that
$p$ is sufficiently regular, all solutions of (\ref{eq:osc}) are bounded.
\begin{prop}
\label{theo:liu}
If
$p $ is of class $C^6,$ and if the function $\Phi_{\alpha,\beta,p} $
is of constant
sign, then all solutions
of equation (\ref{eq:osc}) are such that
\begin{eqnarray*}
\sup_{t \in \mathbb R}\{|x(t)| + |x'(t)|\} < \infty.
\end{eqnarray*}
\end{prop}
In order to gain some understanding of the behavior of solutions
corresponding to large initial
conditions, let us use the change of variables
\begin{equation}%
x(t) = \rho u(t) \varphi_{\alpha,\beta}(t+ \theta(t)), x'(t) =
\rho u(t)\varphi'_{\alpha,\beta}(t+ \theta(t))\,; \label{eq:ch}
\end{equation}%
we will assume that $\rho>0$ is ``large'', but fixed, and consider
initial conditions like $u(0)=1,
\theta(0)=\theta_0.$ That change of variables transforms
(\ref{eq:osc}) into
%
\begin{eqnarray}%
u' &=& \frac{1}{\rho } \,p(t) \varphi'_{\alpha,\beta}(t+\theta) \\
\theta' &=& \frac{1}{\rho u} \,p(t) \varphi_{\alpha,\beta}(t+\theta).
\end{eqnarray}%
The factor $1/\rho$ being small, this can be considered a weakly nonlinear
system
to which
the method of averaging can be applied. The averaged system, for
which the variables
will be denoted $\sigma,\tau,$ writes
\begin{eqnarray}%
\sigma' &=& \frac{1}{\rho } \,\Phi'_{\alpha,\beta,p}(\tau)\label{eq:av1}\\
\tau' &=& - \frac{1}{ \rho \sigma }
\,\Phi_{\alpha,\beta,p}(\tau).\label{eq:av2}
\end{eqnarray}%
If we take the initial conditions $\sigma(0)= u(0), \tau(0)= \theta(0),$
the method of averaging (see \cite{ver}) ensures that $\sigma(t),
\tau(t)$ are approximations of
$u(t),\theta(t),$ with an error which is of the order of $1/\rho,$ on an
interval whose length is of the order of $\rho.$ But,
%
\begin{equation}%
\sigma(t)
\Phi_{\alpha,\beta,p}(\tau(t))= C \label{eq:int}
\end{equation}%
is a first integral for the system (\ref{eq:av1}),
(\ref{eq:av2}). It then follows that, $u(t)
\Phi_{\alpha,\beta,p}(\theta(t))$ will be
close to a constant on an interval of length of the order of $\rho.$ The
following conclusions can be drawn from that observation.
\medbreak
\noindent{\sc $1^\mathrm{st}$ case.} We discuss first the case where
$\Phi_{\alpha,\beta,p}$ is of constant sign. Since $u(t)
\Phi_{\alpha,\beta,p}(\theta(t))$
remains close to a constant $C$ on a interval of the order of $\rho,$
$u(t)$ will, on such an interval, oscillate roughly between
$$ \frac{|C|}{\max |\Phi_{\alpha,\beta,p}|} \mbox{ and }
\frac{|C|}{\min |\Phi_{\alpha,\beta,p}|},$$
the constant $C$ depending on the initial conditions. Since, by
(\ref{eq:ch}), $\rho u$ can be interpreted as the amplitude of
the solutions, this means that a beating phenomenon is observed.
