\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 188, pp. 1--28.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/188\hfil Chaotic oscillations] {Chaotic oscillations of the Klein-Gordon equation with distributed energy pumping and van der Pol boundary regulation and distributed time-varying coefficients} \author[B. Sun, T. Huang \hfil EJDE-2014/188\hfilneg] {Bo Sun, Tingwen Huang} % in alphabetical order \address{Bo Sun \newline Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan, China} \email{sunbo1965@yeah.net} \address{Tingwen Huang \newline Science Program, Texas A\&M University at Qatar, Education City, Doha, Qatar} \email{tingwen.huang@qatar.tamu.edu} \thanks{Submitted August 1, 2014. Published September 10, 2014.} \subjclass[2000]{35L05, 35L70, 58F39, 70L05} \keywords{Chaotic Oscillations; Klein-Gordon equation; \hfill\break\indent distributed energy pumping; van der Pol boundary regulation} \begin{abstract} Consider the Klein-Gordon equation with variable coefficients, a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed \emph{antidamping} term, we use a \emph{variable-substitution} technique together with the analogy with the 1-dimensional wave equation to prove that for the Klein-Gordon equation chaos occurs for a class of equations and boundary conditions when system parameters enter a certain regime. Chaotic and nonchaotic profiles of solutions are illustrated by computer graphics. \end{abstract} \maketitle \numberwithin{equation}{section} \numberwithin{figure}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} During the past decade, progress has been made in dynamical system theory in proving the onset of chaos in the 1D wave equation and the Klein-Gordon equation with a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed \emph{antidamping} term, see \cite{ChHs1,ChHs2,ChHs3,ChHs4,ChHs5,ChSb}. The basic method is \emph{characteristic reflections}, by which discrete dynamical systems are extracted. We first give a quick review of the work mentioned above, where the main motivating interest was the significance in \emph{nonlinear feedback boundary control}. For the wave equation \begin{equation}\label{KGeq1.1} w_{tt}(x,t)-c^2w_{xx}(x,t) = 0,\quad 00,\; c>0, \end{equation} we assume that at the left-end $x=0$, the boundary condition is \begin{equation}\label{KGeq1.2} w_t(0,t) = -\eta w_x(0,t),\quad t>0,\; \eta>0,\; \eta\ne c, \end{equation} and at the right-end $x=1$, the boundary condition is of the van der Pol type: \begin{equation}\label{KGeq1.3} w_x(1,t) = \alpha w_t(1,t) - \beta w^3_t(1,t),\quad t>0,\; 0<\alpha0. \end{equation} Then the energy functional \begin{equation}\label{KGeq1.4} E(t) = \frac12 \int^1_0 \Big[w^2_x(x,t) + \frac1{c^2}w^2_t(x,t)\Big] dx \end{equation} \emph{rises} if $|w_t(1,t)|$ is small, and \emph{falls} if $|w_t(1,t)|$ is large. Thus, the van der Pol boundary condition \eqref{KGeq1.3} has a \emph{self-regulating effect}. This can cause chaos to occur in $w_x$ and $w_t$ if the parameters $\alpha,c$ and $\eta$ enter a certain regime. The treatment in \cite{ChHs3} relies heavily on the \emph{method of characteristics} for linear hyperbolic systems and simple wave-reflecting relations. Let $c=1$, and \begin{equation} u(x,t)=\frac{1}{2}[w_x(x,t)+w_t(x,t)],\quad v(x,t)=\frac{1}{2}[w_x(x,t)-w_t(x,t)]. \end{equation} Then \begin{gather} v(0,t)=G_{\eta}(u(0,t))\equiv \frac{1+\eta}{1-\eta}u(0,t), \\ u(1,t)=F_{\alpha,\beta}(v(1,t)), \end{gather} where $F_{\alpha,\beta}:\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear mapping such that for each given $v\in\mathbb{R}$, $u= F_{\alpha,\beta}(v)$ is the unique solution of the cubic equation \begin{equation} \beta (u-v)^3+(1-\alpha)(u-v)+2v=0. \end{equation} Therefore, $u(x,t)$ and $v(x,t)$ for $x\in [0,1]$ and $t\in (0,\infty)$ are determined by the initial data $u(x,0)$, $v(x,0)$ and the iterative composition of $F_{\alpha,\beta}\circ G_{\eta}$ or $G_{\eta}\circ F_{\alpha,\beta}$. Finally, the chaotic dynamics in the 1D wave equation is reduced to the discrete dynamical system generated by the interval map $F_{\alpha,\beta}\circ G_{\eta}$ or $G_{\eta}\circ F_{\alpha,\beta}$. A generalization of the 1D wave equation is the Klein-Gordon equation described as \begin{equation}\label{KGeq1.5} w_{tt}(x,t) + \eta w_t(x,t) - w_{xx}(x,t) + k^2w(x,t) = 0,\quad 00, \end{equation} where $k\neq 0$, $\eta>0$. A special case is \begin{equation}\label{KGeq1.6} w_{tt}+2kw_t - w_{xx} + k^2w=0, \quad \text{for } (x,t)\in (0,1)\times (0,\infty). \end{equation} We consider, for \eqref{KGeq1.6}, the following boundary condition, \begin{gather}\label{KGeq1.7} w_t(0,t) + kw(0,t) = -\lambda w_x(0,t),\quad t>0, \text{ at $x=0$, for given } \lambda\in \mathbb{R};\\ \label{KGeq2.26} w_x(1,t) = \alpha[w_t(1,t)+kw(1,t)] - \beta[w_t(1,t)+kw(1,t)]^3,\quad t>0, \text{ at } x=1, \end{gather} where $0<\alpha<1$, $\beta>0$; and the energy function \begin{equation}\label{KGeq1.8} \widetilde E(t) = \frac12 \int^1_0 [w^2_x+(w_t+kw)^2]dx. \end{equation} Then $\frac{d}{dt}\widetilde E(t)$ is indefinite, which is the sign of chaos. The simple change of variable \begin{equation} w(x,t) = e^{-kt}W(x,t) \end{equation} leads to \begin{equation} \frac{\partial^2}{\partial x^2} W(x,t) - \frac{\partial^2}{\partial t^2} W(x,t) = 0. \end{equation} Define \begin{equation}\label{KGeq1.9} u=\frac12(w_x+w_t+kw), \quad v=\frac12(w_x-w_t-kw). \end{equation} Then \begin{equation}\label{KGeq1.10} v(0,t) = \frac{1+\lambda}{1-\lambda} u(0,t)\equiv G_\lambda(u(0,t)), \end{equation} where $G_\lambda$ is defined to be the linear operator of multiplication by $(1+\lambda)/(1-\lambda)$. Also we have \begin{equation}\label{KGeq1.11} \beta[u(1,t)-v(1,t)]^3 + (1-\alpha) [u(1,t)-v(1,t)] + 2v(1,t) = 0. \end{equation} For any $v\in {\mathbb{R}}$, define $g(v)$ to be the \emph{unique real} solution to the cubic equation \begin{equation}\label{KGeq1.12} \beta g(v)^3 + (1-\alpha) g(v)+2v=0, \end{equation} and \begin{equation}\label{KGeq1.13} F(v) \equiv F_{\alpha,\beta}(v) \equiv v+g_{\alpha,\beta}(v). \end{equation} Then for each given $v(1,t)$, equation \eqref{KGeq1.11} has a unique solution $u(1,t)$ given by \[ u(1,t) = F_{\alpha,\beta}(v(1,t)). \] It is easy