0$ such that $-2c^2 \le Q'(s) \le -c^2/2$ for all $s \in (-\delta,0)$. Also, since $x_{-}(E) \to 0^-$ as $E \to 0^+$, there exists $E_1$ such that $-\delta< x_{-}(E) \le 0$ for all $E\gamma_2$ and $0 \gamma_2$. For simplicity, we will suppress $\gamma$ in the expressions $U(x,\gamma), x_{+}(e,\gamma)$ and $P(E,\gamma)$. Using the Hamiltonian~\eqref{hamiltonian} and integrating along orbits, we arrive at the familiar formula for the period of the orbit at energy level $E$ (see \cite{chiconejacobs}): \[ P(E)=\frac{2}{\sqrt{2}}\int_{x_{-}(E)}^{x_{+}(E)}\frac{dx}{\sqrt{E-U(x)}} \,. \] The change of variables $s=h(x)$, where $h(x)= \mathop{\rm sgn}(x)\sqrt{2U(x)}$, transforms the integral into \[ P(E)=\frac{2}{\sqrt{2}}\int_{-\sqrt{2E}}^{\sqrt{2E}} \frac{s}{U'(h^{-1}(s))\sqrt{E-\frac{s^2}{2}}}\,ds\,. \] After another change of variables, $s=\sqrt{2E}\sin \theta $, the period function is represented by \begin{equation}\label{period} P(E)=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{d\theta}{h'(h^{-1}(\sqrt{2E} \sin \theta))} =2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(h^{-1})'(\sqrt{2E} \sin \theta) \, d \theta. \end{equation} By differentiating and splitting the integral in two pieces, we have \begin{align*} P'(E)&=\sqrt{\frac{2}{E}}\,\int_{-\frac{\pi}{2}}^0(h^{-1})''(\sqrt{2E} \sin \theta) \sin \theta \, d \theta \\ &\quad +\sqrt{\frac{2}{E}}\,\int_0^{\frac{\pi}{2}}(h^{-1})''(\sqrt{2E} \sin \theta) \sin \theta \, d \theta\\ &=I+II. \end{align*} \begin{lemma}\label{lemma5} Let $E_2$ be as in Lemma~\ref{lemma4}. If $0< E 0$ and $F'(s)<0$ on $(x_{-}(E),0)$, it follows that $I$ is positive and \[ I \le \frac{Q(x_{-}(E))}{\sqrt{2}E}\int_{x_{-}(E)}^0 \frac{-F'(s)}{\sqrt{E-F(s)}}\,ds=\frac{\sqrt{2}Q(x_{-}(E))}{\sqrt{E}} \le \frac{\sqrt{2}\, C \sqrt{E}}{\sqrt{E}}=C\,, \] where we have consolidated constants. \end{proof} \begin{lemma}\label{lemma6} Integral $II$ is negative and $|\,II|>C/({\gamma\sqrt{E}\,})$. \end{lemma} \begin{proof} By defining $\tau(\theta):=U^{-1}(E \sin^2 \theta)$ and proceeding as in Lemma~\ref{lemma5}, we express $II$ in the form \[ II=\frac{1}{\sqrt{2}E}\int_0^{x_+(E)}\frac{(U'(s))^2 -2U(s)U''(s)}{(U'(s))^3}\frac{U'(s)}{\sqrt{E-U(s)}}\,ds\,. \] By Lemma~\ref{lemma1} and the inequality $U'(s) \ge 0$ on $(0,x_+(E))$, the integrand is always negative. Thus, $II<0$ and \[ |\,II| > \frac{M}{2\sqrt{2}E\gamma}\int_0^{x_+(E)}\frac{U'(s)}{\sqrt{E-U(s)}}\,ds =\frac{2 M \sqrt{E}}{2\sqrt{2} E \gamma}=\frac{C}{\gamma \sqrt{E}}\, , \] where we have consolidated constants. \end{proof} To complete the proof of the theorem, we choose $E_*=\min(E_2,C_2^2/(\gamma^2 C_1^2))$ so that $P'(E,\gamma)=I+II<0$ whenever $0 \gamma_2$, then the period function for system~\eqref{maineq} is non-monotone and has at least one critical point in the interval $[E_*,F(-K_1))$. \end{corollary} \begin{corollary} Let $\gamma_2$ be as in Theorem~\ref{theorem1}. If $\gamma>\gamma_2$ and $x_c>0$ is sufficiently small, then the period function for the system \begin{equation}\label{xcperturb} \ddot{x}+f(x)+\gamma H(x-x_c)g(x-x_c)=0 \end{equation} is non-monotone and has at least two critical points. \end{corollary} \begin{proof} Let $T(x_0,x_c)$ be the period of the orbit with initial conditions $x(0)=x_0$ and $\dot{x}(0)=0$ for equation~\eqref{xcperturb} and define \[T(0,x_c)=\lim_{x_0\to 0} T(x_0,x_c).\] Since $T$ is continuous and the function $x_0\mapsto T(x_0,0)$ is decreasing near the origin, there exists $\bar{x}_0$ near $0$ such that $T(\bar{x}_0,0)< T(0,0)$. Hence, $T(\bar{x}_0,\bar{x}_c) 0$, the period function $T(x_0,\bar{x}_c)$ must coincide with the period function of $\ddot{x}+f(x)=0$, which is increasing near the origin. Because the periods of periodic orbits are unbounded in a neighborhood of the homoclinic loop boundary of the period annulus under consideration, we have the desired result. \end{proof} \section{The Impact Pendulum}\label{example} The non-dimensional system $\ddot{x}+\sin x +\gamma H(x)x^{\frac{3}{2}}=0$ is a Hertzian contact model (see~\cite{goldsmith}) for an undamped unforced pendulum striking an elastic barrier at its downward vertical position. The constant $\gamma$ corresponds to the elastic modulus of the barrier (see~\cite{mannpaper}). The functions $f(x)=\sin x$ and $g(x)=x^{3/2}$ satisfy the assumptions in section~\ref{model} for Theorem~\ref{theorem1}, which states that there exists a region near the rest point at the barrier where the period function is decreasing. Using numerical integration techniques with $\gamma = 3.57 \times 10^8$ (an approximate value for an aluminum barrier), we are able to integrate the system numerically and graph its period function. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig2} \end{center} \caption{\label{periodplot} Period function for the impact pendulum~\eqref{impactpendulum} with $\gamma=3.57 \times 10^8$.} \end{figure} A plot of the period function near $E=0$ is given in Fig.