Moreover, since $u'$ is proportional to
$1/\rho$ and since the amplitude is of the order of $\rho,$ the
amplitude fluctuations are slower for larger solutions. Figure
1 represents the actual behavior of a solution for the equation
$$x'' + \alpha x^+ - \beta x^- = -2 + \cos t, $$
with $\alpha=2,$ and $ (\alpha,\beta) \in C_1.$ Using formula
(\ref{eq:fs}) and the remark following it, it is easily
seen that, for the choice of
$\alpha,\beta,p$ made here,
$\Phi_{\alpha,\beta,p}$ is positive.
\begin{figure}[h!] \label{fig:f1}
\begin{center}
\includegraphics[scale=0.65]{beating.eps}
\caption{Beating phenomenon}
\end{center}
\end{figure}
The proof of Proposition \ref{theo:liu} is based on a version of the
so-called ``twist theorem'' of
Kolmogorov-Arnold-Moser. To get an idea of the twist conditions
involved, observe that, by (\ref{eq:av1}), (\ref{eq:av2}),
(\ref{eq:int}) we have
$$ \frac{2\pi}{\rho |C|} \, \min \Phi^2_{\alpha,\beta,p} \leq
|\tau(2\pi)-\tau(0)| \leq \frac{2\pi}{\rho |C|} \, \max
\Phi^2_{\alpha,\beta,p}\,.$$
Denoting $\tau_1,\tau_2$ solutions corresponding respectively to
$\rho_1,\rho_2,$ it is clearly
possible to find $\rho_1,\rho_2,$ with $\rho_2 >
\rho_1,$ such that
$$ |\tau_2(2\pi)-\tau_2(0)|\leq |\tau_1(2\pi)-\tau_1(0)|,$$
independently of $\tau_1(0),\tau_2(0).$ This is, for the averaged
system (\ref{eq:av1}), (\ref{eq:av2}), the desired twist
effect: interpreting $(\rho \sigma,\tau)$ as pseudo-polar
coordinates, it appears that the angular variation
$|\tau(2\pi)-\tau(0)|,$ on the interval $[0,2\pi],$ decreases when
$\rho$ is increased. From there, it can be shown, through
the approximation of
$u,\theta$ by
$\sigma,\tau$ that, for
$\rho$ sufficiently large, the Poincar\'e map
$(x(0),x'(0)) \mapsto (x(2\pi),x'(2\pi)),$ relative to equation
(\ref{eq:osc}) for the period $2\pi,$ also presents a twist
effect for solutions of large amplitude.
The dynamics of the system can be illustrated by looking at
Poincar\'e sections. Figure
2 shows Poincar\'e sections for the equation $$ x'' +
\alpha x^+ - \beta x^- = 2.5+ \cos
t, $$ with $\alpha=2 ,$ and $
(\alpha,\beta) \in C_1.$ For that choice of $\alpha,\beta,p,$ the
function $\Phi_{\alpha,\beta,p}$ is negative everywhere.
One recognizes in Figure
2 invariant curves for the Poincar\'e map associated to
equation (\ref{eq:osc}), and the
so-called ``islands chains'' typical of area-preserving maps, when a
twist condition
is satisfied.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.65]{island.eps}
\caption{Poincar\'e sections for $ x'' + 2 x^+ - \beta x^- = 2.5 +
\cos t, $ with $
(2,\beta) \in C_1.$ }
\end{center}\label{fig:f2b}
\end{figure}
\medbreak
\noindent{\sc $2^\mathrm{nd}$ case.}
Suppose that $\Phi_{\alpha,\beta,p}$ has zeros, all zeros being
simple. Assuming $C \neq 0,$ the
relation $ \sigma(t)
\Phi_{\alpha,\beta,p}(\tau(t))= C $ prevents
$\Phi_{\alpha,\beta,p}(\tau(t))$ from changing sign.
By (\ref{eq:av2}), we see that $\tau'$ will be of constant sign. For
the sake of
definiteness, assume for instance that $\tau$ is decreasing. If
$\tau^{\ast}$ is the
nearest zero of $\Phi_{\alpha,\beta,p}$ at the left of $\tau (0)$, it
is fairly clear
that we must have
$$\lim_{t \rightarrow \infty} \tau (t) = \tau^{\ast}.$$
By (\ref{eq:int}), we then have
$$\lim_{t \rightarrow \infty} \sigma (t) = + \infty.$$
and, by (\ref{eq:av1}),
$$\lim_{t \rightarrow \infty} \sigma'(t) = \frac{1}{\rho}\,
\Phi'_{\alpha,\beta,p}(\tau^{\ast}
),$$ showing that, asymptotically, the growth of $\sigma$ is linear
($\tau^{\ast}$ is
assumed to be a simple zero of $\Phi_{\alpha,\beta,p}).$ Since
$\sigma (t)$ is close to
$u(t)$ on large intervals, the same conclusions will roughly hold for
the amplitude
$u(t).$ The argument can be turned into a proof of Proposition
\ref{theo:ao} (see \cite{fm}).