to check that $e^{kt}u(x,t)$ keeps constant along each characteristics $x+t=c$, and $e^{kt}v(x,t)$ keeps constant along each characteristics $x-t=c$. Therefore, we have \begin{equation}\label{KGeq1.14} \begin{gathered} u(1,t+2) = F_{\alpha,\beta}(G_\lambda(e^{-2k}u(1,t))),\\ v(0,t+2) = G_\lambda(e^{-k}F_{\alpha,\beta}(e^{-k}v(0,t))), \end{gathered} \end{equation} for any $t>0$. Finally, the dynamics of \begin{equation} u=\frac12(w_x+w_t+kw), \quad v=\frac12(w_x-w_t-kw) \end{equation} are determined by the iterative compositions of the functions $F_{\alpha,\beta}(G_\lambda(e^{-2k}\cdot))$ or $G_\lambda(e^{-k}F_{\alpha,\beta}(e^{-k}\cdot))$. One can imagine that there may be more chaos in the Klein-Gordon equation if its constant coefficients are replaced by variable coefficients. So in this paper we consider more general situations and problems below: \begin{equation}\label{KGeq1.15} [\frac{\partial}{\partial t}-a(x)\frac{\partial}{\partial x}+k_1] [\frac{\partial}{\partial t}+b(x)\frac{\partial}{\partial x}+k_2]w(x,t)=0, \quad \text{for} \quad x\in (0,1), \quad t>0 \end{equation} with some linear and cubic nonlinear boundary condition, where $a(x)>0$ and $b(x)>0$ are continuous real functions defined on $[0,1]$. The organization of this article is as follows: In Section 2, we consider a simple case with $a(x)\equiv Kb(x)$, and $k_1=k_2=0$, where $K$ is a positive constant. In Section 3, we consider some more cases with $a(x)\equiv Kb(x)$, and $k_1$, $k_2\in \mathbb{R}$. In Section 4, we prove the bifurcation from a stable fixed point to chaos. In Section 5, we consider some more general cases. \section{Klein-Gordon Equation with variable coefficients but no state term} Let us consider a simple case of \eqref{KGeq1.15} with $k_1=k_2=0$; i.e., \begin{equation}\label{2.1} [\frac{\partial}{\partial t}-a(x)\frac{\partial}{\partial x}] [\frac{\partial}{\partial t}+b(x)\frac{\partial}{\partial x}]w=0, \quad \text{for } x\in (0,1),\; t>0. \end{equation} Let \begin{equation}\label{variableSubstitution} \xi=\int_0^x\frac{dx}{a(x)}, \quad \eta=\int_0^x\frac{dx}{b(x)}, \end{equation} then it follows from $a=Kb$ that $\eta =K\xi$, and thus \begin{equation*} \frac{\partial}{\partial \xi}=K\frac{\partial}{\partial \eta}. \end{equation*} Let $\tilde{w}(\eta,t)=w(x,t)$, then \eqref{2.1} is equivalent to \begin{equation} [\frac{\partial}{\partial t}-K\frac{\partial}{\partial \eta}] [\frac{\partial}{\partial t}+\frac{\partial}{\partial \eta}]\tilde{w}=0, \end{equation} or \begin{equation} [\frac{\partial}{\partial t}+\frac{\partial}{\partial \eta}] [\frac{\partial}{\partial t}-K\frac{\partial}{\partial \eta}]\tilde{w}=0. \end{equation} It follows immediately that \begin{gather*} \tilde{w}_{\eta}+\tilde{w}_t=f(\eta +Kt),\\ K\tilde{w}_{\eta}-\tilde{w}_t=g(t-\eta) \end{gather*} for some functions $f$ and $g$ depending on the initial data. Let \begin{gather*} \tilde{u}(\eta,t)=\frac{1}{2}[\tilde{w}_{\eta}+\tilde{w}_t],\\ \tilde{v}(\eta,t)=\frac{1}{2}[K\tilde{w}_{\eta}-\tilde{w}_t]. \end{gather*} Then $\tilde{u}$ keeps constant along lines $\eta +Kt=c$, and $\tilde{v}$ keeps constant along lines $\eta-t=c$. We impose nonlinear boundary conditions as \begin{gather} \tilde{w}_t(0,t)=-\lambda \tilde{w}_{\eta}(0,t),\label{zuobianjie} \\ \tilde{w}_{\eta}(L,t)=\alpha \tilde{w}_t(L,t) -\beta \tilde{w}_t^3(L,t),\label{youbianjie} \end{gather} where $L=\int_0^1\frac{dx}{b(x)}$. Then \begin{gather} \tilde{v}(0,t)=G_{K,\lambda}(\tilde{u}(0,t)) =\frac{K+\lambda}{1-\lambda}\tilde{u}(0,t),\\ \tilde{u}(L,t)=F_{K,\alpha,\beta}(\tilde{v}(L,t)), \end{gather} where $u=F_{K,\alpha,\beta}(v)$ is the unique real solution of the cubic equation \begin{align}\label{cubicEquation} \frac{4\beta}{(K+1)^2}(Ku-v)^3+(\frac{1}{K}-\alpha )(Ku-v)+(\frac{1}{K}+1)v=0. \end{align} It follows from \eqref{variableSubstitution} that \begin{equation} d\xi=\frac{dx}{a(x)}, \quad d\eta=\frac{dx}{b(x)}, \end{equation} and thus \begin{gather} \frac{\partial w}{\partial x} =\frac{\partial \tilde{w}}{\partial \xi}\frac{d\xi}{dx}=\frac{1}{a(x)}\frac{\partial \tilde{w}}{\partial \xi}, \\ \frac{\partial w}{\partial x} =\frac{1}{b(x)}\frac{\partial \tilde{w}}{\partial \eta}, \end{gather} or \begin{gather} \frac{\partial \tilde{w}}{\partial \xi}=a(x)\frac{\partial w}{\partial x}, \\ \frac{\partial \tilde{w}}{\partial \eta}=b(x)\frac{\partial w}{\partial x}. \end{gather} So the boundary condition \eqref{zuobianjie}-\eqref{youbianjie} is equivalent to \begin{gather} w_t(0,t)=-\lambda b(0)w_x(0,t), \label{2.1leftBoundary}\\ b(1)w_x(1,t)=\alpha w_t(1,t)-\beta w_t^3(1,t). \label{2.1rightBoundary} \end{gather} Let \begin{gather*} u(x,t)=\tilde{u}(\eta,t)=\frac{1}{2}[b(x)w_x+w_t], \\ v(x,t)=\tilde{v}(\eta,t)=\frac{1}{2}[a(x)w_x-w_t], \end{gather*} then \begin{gather} v(0,t)=G_{K,\lambda}(u(0,t))=\frac{K+\lambda}{1-\lambda}u(0,t),\\ u(1,t)=F_{K,\alpha,\beta}(v(1,t)). \end{gather} Therefore \begin{gather} u(1,t)=F_{K,\alpha,\beta}(G_{K,\lambda}(u(1,t-L-\frac{L}{K})),\label{reflection2.1} \\ v(0,t)=G_{K,\lambda}(F_{K,\alpha,\beta}(v(0,t-L-\frac{L}{K})).\label{reflection2.2} \end{gather} The dynamics of $u$ and $v$ are reduced to the properties of $G\circ F$ and $F\circ G$. Define a function $\psi(x)=\int_0^x\frac{dx}{b(x)}$ on $[0,1]$, then $\psi$ is one to one. \begin{lemma}[Solution representations for $u(x,t)$ and $v(x,t)$]\label{Lemma2.1} Assume \eqref{2.1}, \\ \eqref{2.1leftBoundary} and \eqref{2.1rightBoundary}. Then for any $x$: $00$, we have, for $t$: $t=(1+\frac{1}{K})jL+\tau$, $j=0,1,2,\dots$, $\tau >0$, \begin{gather*} u(x,t) =\begin{cases} (F_{\alpha,\beta}\circ G_{\lambda,K})^j (F_{\alpha,\beta} (v_0(\psi^{-1}(1+\frac{1}{K})L-\tau-\frac{\eta}{K}))), \\ \quad \text{if } L\le K\tau +\eta \le (K+1)L;\\[4pt] (F_{\alpha,\beta}\circ G_{\lambda,K})^{j+1}(u_0(\psi^{-1}(K\tau+\eta-(K+1)L)))),\\ \quad \text{if } (K+1)L \le K\tau +\eta \le (K+2)L; \end{cases} \\[4pt] v(x,t) =\begin{cases} (G_{\lambda,K}\circ F_{\alpha,\beta}))^j (G(u_0(\psi^{-1}(K(\tau-\eta))))),\\ \quad \text{if } t=(1+\frac{1}{K})jL+\tau, \quad 0\le \tau -\eta \le \frac{L}{K}; \\[4pt] (G_{\lambda,K}\circ F_{\alpha,\beta}))^{j+1} (v_0(\psi^{-1}(1+\frac{1}{K})L-\tau+\eta)), \\ \quad \text{if } \frac{L}{K} \le \tau -\eta \le (1+\frac{1}{K})L. \end{cases} \end{gather*} \end{lemma} \begin{lemma}[Derivative Formulas]\label{Lemma2.2} Let $0<\alpha\le \frac{1}{K}$, $\beta >0$ and $\eta >0$, $\eta \neq 1$, where $\alpha$ and $\beta$ are given and fixed, but $\eta$ is a varying parameter. Define \[ f_1(v,\eta)=G\circ F(v)=\frac{K+\eta}{1-\eta}F(v),\quad f_2(v,\eta)=F\circ G(v)=F(\frac{K+\eta}{1-\eta}v),\quad v\in \mathbb{R}. \] Let $g(v)$ be the unique real solution of the cubic equation \begin{equation}\label{g} \frac{4\beta}{(K+1)^2}g(v)^3+(\frac{1}{K}-\alpha)g(v)+(\frac{1}{K}+1)v=0, \end{equation} for a given $v\in \mathbb{R}$. Then \begin{gather} \frac{\partial}{\partial v}f_1(v,\eta) =\frac{K+\eta}{K(1-\eta)}[1-\frac{K+1} {\frac{12K\beta}{(K+1)^2}g(v)^2+1-K\alpha}],\label{Derivative1} \\ \frac{\partial}{\partial v}f_2(v,\eta) =\frac{K+\eta}{K(1-\eta)}[1-\frac{K+1}{\frac{12K\beta}{(K+1)^2} g(\frac{K+\eta}{1-\eta}v)^2+1-K\alpha}], \notag \\ \frac{\partial}{\partial \eta}f_1(v,\eta) =\frac{1+K}{K(1-\eta)^2}[v+g(v)], \notag \\ \frac{\partial}{\partial \eta}f_2(v,\eta) =\frac{1+K}{K(1-\eta)^2}[1-\frac{K+1}{\frac{12K\beta}{(K+1)^2} g(\frac{K+\eta}{1-\eta}v)^2+1-K\alpha}]v, \notag \\ \frac{\partial^2}{\partial \eta \partial v}f_1(v,\eta) =\frac{K+1}{K(1-\eta)^2} [1-\frac{K+1}{\frac{12K\beta}{(K+1)^2}g(v)^2+1-K\alpha}], \notag \\ \frac{\partial^2}{\partial v^2}f_1(v,\eta) =\frac{K+\eta}{(1-\eta)}(-24)\beta\cdot \frac{g(v)}{[\frac{12K\beta}{(K+1)^2} g(v)^2+1-K\alpha]^3}, \notag \\ \frac{\partial^3}{\partial v^3}f_1(v,\eta) =\frac{K+\eta}{1-\eta}24(K+1)\beta\frac{ 1-K\alpha -\frac{60K\beta}{(K+1)^2} g(v)^2}{[\frac{12K\beta}{(K+1)^2}g(v)^2+1-K\alpha]^5}. \end{gather} \end{lemma} \begin{lemma} [Intersections with the Lines $u-v=0$ and $u+v=0$]\label{Lemma2.3} Let $0<\alpha\le 1/K$, $\beta >0$, $\eta >0$, $\eta \neq 1$ be given. Then {\rm (i)} $u=G\circ F(v)$ intersects the line $u=v$ at the points \begin{align*} (u,v)&=(-\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}, -\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}), \\ &\quad (0,0), \\ &\quad (\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}, \frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}); \end{align*} {\rm (ii)} $u=G\circ F(v)$ intersects the line $u=-v$ at the points \begin{align*} (u,v)&=(-\frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]} \sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}},\\ &\quad \frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}}), \\ &\quad (0,0), \\ &\quad (\frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}},\\ &\quad -\frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}}); \end{align*} {\rm (iii)} $u=F\circ G(v)$ intersects the line $u=v$ at the points \begin{align*} (u,v)&=(-\frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}, -\frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}), \\\ &\quad(0,0), \\ &\quad (\frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}, \frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta} {\beta\eta}}); \end{align*} {\rm (iv)} $u=F\circ G(v)$ intersects the line $u=-v$ at the points \begin{align*} (u,v)&=(-\frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]}\sqrt{\frac{(1+2\alpha) K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}}, \\ &\quad \frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}}), \\ &\quad (0,0), \\ &\quad (\frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}},\\ &\quad -\frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}}). \end{align*} \end{lemma} \begin{remark} \label{rmk2.4} \rm Conclusions (ii) and (iv) are based on the assumption that \begin{equation} \frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]} \geq 0. \end{equation} So we assume $K\le 1$ and $(1+2\alpha)K\geq 1$ for conclusions (ii) and (iv). We also make this assumption for some related results below, e.g., (ii) and (iv) in Lemma \ref{boundedInvariantSet}. \end{remark} \begin{lemma}[$v$-axis Intercepts]\label{lem2.4} Let $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta >0$, $\eta \neq 1$ be given. Then {\rm (i)} $u=G\circ F(v)$ has $v$-axis intercepts $v=-\frac{K+1}{2}\sqrt{\frac{1+\alpha}{\beta}}$, $0$, $\frac{K+1}{2}\sqrt{\frac{1+\alpha}{\beta}}$; {\rm (ii)} $u=F\circ G(v)$ has $v$-axis intercepts \[ v=-\frac{(K+1)(1-\eta)}{2(K+\eta)}\sqrt{\frac{1+\alpha}{\beta}}, \quad 0, \quad \frac{(K+1)(1-\eta)}{2(K+\eta)}\sqrt{\frac{1+\alpha}{\beta}}. \] \end{lemma} \begin{lemma}[Local Maximum, Minimum and Piecewise Monotonicity] Let $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta >0$, $\eta \neq 1$ be given. Then {\rm (i)} If $0<\eta<1$, then $G\circ F$ has local extremal values \begin{gather*} M=G\circ F(-v_c)=\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3} \sqrt{\frac{1+\alpha}{3\beta}}, \\ m=G\circ F(v_c)=-\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3} \sqrt{\frac{1+\alpha}{3\beta}}, \end{gather*} where $v_c=\frac{3+(1-2\alpha)K}{6}\sqrt{\frac{1+\alpha}{3\beta}}$, and $M$, $m$ are, respectively, the local maximum and minimum of $G\circ F$. The function $G\circ F$ is strictly increasing on $(-\infty, -v_c)$ and $(v_c, \infty)$, but strictly decreasing on $(-v_c, v_c)$. On the other hand, if $\eta>1$, then $G\circ F$ has local minimum ($m$) and maximum ($M$) values \begin{gather*} m=G\circ F(-v_c)=\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3} \sqrt{\frac{1+\alpha}{3\beta}}, \\ M=G\circ F(v_c)=-\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3} \sqrt{\frac{1+\alpha}{3\beta}}, \end{gather*} where $v_c$ is the same as above. The function $G\circ F$ is strictly decreasing on $(-\infty, -v_c)$ and $(v_c,\infty)$, but strictly increasing on $(-v_c,v_c)$. {\rm (ii)} If $0<\eta<1$, then $F\circ G$ has local extremal values \begin{gather*} M=F\circ G(-\tilde{v}_c)=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \\ m=F\circ G(\tilde{v}_c)=-\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \end{gather*} where $\tilde{v}_c=\frac{1-\eta}{K+\eta}\frac{3+(1-2\alpha)K}{6} \sqrt{\frac{1+\alpha}{3\beta}}$, and $M$, $m$ are, respectively, the local maximum and minimum of $F\circ G$. The function $F\circ G$ is strictly increasing on $(-\infty, -\tilde{v}_c)$ and $(\tilde{v}_c, \infty)$, but strictly decreasing on $(-\tilde{v}_c, \tilde{v}_c)$. On the other hand, if $\eta>1$, then $F\circ G$ has local extremal values \begin{gather*} m=F\circ G(-\tilde{v}_c)=-\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}},\\ M=F\circ G(\tilde{v}_c)=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}. \end{gather*} The function $F\circ G$ is strictly decreasing on $(-\infty, -\tilde{v}_c)$ and $(\tilde{v}_c, \infty)$, but strictly increasing on $(-\tilde{v}_c, \tilde{v}_c)$. \end{lemma} \begin{lemma}[Bounded Invariant Intervals]\label{boundedInvariantSet} Let $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta >0$, $\eta \neq 1$. {\rm (i)} If $0<\eta<1$ and \[ M=\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}} \le\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}, \] then the iterates of every point in the set \[ U\equiv (-\infty,-\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}) \cup (\frac{K+\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}},\infty) \] escape to $\pm \infty$, while those of any point in $\mathbb{R}\backslash\bar{U}$ are attracted to the bounded invariant interval $\mathcal{I}\equiv [-M,M]$ of $G\circ F$. {\rm (ii)} If $\eta >1$ and \begin{align*} M&=-\frac{K+\eta}{1-\eta}\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}\\ &\le \frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha)K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}}, \end{align*} then the same conclusion as in (i) holds, with \begin{align*} U&\equiv (-\infty, -\frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha)K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}})\\ &\cup (\frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha)K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}},\infty) \end{align*} and ${\mathcal{I}}\equiv [-M,M]$ for $G \circ F$. {\rm (iii)} If $0<\eta<1$ and $M=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}} \le \frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}}$, then the same conclusion holds, with \[ U\equiv (-\infty, -\frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}})\cup (\frac{1-\eta}{2\eta}\sqrt{\frac{1+\alpha\eta}{\beta\eta}},\infty) \] and ${\mathcal{I}}=[-M,M]$ for $F\circ G$. {\rm (iv)} If $\eta>1$ and \[ M=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}\le -\frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}}), \] then the same conclusion holds, with \begin{align*} U&\equiv (-\infty, \frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}})\\ &\quad \cup (-\frac{(1-\eta)(K+1)}{2[2K+ (1-K)\eta]} \sqrt{\frac{(1+2\alpha)K-1+[2+\alpha (1-K)]\eta}{\beta [2K+(1-K)\eta]}}) ,\infty) \end{align*} and $\mathcal{I}\equiv [-M,M]$ for $F\circ G$. \end{lemma} Now we try to set a period-doubling bifurcation theorem similar to our earlier work. \begin{theorem}[Period-Doubling Bifurcation Theorem] Let $\alpha$: $0<\alpha\le \frac{1}{K}$, $\beta >0$ be fixed, and let $\eta$: $0<\eta\le \underline{\eta}$ be a varying parameter. Let $h_1(v,\eta)=-G\circ F(v)$. Then {\rm (i)} $v_0= \frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]}\sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}}$ is a curve of fixed points of $h_1$. {\rm (ii)} The algebraic equation \begin{equation} \label{algebraicEquation} \begin{aligned} &\frac{K-2K\alpha\eta+3\eta}{6\eta}\sqrt{\frac{1+\alpha\eta}{3\beta\eta}}\\ &= \frac{(K+1)(K+\eta)}{2[2K+(1-K)\eta]} \sqrt{\frac{(1 +2\alpha )K-1 +[2+\alpha(1-K)]\eta}{[2K+(1-K)\eta]\beta}} \end{aligned} \end{equation} has a solution $\eta =\eta_0$ in $(0,1)$ for any given $\alpha$: $0<\alpha \le \frac{1}{K}$ and $\beta >0$. We have \[ \frac{\partial}{\partial v}h_1(v_0,\eta_0)=-1. \] {\rm (iii)} For $\eta =\eta_0$ satisfying \eqref{algebraicEquation}, we have \begin{align*} A&=\frac{\partial ^2}{\partial\eta\partial v}h_1 +\frac{1}{2}\frac{\partial h_1}{\partial \eta}\frac{\partial ^2h_1}{\partial v^2}\\ &=-\Big((K+1)\{[(K+1)\alpha(2\alpha+3)+3]\eta_0^3+(6K+\alpha K+\alpha -3)\eta_0^2 \\ &\quad -(7K-2)\eta_0 +3K\}\Big)/\Big(3(1-\eta_0)^3(K+\eta_0)^2\Big). \end{align*} for $\eta=\eta_0$ and $v=v_0(\eta_0)$. {\rm (iv)} \begin{align*} B&=\frac{1}{3}\frac{\partial ^3h_1} {\partial v^3}+\frac{1}{2}(\frac{\partial ^2h_1}{\partial v^2})^2\\ &=\frac{8(K+1)\beta \eta^4}{(1-\eta_0)^2(K+\eta_0)^5} \Big[(3\alpha -3K\alpha +1)\eta_0^3+(9K\alpha -3K^2\alpha-K+2)\eta_0^2\\ &\quad +K(6K\alpha-2K+7)\eta_0+5K^2\Big]. \end{align*} \end{theorem} \begin{proof} (i) This is an immediate consequence of Lemma \ref{Lemma2.3}. (ii) We first determine the point(s) $v>0$ such that $\frac{\partial h_1}{\partial v}=-1$. By \eqref{Derivative1}, with a change of sign for $f_1$, we obtain \[ \frac{K+\eta}{K(1-\eta)}[1-\frac{K+1} {\frac{12K\beta}{(K+1)^2}g(v)^2+1-K\alpha}]=1. \] Therefore, \begin{gather} \frac{12K\beta}{(K+1)^2}g(v)^2=\frac{K}{\eta}+K\alpha, \notag \\ \label{Bifurcation1} g(v)=\pm \frac{K+1}{2}\sqrt{\frac{1+\alpha\eta}{3\beta\eta}}. \end{gather} We choose positive $v$ and thus the ``$-$'' sign in \eqref{Bifurcation1}. Hence \begin{align}\label{Bifurcation2} g(v)=-\frac{K+1}{2}\sqrt{\frac{1+\alpha\eta}{3\beta\eta}}. \end{align} Since $g(v)$ satisfies \eqref{g}, from \eqref{Bifurcation2} we obtain \begin{equation}\label{58} \begin{aligned} v&=-\frac{K}{K+1}[\frac{4\beta}{(K+1)^2}g(v)^3+(\frac{1}{K}-\alpha)g(v)]\\ &=\frac{K-2K\alpha\eta+3\eta}{6\eta} \sqrt{\frac{1+\alpha\eta}{3\beta\eta}}\\ &= \text{LHS of \eqref{algebraicEquation}}. \end{aligned} \end{equation} Further setting \eqref{58} equal to $v_0(\eta)$ in (i), we obtain the RHS of \eqref{algebraicEquation}. Now we show that \eqref{algebraicEquation} has a solution. It is easy to see that the LHS tends to $+ \infty$, but the RHS keeps bounded as $\eta \rightarrow 0^{+}$. So the LHS is greater than the RHS for some $\eta$ close to $0$. On the other hand, it is easy to verify that the LHS is smaller than the RHS for some $\eta$ close to $1$. It follows from the Mean Value Theorem of continuous functions that \eqref{algebraicEquation} has a solution in $(0,1)$. (iii) and (iv) are also immediate consequences of Lemma \ref{Lemma2.3}. However, it is hard to judge whether $A\neq 0$ in (iii) and $B\neq 0$ in (iv) without knowledge of $\eta_0$. So we can not conclude the period-doubling bifurcation so far. We will try other methods in next section. \end{proof} \begin{theorem}[Homoclinic Orbits for the Case $0<\eta <1$] Let $K>0$, $\alpha$: $0<\alpha \le \frac{1}{K}$ and $\beta >0$ be fixed, and let $\eta \in (0,1)$ be a varying parameter such that \begin{align} \frac{K+1}{2}\sqrt{\frac{1+\alpha}{\beta}}<\frac{K+\eta}{1-\eta} \frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}},\label{homoclinicCondition} \end{align} then the repelling fixed point $0$ of $G\circ F$ and $F\circ G$ has homoclinic orbits. \end{theorem} \begin{proof} By \eqref{Derivative1} and \eqref{homoclinicCondition}, we easily obtain \begin{align*} \frac{\partial}{\partial v}f_i(v,\eta)|_{v=0} &=\frac{K+\eta}{K(1-\eta)}(1-\frac{K+1}{1-K\alpha})\\ &=-\frac{(\alpha +1)(K+\eta)}{(1-\eta)(1-K\alpha)} < -1, \quad i=1,2. \end{align*} Therefore $0$ is a repelling fixed point of $G\circ F$ and $F\circ G$. For a homoclinic to exist, the local maximum of $G\circ F$ (resp., $F\circ G$) must be larger than the positive $v$-axis intercept of $G\circ F$ (resp., $F\circ G$); i.e., \begin{align} \frac{K+1}{2}\sqrt{\frac{1+\alpha}{\beta}}<\frac{K+\eta}{1-\eta} \frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}. \end{align} This is exactly \eqref{homoclinicCondition}. \end{proof} \section{Klein-Gordon equation with variable coefficients} In this section, we consider \eqref{KGeq1.15} with $a(x)\equiv Kb(x)$, where $K>0$ is a constant. Let $W=e^{-ct-d\eta}w$, where $w$ satisfies \eqref{2.1}. Then $w=e^{ct+d\eta}W$, and \begin{gather*} w_t=e^{ct+d\eta}(W_t+cW), \\ w_{\eta}=e^{ct+d\eta}(W_{\eta}+dW). \end{gather*} Then it follows immediately that \begin{equation}\label{3.1} [\frac{\partial}{\partial t}-K\frac{\partial}{\partial \eta}+c-Kd] [\frac{\partial}{\partial t}+\frac{\partial}{\partial \eta}+c+d]W=0. \end{equation} Let $k_1=c-Kd$, $k_2=c+d$, then \eqref{3.1} becomes \begin{equation}\label{3.2} [\frac{\partial}{\partial t}-a(x)\frac{\partial}{\partial x}+k_1] [\frac{\partial}{\partial t}+b(x)\frac{\partial}{\partial x}+k_2]W=0. \end{equation} Reversely, for any $k_1$ and $k_2$, there exist a unique pair of $c$ and $d$ such that $k_1=c-Kd$, $k_2=c+d$. Note that \eqref{3.2} is just our original problem \eqref{KGeq1.15}. It follows immediately that \[ \frac{1}{2}(w_{\eta}+w_t)=e^{ct+d\eta} \frac{W_{\eta}+W_t+k_2W}{2} = c'_1, \] along each characteristics $ \eta +Kt=c_1$; and \[ \frac{1}{2}(Kw_{\eta}-w_t)=e^{ct+d\eta}\frac{KW_{\eta}-W_t-k_1W}{2} = c'_2, \] along each characteristics $\eta -t=c_2$. Let \begin{gather*} u=\frac{W_{\eta}+W_t+k_2W}{2}=\frac{b(x)W_x+W_t+k_2W}{2},\\ v=\frac{KW_{\eta}-W_t-k_1W}{2}=\frac{a(x)W_x-W_t-k_1W}{2}. \end{gather*} Then $e^{ct+d\eta}u$ keeps constant along lines $\eta +Kt=c$, and $e^{ct+d\eta}v$ keeps constant along lines $\eta -t=c$. We impose boundary conditions such that \begin{gather*} v(0,t) =G_{K,\lambda}(u(0,t))=\frac{K+\lambda}{1-\lambda}u(0,t), \\ u(1,t) =F_{K,\alpha,\beta}(v(1,t)). \end{gather*} Then it is easy to deduce the corresponding boundary conditions: \begin{gather} W_t(0,t) =-\lambda b(0)W_x(0,t)-\frac{(1-\lambda)k_1+(K+\lambda)k_2}{K+1}W(0,t), \label{3.1leftboundary} \\ \begin{aligned} &\frac{(K+1)b(1)W_x(1,t)+(k_2-k_1)W(1,t)}{2}\\ & =\alpha \frac{(K+1)W_t(1,t)+(k_1+Kk_2)W(1,t)}{2} \\ &\quad -\frac{4\beta}{(K+1)^2}[\frac{(K+1)W_t(1,t)+(k_1+Kk_2)W(1,t)}{2}]^3\,. \end{aligned} \label{3.1rightboundary} \end{gather} It is easy to verify the following reflective iterations: \begin{gather} u(1,t) =F(G(e^{-(1+\frac{1}{K})cL}u(1,t-L-\frac{L}{K})),\label{reflection3.1} \\ v(0,t) =G(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}v(0,t-L-\frac{L}{K})).\label{reflection3.2} \end{gather} \begin{lemma}[Solution representations for $u(x,t)$ and $v(x,t)$]\label{Lemma3.1} Assume \eqref{3.1}, \eqref{3.1leftboundary} and \eqref{3.1rightboundary}. Then for any $x$: $00$, we have, for $t=(1+\frac{1}{K})jL+\tau$, $j=0,1,2,\dots$, $\tau >0$, \begin{gather*} u(x,t) = \begin{cases} \big(F(G(e^{-(1+\frac{1}{K})cL}\cdot))\big)^j \Big(F(e^{-(c+d)(\tau-\frac{L-\eta}{K})}v_0(\psi^{-1}(1+\frac{1}{K})L -\tau-\frac{\eta}{K}))\Big), \\ \quad \text{if } L\le K\tau +\eta \le (K+1)L; \\[4pt] \big(F(G(e^{-(1+\frac{1}{K})cL}\cdot))\big)^j \Big(F_{\alpha,\beta}\Big(e^{-(c+d)L}G(e^{(Kd-c) (\tau+\frac{\eta}{K}-(1+\frac{1}{K})L)}\\ \times u_0(\psi^{-1}(K\tau+\eta-(K+1)L)))\Big)\Big),\\ \quad \text{if } (K+1)L \le K\tau +\eta \le (K+2)L; \end{cases} \\[4pt] v(x,t) =\begin{cases} \big(G(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}\cdot))\big)^j \Big(G\Big(e^{(Kd-c)(\tau-\eta)}u_0(\psi^{-1}(K(\tau-\eta)))\Big)\Big),\\ \quad \text{if } 0\le \tau -\eta \le \frac{L}{K}; \\[4pt] \big(G(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}\cdot))\big)^j \Big(Ge^{(d-\frac{c}{K})L}F\Big(e^{-(c+d)(\tau-\eta-\frac{L}{K})}\\ \times v_0(\psi^{-1} ((1+\frac{1}{K})L-\tau+\eta))\Big)\Big), \\ \quad \text{if } \frac{L}{K} \le \tau -\eta \le (1+\frac{1}{K})L. \end{cases} \end{gather*} \end{lemma} Over all, the dynamics of $u$ and $v$ are determined by iterative compositions of functions $f_1$ and $f_2$ as: \begin{gather*} f_1(v,\eta) =G_{\eta}(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}v)) =\frac{K+\eta}{1-\eta}e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}v)), \\ f_2(v,\eta)=F(G(e^{-(1+\frac{1}{K})cL}v)) =F(\frac{K+\eta}{1-\eta}(e^{-(1+\frac{1}{K})cL}v)), \end{gather*} where $F=F_{K,\alpha,\beta}$ is as defined in previous section. The proof of bifurcations depend on the analysis of the derivatives of $f_1$ or $f_2$ with respect to $v$ and $\eta$. One can imagine that it is a hard work, since the formulations of $f_1$ and $f_2$ are so complicated. Chaos and bifurcations are determined by the reflection maps $F(G(e^{-(1+\frac{1}{K})cL}\cdot))$ or $G(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}\cdot))$. Since the two maps are conjugate to each other, it suffices to consider either one of them. Let us look at $F(G(e^{-(1+\frac{1}{K})cL}\cdot))$. Given $\alpha$, $\beta$ and $K$, $F$ is fixed, then the map varies with $G(e^{-(1+\frac{1}{K})cL}\cdot)$. In a word, the dynamics depend on the value of $\frac{K+\lambda}{1-\lambda}e^{-(1+\frac{1}{K})cL}$. So it may be reasonable to define the factor as a parameter. Let $\eta = \frac{K+\lambda}{1-\lambda}e^{-(1+\frac{1}{K})cL}$, then the reflective map $f_2$ can be simplified as \begin{align} f_2(v)=F_{\alpha,\beta,K}(\eta v). \end{align} Of course, it should be much easier to calculate the derivatives of $f_2$ with respect to $v$ and the new parameter $\eta$, so we will easily get the regime of $\eta$ where $f_2$ is chaotic or bifurcated. Then the corresponding regime of $\lambda$ can be obtained by simple calculations. Let us do it as follows: \begin{lemma}[Derivative Formulas]\label{DerivativeFormulas} Let $0<\alpha\le \frac{1}{K}$, $\beta >0$ and $\eta \in \mathbb{R}$, where $\alpha$ and $\beta$ are given and fixed, but $\eta$ is a varying parameter. Define $f(v,\eta)=F_{\alpha,\beta,K}(\eta v)$, $v\in \mathbb{R}$. Let $g(v)$ be the unique real solution of the cubic equation \begin{equation} \frac{4\beta}{(K+1)^2}g(v)^3+(\frac{1}{K}-\alpha)g(v)+(\frac{1}{K}+1)v=0, \end{equation} for a given $v\in \mathbb{R}$. Then \begin{itemize} \item[(i)] $\frac{\partial}{\partial v}f(v,\eta) =\eta\frac{12\beta g(\eta v)^2-(\alpha +1)(K+1)^2}{12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2}$, \item[(ii)] $\frac{\partial}{\partial \eta}f(v,\eta) =v\frac{12\beta g(\eta v)^2-(\alpha +1)(K+1)^2}{12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2}$, \item[(iii)] $\frac{\partial^2}{\partial \eta \partial v}f_(v,\eta) =\frac{12\beta g(\eta v)^2-(\alpha +1)(K+1)^2}{12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2}-\frac{24\beta (K+1)^6\eta g(\eta v)v}{[12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2]^3}$, \item[(iv)] $\frac{\partial^2}{\partial v^2}f(v,\eta) =-\frac{24\beta (K+1)^6\eta ^2g(\eta v)}{[12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2]^3}$, \item[(v)] $\frac{\partial^3}{\partial v^3}f(v,\eta) =-24\beta (K+1)^9\eta ^3\frac{60K\beta g(\eta v)^2-(1-K\alpha)(K+1)^2}{[12K\beta g(\eta v)^2+(1-K\alpha)(K+1)^2]^5}$. \end{itemize} \end{lemma} \begin{lemma}[Intersections with the Lines $u-v=0$ and $u+v=0$] \label{LemmaUVIntersects} Let $\alpha$: $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta \in\mathbb{R}$ be given. Then {\rm (i)} If $\eta >K$ or $\eta <-\frac{1-K\alpha}{1+\alpha}$, then $u=f(v)$ intersects the line $u=v$ at the points \[ (u,v)=(-\frac{K+1}{2(\eta -K)}\sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}}, -\frac{K+1}{2(\eta -K)}\sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}}), \] (0,0), \[ (\frac{K+1}{2(\eta -K)}\sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}}, \frac{K+1}{2(\eta -K)}\sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}}); \] {\rm (ii)} If $\eta <-K$ or $\eta >\frac{1-K\alpha}{1+\alpha}$, then $u=f(v)$ intersects the line $u=-v$ at the points \[ (u,v)=(-\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}, \frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}), \] $(0,0)$, \[ (\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}, -\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}). \] \end{lemma} \begin{lemma}[$v$-aixs Intercepts]\label{vAxisIntercept} Let $\alpha$: $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta >0$, $\eta \neq 1$ be given. Then $u=f(v)$ has $v$-axis intercepts \[ v=-\frac{K+1}{2\eta}\sqrt{\frac{1+\alpha}{\beta}}, \quad 0, \quad \frac{K+1}{2\eta}\sqrt{\frac{1+\alpha}{\beta}}.\] \end{lemma} \begin{lemma}[Local Maximum, Minimum and Piecewise Monotonicity]\label{localM} Let $\alpha$: \\ $0<\alpha\le \frac{1}{K}$, $\beta >0$, $\eta \in \mathbb{R}$ be given. Then If $\eta >0$, then $f$ has local extremal values \begin{gather*} M=f(-v_c)=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \\ m=f(v_c)=-\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \end{gather*} where $v_c=\frac{3+(1-2\alpha)K}{6\eta}\sqrt{\frac{1+\alpha}{3\beta}}$, and $M$, $m$ are, respectively, the local maximum and minimum of $f$. The function $f$ is strictly increasing on $(-\infty, -\tilde{v}_c)$ and $(\tilde{v}_c, \infty)$, but strictly decreasing on $(-\tilde{v}_c, \tilde{v}_c)$. On the other hand, if $\eta<0$, then $f$ has local extremal values \begin{gather*} m=f(-v_c)=-\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \\ M=f(v_c)=\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}. \end{gather*} The function $f$ is strictly decreasing on $(-\infty, -v_c)$ and $(v_c, \infty)$, but strictly increasing on $(-v_c, v_c)$. \end{lemma} \begin{lemma}[Bounded Invariant Intervals]\label{boundedInvariantInterval} Let $0<\alpha\le 1/K$, $\beta >0$, $\eta \in \mathbb{R}$. {\rm (i)} If $\eta >K$, and $$ \frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}<\frac{K+1}{2(\eta -K)} \sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}}, $$ then the iterates of every point in the set \begin{align*} U&\equiv (-\infty,-\frac{K+1}{2(\eta -K)} \sqrt{\frac{1-K\alpha +(\alpha +1)\eta}{\beta(\eta -K)}})\\ &\quad \cup (\frac{K+1}{2(\eta -K)}\sqrt{\frac{1-K\alpha +(\alpha +1)\eta} {\beta(\eta -K)}},\infty) \end{align*} escape to $\pm \infty$, while those of any point in $\mathbb{R}\backslash \bar{U}$ are attracted to the bounded invariant interval ${\mathcal{I}}\equiv [-M,M]$ of $f$; {\rm (ii)} If $\eta <-K$, and $$ \frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}} <-\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}, $$ then the iterates of every point in the set \begin{align*} U&=(-\infty,\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}})\\ &\quad \cup (-\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1} {\beta(\eta +K)}},\infty) \end{align*} escape to $\pm\infty$, while those of any point in $\mathbb{R}\backslash \bar{U}$ are attracted to the bounded invariant interval ${\mathcal{I}}\equiv [-M,M]$ of $f$. \end{lemma} \begin{theorem}[Period-Doubling Bifurcation Theorem]\label{periodDoublingBifurcation} Let $K>0$, $\alpha$: $0<\alpha\le \frac{1}{K}\le 2\alpha +1$, $\beta >0$ be fixed, and let $\eta$: $\eta >0$ be a varying parameter. Then {\rm (i)} For $0<\eta <\frac{1-K\alpha}{1+\alpha}$, $0$ is the unique fixed point of $f$, and it is stable; {\rm (ii)} With the same $\alpha$, $\beta$ and $K$ as in (i), but $\eta >\frac{1-K\alpha}{1+\alpha}$, then $0$ becomes unstable, and there appear stable period-$2$ orbit \[ \big\{\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}},-\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}\big\} \] of $f$; {\rm (iii)} The curve of the period-$2$ points: $$ v=\pm\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1} {\beta(\eta +K)}} $$ is smooth in the $\eta$-$v$ plane, and tangent to the line $\{\frac{1-K\alpha}{1+\alpha}\}\times \mathbb{R}$ at point $(\frac{1-K\alpha}{1+\alpha},0)$; {\rm (iv)} The period-$2$ orbit becomes unstable when $\eta$ increases through \[ \frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-K\alpha)K}}{\alpha +1}. \] \end{theorem} \begin{proof} (i) It follows from Lemma \ref{DerivativeFormulas} (i) that \begin{align*} f'(0) & =\eta\frac{12\beta g(0)^2-(\alpha +1)(K+1)^2}{12K\beta g(0)^2+(1-K\alpha)(K+1)^2}\\ & =\eta\frac{-(\alpha +1)(K+1)^2}{(1-K\alpha)(K+1)^2}\\ & =-\eta \frac{\alpha +1}{1-K\alpha}. \end{align*} So $-12(1-\alpha K)+2\alpha K-K-3 >-K-1, \] and thus \begin{equation} |2(\alpha +1)\eta +2\alpha K-K-3|2K(1-\alpha K)+(K+1)\eta +2(\alpha K-1)K \\ & >(K+1)\eta , \end{aligned}\label{3.17} \end{equation} for $\eta $ greater than $\frac{1-\alpha K}{\alpha +1}$. Combining \eqref{3.15}, \eqref{3.16} and \eqref{3.17}, we have \[ |f'(v_0)|<1 \] for $\eta $ greater than $\frac{1-\alpha K}{\alpha +1}$ and close to $\frac{1-\alpha K}{\alpha +1}$ enough. By similar arguments we have \[ |f'(-v_0)|<1 \] for $\eta $ greater than $\frac{1-\alpha K}{\alpha +1}$ and close to $\frac{1-\alpha K}{\alpha +1}$ enough. Combining the two aspects above, we conclude that the new emerging period-$2$ orbit ares stable. This completes the proof of the period-$2$ bifurcations of $f$ at the origin. (iii) It is easy to verify that \[ v=\pm\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}} \] is differentiable with respect to $\eta$ for $\eta $ in $(\frac{1-\alpha K}{\alpha +1}, \infty)$. The derivative is $\infty$ at $\eta =\frac{1-\alpha K}{\alpha +1}$. This shows that the curve is smooth in the $\eta$-$v$ plane, and tangent to line $\{\frac{1-K\alpha}{1+\alpha}\}\times \mathbb{R}$ at point $(\frac{1-K\alpha}{1+\alpha},0)$. (iv) Let \[ \frac{\partial f}{\partial v}=\eta \frac{2(\alpha +1)\eta +2\alpha K-K-3} {(2\alpha K+3K+1)\eta +2(\alpha K-1)K}>1, \] and then it follows that \[ \eta >\frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-\alpha K)K}}{\alpha +1}, \] or \[ \eta <\frac{K+1-\sqrt{(K+1)^2-(\alpha +1)(1-\alpha K)K}}{\alpha +1}. \] So the period-$2$ orbit becomes unstable when $\eta$ increases through \[ \frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-\alpha K)K}}{\alpha +1}. \] \end{proof} \begin{remark} \label{rmk3.1} \rm We just conclude that $0$ is stable by $|f'(0)|<1$ when $0<\eta < \frac{1-K\alpha}{1+\alpha}$. However, $|f'(0)|<1$ just implies the local stability of $0$. In fact, it follows from Lemma \ref{LemmaUVIntersects} that $u=f(v)$ does not intersect with line $u=v$ or $u=-v$ at other points except the origin, so we have $|f(v)|<|v|$ for $v \neq 0$. Therefore, $0$ attracts $(-\infty, +\infty)$, illustrated by Figure \ref{globalAttract}: \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3-1} \end{center} % 0Attract.eps \caption{Global attraction diagram of $0$ for $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=0.4$.} \label{globalAttract} \end{figure} \end{remark} \begin{remark} \label{rmk3.2} \rm The stable period-$2$ orbit in Theorem \ref{periodDoublingBifurcation} (ii) attracts $(-\infty,0)\cup (0,+\infty)$ for $\eta$ larger than and close to $\frac{1-K\alpha}{1+\alpha}$. Since $-f(-f(v))=f(f(v))$, so the period-$2$ stability under $f$ is equivalent to that under $-f$. The global attraction of its period-$2$ orbit can be easily illustrated by its graph, e.g., Figure \ref{Fig.3.2}. \end{remark} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3-2} \end{center} % period2.eps \caption{Global attraction diagram of the period-$2$ orbit for $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=0.8$.} \label{Fig.3.2} \end{figure} \begin{theorem}[Homoclinic Orbits for the Case $\eta >0$] \label{HomoclinicOrbits} Let $K>0$, $\alpha$: $0<\alpha \le 1/K$ and $\beta >0$ be fixed, and $\eta \geq \frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$, then the repelling fixed point $0$ of $f$ has homoclinic orbits. \end{theorem} \begin{proof} For a homoclinic orbit of $0$ to exist, the local maximum of $f$ must be no less than the positive $v$-axis intercept of $f$, i.e., \[ \frac{K+1}{2\eta}\sqrt{\frac{1+\alpha}{\beta}} <\frac{1+\alpha}{3}\sqrt{\frac{1+\alpha}{3\beta}}, \] which is equivalent to \begin{equation} \eta > \frac{3\sqrt{3}(K+1)}{2(1+\alpha)}.\label{homoclinicOrbitCondition} \end{equation} On the other hand, it follows from Lemma \ref{DerivativeFormulas} (i) that \[ \frac{\partial}{\partial v}f(v,\eta)|_{v=0} =-\eta\frac{\alpha +1}{1-K\alpha}. \] Therefore $0$ is a repelling fixed point of $f$ for $\eta$ larger than $\frac{1-K\alpha}{\alpha +1}$, which is implied by \eqref{homoclinicOrbitCondition}. This completes the proof. \end{proof} For $\eta =\frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$, $v_c$ (or $-v_c$) lies on a degenerated homoclinic orbit. When $\eta <\frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$, $f$ has maximum less than the $v$-axis intercept. Hence there are no points homoclinic to $0$ for these $\eta$-values. On the other hand, when $\eta >\frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$, there are infinitely many distinct homoclinic orbits. Consequently, $f$ is not structurally stable when $\eta =\frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$, i.e., a small change in $f$ can change the number of homoclinic orbits. \begin{example} \label{examp3.1} \rm The parameters chosen are $\alpha=0.5$, $\beta=1$, $\lambda=0.85$, $k_1=k_2=0.7$, $K=0.7$, $b(x)=1+3x^2$, $$ w(x,0)=\sin^2(\pi x), \quad w_t(x,0)=0. $$ Figures \ref{Fig3.3}--\ref{Fig3.6} show the spatiotemporal profiles of $u$, $v$, $w_x$ and $w_t$ for $x\in [0,1]$ and $t\in [7.34,8.80]$ respectively; Figures \ref{Fig3.7} and \ref{Fig3.8} illustrate the reflection maps $F(G(e^{-(1+\frac{1}{K})cL}\cdot))$ and $G(e^{(d-\frac{c}{K})L}F(e^{-(c+d)L}\cdot))$ respectively. \end{example} \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{fig3-3} \end{center} % fig3_1.eps \caption{The spatiotemporal profile of $u(x,t)$ for $x\in [0,1]$ and $t\in [7.34, 8.80]$.