~\ref{periodplot}, which confirms that the period function is decreasing near $E=0$. The interval of decrease is small in this case because $\gamma$ is large. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3} \end{center} \caption{\label{wallnotzero} Period function for the impact pendulum~\eqref{notzeroequ} with $x_c=0.281$ and $\gamma=3.57 \times 10^8$.} \end{figure} Numerical experiments suggest that a version of the decreasing period phenomenon persists in case the wall is positioned at some positive angle relative to the downward vertical. The Hertzian contact model for the impact pendulum with wall angle $x_c>0$ is \begin{equation}\label{notzeroequ} \ddot{x}+\sin x +\gamma H(x-x_c) (x-x_c)^{3/2}=0. \end{equation} Theorem~\ref{theorem1} does not apply to the impact pendulum in this case because our hypotheses are not satisfied. In fact, due to the smoothness of the period function and its positive derivative in the region of small oscillation with no impacts, there must exist an interval containing the contact point on which the period function increases. Our numerical experiments verify this fact and suggest that the period function will decrease for an interval corresponding to more energetic impacts, reach a minimum value, and then increase as the energies of the periodic orbits approach the energy of the homoclinic loop. This scenario is illustrated in Fig.~\ref{wallnotzero}. A natural prediction (cf.~\cite[Ch. 5]{chiconebook}) is that harmonic motions of the periodically forced and damped pendulum with impacts will correspond to low-order resonances between the forcing period and the available periods of the conservative impact pendulum studied in this paper. In experiments, where only relatively small oscillations are feasible, the interval of available periods is the interval corresponding to the local maximum and local minimum in Fig.~\ref{wallnotzero}. By approximating these values, the range of $(1:1)$-period locking (harmonic motions) has been predicted and verified by physical experiments (see \cite{impactoscillator}). \appendix \section{Derivation of the Impact Pendulum Model}\label{sec:mathmod} Consider a pendulum that encounters a barrier when the pendulum's angular position $x$ (measured counterclockwise relative to the downward vertical) is $x_c$. The kinetic energy for the pendulum is \[ T = \frac{1}{2}m \Big[ \Big(L \dot{x}\cos x \Big)^2 + \Big( L \dot{x} \sin x \Big)^2 \Big] , \] where $m$ is the pendulum mass and $L$ is the pendulum effective length. Here, the pendulum effective length refers to the distance between the pendulum pivot point and the pendulum center of mass. Because the mass in this physical system is distributed along the pendulum shaft and bob, the effective length differs from the total length, $l$, which is the distance from the pivot to the sphere center of mass. The pendulum's potential energy when not in contact with the barrier is \[ V = m g L(1-\cos x). \] Using Lagrange's equation \[ \frac{d}{dt} \Big( \frac{\partial T}{\partial \dot{x} } \Big) - \frac{\partial T}{\partial x} + \frac{\partial V}{\partial x} = 0, \] the equation of motion for the pendulum during the contact and non-contact regimes is \begin{equation} \label{system} \ddot{x} + \omega^2 \sin x + \frac{ l }{m L^2} H(x-x_c) F_c (x-x_c) = 0. \end{equation} where $H$ is the Heaviside function, $F_c$ is the contact force function that occurs at distance $l$ from the pendulum pivot point, and $\omega^2=g/L$ is the square of the pendulum's natural frequency. The Hertzian contact force is given by \[ F_c(x) = \frac{4}{3} E \sqrt{R} ( l \sin x )^{3/2}, \] where $E$ is the elastic modulus of the barrier and $R$ is the radius of the sphere that impacts the barrier (see~\cite{goldsmith}). The equation of motion~\eqref{system} is made non-dimensional by changing the time-scale via $t\mapsto t/\omega$. After simplifying and replacing the sine function in the contact term by the first term of its Taylor series centered at $x_c$ (which is justified by the small penetration depth), we obtain the smooth dimensionless model equation \begin{equation}\label{sys1} \nonumber \ddot{x}+ \sin x + \gamma(x-x_c)^{3/2} H(x-x_c)=0, \end{equation} where \[\gamma=\frac{4 l^{5/2} E R^{1/2}}{3\omega^2 m L^2}. \] While the equation of motion incorporates the discontinuous Heaviside function, we note that the contact term is class $C^1$ due to the presence of the Hertzian penetration function given in the model equation~\eqref{sys1} by $(x-x_c)^{3/2}$. \subsection*{Acknowledgments} The authors thank the anonymous referee for carefully reading this paper and making valuable suggestions for improvements. \begin{thebibliography}{0} \bibitem{chiconebook} C. Chicone, \emph{Ordinary Differential Equations with Applications}, 2nd Ed. (New York: Springer-Verlag), 2006. \bibitem{chiconejacobs} C. Chicone and M. 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