An example of solutions whose
amplitude grow exactly linearly is provided by the equation
$$ x'' + \alpha x^+- \beta x^-=2 \varphi'_{\alpha,\beta}(t),$$
with $(\alpha,\beta) \in C_1.$ Indeed, it is easy to check that $x(t)
= (t-t_0) \varphi_{\alpha,\beta}(t)$ is a solution of that
equation for $t\geq t_0.$
\section{Forced oscillator with damping}
We consider now the equation with damping
\begin{equation}
x'' + \varepsilon x'+\alpha x^+- \beta x^-=p(t).\label{eq:dam2}
\end{equation}
We will assume that the damping coefficient is small and will study
the following questions:
\begin{itemize}
\item
For which $p$ does equation (\ref{eq:dam2}) have $2\pi$-periodic
solutions whose amplitude grows to infinity when
$\varepsilon$ goes to $0?$
\item
In the opposite case, what is the behavior of solutions with large
initial conditions?
\end{itemize}
The following proposition \cite{fafo,fafo2} shows basically that
large amplitude $2\pi$-periodic
solutions are present when $\Phi_{\alpha,\beta,p}$ has simple zeros,
whereas the set of $2\pi$-periodic
solutions is bounded, independently of $\varepsilon$ close to $0,$
when $\Phi_{\alpha,\beta,p}$ is
of constant sign.
\begin{prop} Let $\alpha,\beta$ satisfy (\ref{eq:fuc}).\\
(i) Assume that $\theta^*$ is a simple zero of
$\Phi_{\alpha,\beta,p}.$ Let $u^*:=2
|\Phi'_{\alpha,\beta,p}(\theta^*)|.$
Then, for $\varepsilon \Phi'_{\alpha,\beta,p}(\theta^*) >0,$ with
$\varepsilon$ small enough, the equation (\ref{eq:dam2}) has
a $2 \pi$-periodic solution
$x_\varepsilon$ such that
$$
(x_\varepsilon(t),x_\varepsilon'(t))=\frac{1}{|\varepsilon|}u_{\varepsilon}(t)
(\varphi_{\alpha,\beta}(t+\theta_{\varepsilon}(t)),\varphi'_{\alpha,\beta}(t+
\theta_{\varepsilon}(t)))\,,
$$
the functions $u_{\varepsilon}, \theta_{\varepsilon}$ being
$2\pi$-periodic and such that
$$
\lim_{\varepsilon \to 0}u_{\varepsilon}(t)=u^*\;,\lim_{\varepsilon
\to 0} \theta_{\varepsilon}(t)=\theta^*\;,
$$
uniformly in $t.$ Moreover, the solution $x_\varepsilon$ is
asymptotically stable if $\varepsilon>0,$ unstable if $\varepsilon<0.$\\
(ii) If $\Phi_{\alpha,\beta,p}$ has $2z$ zeros in $[0,2\pi/n),$ all
simple, there exists $\varepsilon_0>0$ and $R>0$ such that,
if
$0<|\varepsilon| \leq \varepsilon_0,$ there are exactly $z$
$2\pi$-periodic solutions of (\ref{eq:dam2}) having $\|\cdot
\|_{\infty}$-norm larger than $R.$ If $z=1$ and if problem
(\ref{eq:osc}) has no $2\pi$-periodic solution, problem
(\ref{eq:dam2}) has a unique $2\pi$-periodic solution for
$0<|\varepsilon| \leq \varepsilon_0.$\\
(iii) If $\Phi_{\alpha,\beta,p}$ is of
constant sign (and does not vanish), the set of $2 \pi$-periodic
solutions of (\ref{eq:dam2}) is bounded, independently of
$\varepsilon$ in some neighborhood of $0.$
\end{prop}
When $\Phi_{\alpha,\beta,p}$ is of constant sign, the information
provided by the above theorem, concerning the $2
\pi$-periodic solutions, can be completed by a result concerning all
solutions of (\ref{eq:dam2}). Indeed, as stated
in the next proposition \cite{fab}, when $\Phi_{\alpha,\beta,p}$ is
of constant sign, all solutions end up ultimately in a set
whose size is
$o(\varepsilon)$ for
$\varepsilon $ going to $0$ by positive values.
\begin{prop} \label{prop:dam2}
Let $\alpha,\beta$ satisfy (\ref{eq:fuc}). If $\Phi_{\alpha,\beta,p}$
of constant sign (and does not vanish), and if $x_{\varepsilon} $
denotes any solution of (\ref{eq:dam}),
$$ \lim_{\varepsilon \to 0_+} \,\limsup_{t \to \infty} \varepsilon
(|x_{\varepsilon}(t)|+|x_{\varepsilon}'(t)|) =0.$$
\end{prop}
In Figure 3, we show the asymptotic behavior of the
Poincar\'e sections for the damped
oscillator represented by the equation
$$ x'' + \varepsilon x' + \alpha x^+ - \beta x^- = 2.5+ \cos t, $$
with $\alpha=2
,$$
(\alpha,\beta) \in C_1, \varepsilon=0.001.$ As indicated above, the function
$\Phi_{\alpha,\beta,p}$ is then negative everywhere. An analysis of
the numerical results shows the presence of
four periodic solutions, three of them being subharmonic solutions of
order $3,4$ and $7$
respectively; all solutions tend towards those periodic solutions when
$t \to
\infty.$ This damped oscillator thus apparently has several
asymptotically stable periodic solutions.
\begin{figure}[h!]\label{fig:f3}
\begin{center}
\includegraphics[scale=0.65]{damped.eps}
\caption{Poincar\'e sections for $ x'' + 0.001 x' + 2 x^+ - \beta x^-
= 2.5 + \cos t, $ with $
(2,\beta) \in C_1.$ }
\end{center}
\end{figure}
\section{Frequency-response curves}
In this section, we study the damped equation, with a forcing term
whose frequency is close, but not equal to the frequency
of the free oscillations, i.e. we assume that $(\alpha,\beta)$
satisfy (\ref{eq:fuc}) and consider the equation
$$
x'' + \varepsilon x'+\alpha x^+- \beta x^-=p(t)\,,
$$
with $\varepsilon$ ``small'' and $p$ of period $T,$ with
$T=2\pi+O(\varepsilon),$ for $\varepsilon \to 0.$ By means of a change
of time scale, it is basically equivalent to consider the problem
\begin{equation}
x'' + \varepsilon x'+(1+\varepsilon k) (\alpha x^+- \beta x^-)=p(t),
\label{eq:dam3}
\end{equation}
where $(\alpha,\beta)$ still satisfy (\ref{eq:fuc}), $p$ is of period
$2\pi$ and $k $ is a given constant.
That equation has been studied in \cite{fab2}; we recall here the main results.
It is shown in \cite{fab2} that, when $2\pi$-periodic solutions
having an amplitude
of the order of $1/\varepsilon$ are present, they can be
written under the form
$$
(x_{\varepsilon}(t),x'_{\varepsilon}(t))=\frac{1}{|\varepsilon|}u_{\varepsilon}(
t)
(\varphi(t+\theta_{\varepsilon}(t)),\varphi'(t+\theta_{\varepsilon}(t)),
$$
with
\begin{equation}
\lim_{\varepsilon \to 0} \theta_{\varepsilon}(t) = \theta^{*},
\lim_{\varepsilon \to
0}u_{\varepsilon}(t) = u^* =2 |\Phi'_{\alpha,\beta,p} (\theta^{*})|,
\label{eq:tu}
\end{equation}
$\theta^*$ being a solution of
\begin{eqnarray}
\varepsilon \Phi'_{\alpha,\beta,p} (\theta^{*})>0 &,& \quad
k\Phi'_{\alpha,\beta,p}(\theta^{*})
- \Phi_{\alpha,\beta,p}(\theta^{*}) = 0.\label{eq:th} % \\
\end{eqnarray}
Conditions (\ref{eq:th}) are easily studied by looking, in the
$xy$-plane, at the intersection of the line $y=x/k$ with the
curve parametrized by $$ \theta \mapsto
(\Phi_{\alpha,\beta,p}(\theta), \Phi'_{\alpha,\beta,p}(\theta)).$$
The fact that
$\Phi_{\alpha,\beta,p}$ admits zeros or not again makes a difference;
it is obvious that, when
$\Phi_{\alpha,\beta,p}$ admits zeros, all of them being simple, a
value of $\theta^*$ can always be
found such that conditions (\ref{eq:th}) are satisfied. This
observation is the basis of the next
proposition, which is limited, for the sake of simplicity, to the case $n=1.$
\begin{prop} Assume that (\ref{eq:fuc}) is satisfied with $n=1.$ Let
the sign of $\varepsilon$ be
fixed. If
$\Phi_{\alpha,\beta,p}$ has $z$ simple zeros $\theta_1,
\ldots,\theta_z \in [0,2\pi)\;$ with $\varepsilon
\Phi'_{\alpha,\beta,p}(\theta_i)>0$ for
$i=1,\ldots,z,$ then, for any
$k \in \mathbb R,$ problem
(\ref{eq:dam3}) has for $\varepsilon$ sufficiently small, at least $z$ periodic
solutions of period $2\pi,$ whose amplitude is $O(1/\varepsilon)$ for
$\varepsilon \to 0\,.$ Moreover, if $\varepsilon>0$ and if
\begin{equation}
k\Phi'_{\alpha,\beta,p}(\theta) -\Phi_{\alpha,\beta,p}(\theta) = 0
\Longrightarrow
k\Phi_{\alpha,\beta,p}''(\theta) -\Phi'_{\alpha,\beta,p}(\theta) \neq 0,
\label{eq:nd}
\end{equation}
those $z$ solutions are asymptotically stable.
\end{prop}
By contrast to the above case, when $\Phi_{\alpha,\beta,p}$ is of
constant sign, the existence of solutions for
(\ref{eq:th}) and hence, the presence of $2\pi$-periodic solutions having an
amplitude of the order of $1/\varepsilon$ for (\ref{eq:dam3}), will
depend on the value of
$k.$ It is fairly immediate that, for $\varepsilon >0,$ a value of
$\theta^*$ satisfying conditions
(\ref{eq:th}) can be found only if
$sk \geq k_{{\rm crit}},$ where $s$ is the sign of $\Phi_{\alpha,\beta,p}$ and
$$
k_{{\rm crit}}=1\left /\max\left\{\frac{\Phi'_{\alpha,\beta,p}
(\theta)}{|\Phi_{\alpha,\beta,p}(\theta)|} \mid \theta \in
[0,2\pi/n)\right\}\right..
$$
This explains the following proposition.
\begin{prop} \label{prop:7}
Let $\varepsilon>0$ and let $\Phi_{\alpha,\beta,p}$ be nonconstant,
but of constant sign
\linebreak $s \;(=\pm 1)$ (and not vanishing) on
$[0,2\pi)\;.$ Then,
%
\begin{itemize}%
\item
for $skk_{crit},$ equation (\ref{eq:dam3}) has, for $\varepsilon$
sufficiently small,
at least two
$2\pi$-periodic solutions having an amplitude
$O(1/\varepsilon)$ for $\varepsilon\rightarrow 0_+\,;$ moreover, if
(\ref{eq:nd}) is satisfied,
there is at least one asymptotically stable $2\pi$-periodic solution
and one unstable
$2\pi$-periodic solution of amplitude
$O(1/\varepsilon)$ for $\varepsilon\rightarrow 0_+\,.$
%
\end{itemize}%
\end{prop}
The amplitude of the solutions of the $2\pi$-periodic solutions can
be estimated by (\ref{eq:tu}). This has been done in
\cite{fab2} for $p(t) = a + \cos t.$ With $n=1,$ the function
$\Phi_{\alpha,\beta,p}$ is then of the form
$$\Phi_{\alpha,\beta,p}=a
c_0 +c_1
\cos(\theta -\theta_0) \mbox{ for some }\theta_0\,;$$
$c_0$ and $c_1$ can be computed from (\ref{eq:for}), which gives
\begin{eqnarray*}
c_0&=&\frac{1}{\pi}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\,,\\
c_1&=& \frac{1}{\pi} \left|
\cos\left(\frac{\pi}{2\sqrt{\alpha}}\right) \right|
\frac{|\beta-\alpha|}{|\beta -1|\,|\alpha-1|}\,.
\end{eqnarray*}
From (\ref{eq:th}), it is easy to show that the value of $u^*,$ which
determines the amplitude, satisfies the
relation
$$
\left(\frac{u^*}{2}\right)^2 + \left(sk\frac{u^*}{2}- a c_0 \right)^2
=c_1^2\,,
$$
(we assume $\varepsilon>0).$
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.7]{freq_response.eps}
\caption{Large periodic solutions of
$x'' + \varepsilon x' + (1+\varepsilon k) (\alpha
x^+ - \beta x^-) = a +\cos t$\label{fig:f4b}}
\end{center}
\end{figure}
Figure 4 represents, for $n=1, \alpha=2, (\alpha,\beta)
\in C_1, \varepsilon >0,$ the
amplitude $u^*$ as a function of
$k,$ for $a=1$ and for $a=2.$ The last value corresponds to a
function $\Phi_{\alpha,\beta,p}$ of constant (negative) sign,
whereas the former leads to a function $\Phi_{\alpha,\beta,p}$ having
zeros. It follows from Proposition \ref{prop:7} that
for $k>-k_{{\rm
crit}},$ the set of $2\pi$-periodic solutions is bounded,
independently of $\varepsilon,$ for $ \varepsilon$ small. The value
$k_{{\rm crit}}$ is easily computed to be
$$
k_{{\rm crit}}=
\frac{\sqrt{a^2c_0^2-c_1^2}}{c_1}\simeq 1.13\,.
$$
The diagram in Figure 4 shows the amplitude in the limiting
situation
$\varepsilon \to 0_+.$ It must be pointed out that the limit need not be
uniform in $k, $ and consequently, the asymptotic behavior suggested
by that figure
for $k\to \pm \infty,$ might not correspond to what is observed for a
particular
value of $\varepsilon.$ It is therefore interesting to look at actual
frequency-response curves.
Such curves are represented in Figures 5 and
6. The equation considered is
$$ x'' + 0.1 x' + \alpha
x^+ - \beta x^- = a +\cos \omega t, \mbox{ with } \alpha=2,
(\alpha,\beta) \in C_1.$$
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.7]{actual_response.eps}
\caption{Frequency-response curve for
$x'' + 0.1 x' + \alpha
x^+ - \beta x^- = 0.5 +\cos \omega t $ \label{fig:rep1}}
\end{center}
\end{figure}
The curves in the two diagrams show, as a function of $\omega,$ the
norm of the vector of initial
conditions for $2\pi$-periodic solutions, respectively when $a=0.5 $
and when $a=5.$ In Figure
5, the frequency-response curve is similar to
that of a linear oscillator whereas, in the second case, where the function
$\Phi_{\alpha,\beta,p}$ is of constant (negative) sign, a
``foldover'' of the curve is observed. The dashed part in the
frequency-response curve corresponds to an unstable periodic
solution, meaning that an hysteresis phenomenon is present, as
in Duffing's equation with a forcing term. When the pulsation
$\omega$ is decreased, starting from a value $\omega_0 >1.5,$ for
instance, the amplitude of the response will
increase, until the value $\omega_1$ is reached; at that point, the
amplitude will jump abruptly to a smaller
value.
\begin{figure} [h!]
\begin{center}
\includegraphics[scale=0.7]{actual_response2.eps}
\caption{Frequency-response curve for
$x'' + 0.1 x' + \alpha
x^+ - \beta x^- = 5 +\cos \omega t $ \label{fig:rep4}}
\end{center}
\end{figure}
This phenomenon is illustrated by Figure 7, where
a solution of the equation
$$ x'' + 0.1 x' + \alpha
x^+ - \beta x^- = 5 +\cos (1.6 t - t^2/1500) $$ is plotted; as
before, we assume that $(\alpha, \beta) $ belongs
to
$C_1, $ with $\alpha=2.$ On the contrary, if the pulsation is
increased starting from a value
$\omega<\omega_1, $ the largest amplitudes are not reached.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.7]{hysteresis.eps}
\caption{A solution of
$x'' + 0.1 x' + \alpha
x^+ - \beta x^- = 5 +\cos (1.6 t - t^2/1500) $ \label{fig:hysteresis}}
\end{center}
\end{figure}
\begin{thebibliography}{xx}
\bibitem{ao} J. M. Alonso and R. Ortega, Roots of unity and unbounded motions
of an asymmetric oscillator, J. Differential Equations, 143 (1998),
201-220.
\bibitem{dan1} E. N. Dancer, Boundary-value problems for
weakly nonlinear ordinary differential equations, Bull. Austral.
Math. Soc. 15 (1976), 321-328.
\bibitem{dan2} E. N. Dancer, Proofs of the results in "Boundary-value
problems for weakly nonlinear ordinary differential equations",
unpublished.
\bibitem{fab} C. Fabry, Aspects of resonance in nonlinear oscillators, to
appear.
\bibitem{fab2} C. Fabry, Large-amplitude oscillations of a nonlinear
asymmetric oscillator with damping,
Nonlinear Anal., to appear.
\bibitem{fafo} C. Fabry, A. Fonda, Nonlinear resonance in
asymmetric oscillators,
J. Differential Equations, 147 (1998), 58-78.
\bibitem{fafo2} C. Fabry, A. Fonda, Bifurcations from infinity in
asymmetric nonlinear
oscillators, NoDEA 7 (2000), 23-42.
\bibitem{fm} C. Fabry, J. Mawhin, Oscillations of a forced asymmetric
oscillator at resonance, Nonlinearity, 13 (2000), 493-505.
\bibitem{fu1} S. Fu\v c\'\i k, Boundary value problems with jumping
nonlinearities, \v Casopis pro p\v est. mat. 101 (1976), 69--87.
\bibitem{fu2} S. Fu\v c\'\i k, Solvability of Nonlinear Equations
and Boundary Value Problems, Reidel, Boston, 1980.
\bibitem{lm1}
A. C. Lazer and P. J. McKenna, Existence,
uniqueness, and stability of oscillations in differential equations
with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721-739.
\bibitem{lm2}
A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations
in suspension bridges : some new connections with nonlinear
analysis, SIAM Rev. 32 (1990), 537-578.
\bibitem{liu} Liu Bin, Boundedness in asymmetric oscillations, J.
Math. Anal. Appl.
231 (1999), 355-373.
\bibitem{mas} J. Massera, The existence of periodic solutions of systems of
differential equations, Duke Math. J., 17 (1950), 457-475.
\bibitem{ort} R. Ortega, Asymmetric oscillators and twist mappings,
J. London Math. Soc. 53(1996), 325-342.
\bibitem{ver} F. Verhulst, Nonlinear Differential Equations and Dynamical
Systems, Springer, Berlin, 1990.
\bibitem{wan} Wang Zaihong, Periodic solutions of the second order
differential equations
with jumping nonlinearities, to appear.
\end{thebibliography}
\noindent{\sc Christian Fabry} \\
Universit\'e Catholique de Louvain \\
Institut de Math\'ematique Pure et Appliqu\'ee, \\
Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Belgium\\
e-mail: fabry@amm.ucl.ac.be
\end{document}