} \label{Fig3.3} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{fig3-4} \end{center} % fig3_2.eps \caption{The spatiotemporal profile of $v(x,t)$ for $x\in [0,1]$ and $t\in [7.34,8.80]$.} \label{Fig3.4} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{fig3-5} \end{center} % fig3_3.eps \caption{The spatiotemporal profile of $w_x(x,t)$ for $x\in [0,1]$ and $t\in [7.34, 8.80]$.} \label{Fig3.5} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.8\textwidth]{fig3-6} \end{center} % fig3_4.eps \caption{The spatiotemporal profile of $w_t(x,t)+kw(x,t)$ for $x\in [0,1]$ and $t\in [7.34, 8.80]$.} \label{Fig3.6} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3-7} \end{center} % fig3_5.eps \caption{Orbits of $H_1=F(G(e^{-(\frac{k_1}{K}+k_2)L}\cdot))$, $\alpha=0.5$, $\beta=1$, $\lambda=0.85$, $k_1=k_2=0.7$, $K=0.7$.} \label{Fig3.7} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3-8} \end{center} % fig3_6.eps \caption{Orbits of $H_2=G(e^{-\frac{k_1}{K}L}F(e^{-k_2L}\cdot))$, $\alpha=0.5$, $\beta=1$, $\lambda=0.85$, $k_1=k_2=0.7$, $K=0.7$.} \label{Fig3.8} \end{figure} Figures \ref{Fig3.7} and \ref{Fig3.8} show that $H_1$ and $H_2$ are topologically transitive, so probably they are chaotic according to Devaney's definition \cite{Saber}. \section{Period doubling bifurcation and pitchfork bifurcation route to chaos} The mapping $f_{\eta}$ (or $H_1$, $H_2$) has a unique fixed point or periodic point ($0$), which is stable when $\eta>0$ is small enough. As $\eta$ increases, the fixed point $0$ becomes unstable, and there appears a stable periodic-$2$ orbit, then the period-$2$ orbit becomes unstable, too. Finally, homoclinic orbits appear when $\eta$ is large enough. We have proved these facts in Theorem \ref{periodDoublingBifurcation} and \ref{HomoclinicOrbits}. In this section, we try to explore more about the bifurcation routes. Let us start by a bifurcation diagram, we take $\alpha=0.5$, $\beta=1$, $K=0.7$, and let $\eta$ vary from $0.4$ to $3$. The stable fixed point $0$ bifurcates into a stable symmetric period-$2$ orbit at $\eta \approx 0.43$, then the symmetric period-$2$ orbit bifurcates into two new stable period-$2$ orbits at $\eta \approx 2.2$, then they bifurcate into two period-$4$ orbits at $\eta \approx 2.6$. The bifurcations are illustrated by Figures \ref{Fig.4.1} and \ref{Fig.4.2}. To distinguish the pitchfork period-$2$ bifurcation from the period doubling bifurcation of period-$4$, we start our iteration at $v=0.3$ and $v=-0.3$ respectively, and found that they are stablized by different period-$2$ orbits. It is easy to see that there is a pitchfork bifurcation of period-$2$ following the period doubling bifurcation of period-$2$ described by Theorem \ref{periodDoublingBifurcation}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-1} \end{center} % fenchatu1.eps \caption{Bifurcation diagram of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, iteration starts at $v=0.3$.} \label{Fig.4.1} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-2} \end{center} % fenchatu2.eps \caption{Bifurcation diagram of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, iteration starts at $v=-0.3$.} \label{Fig.4.2} \end{figure} Let us compare this experiment results with Theorem \ref{periodDoublingBifurcation}. $\bullet$ The first bifurcation: from the fixed point to period-$2$ orbit. According to Theorem \ref{periodDoublingBifurcation} (i)-(ii), the first bifurcation parameter value is \begin{align}\label{bifurcationParameter1} \eta = \frac{1-K\alpha}{1+\alpha}. \end{align} Substitute the experiment parameter values $\alpha=0.5$ and $K=0.7$ to \eqref{bifurcationParameter1}, we obtain \begin{align*} \eta=\frac{1-K\alpha}{1+\alpha}=0.4333, \end{align*} which agrees with the bifurcation diagrams. $\bullet$ The second bifurcation: from the symmetric period-$2$ orbit to the nonsymmetric period-$2$ orbits. According to Theorem \ref{periodDoublingBifurcation} (iv), the second bifurcation parameter value is \begin{align}\label{secondBifurcation} \eta = \frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-K\alpha)K}}{\alpha +1}. \end{align} Substitute the experiment parameters to \eqref{secondBifurcation}, we obtain $\eta =2.1238$, which agrees with the bifurcation diagrams. The \emph{ old} period-$2$ points $\pm\frac{K+1}{2(\eta +K)}\sqrt{\frac{(\alpha +1)\eta +\alpha K-1} {\beta(\eta +K)}}$ of $f$ are fixed points of $-f$. Suppose $-f$ has a period-doubling bifurcation at \[ \eta=\frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-K\alpha)K}}{\alpha +1}, \] then the period-$2$ orbits of $-f$ are just the \emph{ new} period-$2$ orbits of $f$. Let us check it as follows. \begin{proof} Let $h=-f$, we have found the parameter value and the fixed point for which $\frac{\partial h}{\partial v} =-1$. So it suffices to verify that \begin{gather*} A\equiv \Big[\frac{\partial ^2h}{\partial\eta\partial v} +\frac{1}{2}(\frac{\partial h}{\partial\eta})\frac{\partial^2h}{\partial v^2} \Big] \neq 0, \\ B\equiv \frac{1}{3}\frac{\partial^3h}{\partial v^3} +\frac{1}{2}(\frac{\partial^2h}{\partial v^2})^2\neq 0, \end{gather*} for \begin{gather}\label{periodTwoPoint} v= v_0\triangleq \frac{K+1}{2(\eta +K)} \sqrt{\frac{(\alpha +1)\eta +\alpha K-1}{\beta(\eta +K)}}, \\ \label{periodTwoParameter} \eta= \eta_0\triangleq \frac{K+1+\sqrt{(K+1)^2-(\alpha +1)(1-K\alpha)K}}{\alpha +1}. \end{gather} It follows from Theorem \ref{DerivativeFormulas} that \begin{equation} \begin{aligned} A &=-\frac{12\beta g(\eta_0 v_0)^2-(\alpha +1)(K+1)^2}{12K\beta g(\eta_0 v_0)^2+(1-K\alpha)(K+1)^2} \\ &\quad +\frac{24\beta (K+1)^6\eta_0 g(\eta_0 v_0)v_0}{[12K\beta g(\eta_0 v_0)^2 +(1-K\alpha)(K+1)^2]^3} \\ &\quad -\frac{12\beta (K+1)^6\eta_0 ^2g(\eta_0 v_0)v_0 [12\beta g(\eta_0 v_0)^2-(\alpha +1)(K+1)^2]} {[12K\beta g(\eta_0 v_0)^2+(1-K\alpha)(K+1)^2]^4}, \label{A} \end{aligned} \end{equation} and \begin{align*} B&=8\beta(K+1)^9\eta_0^3\frac{60K\beta g(\eta_0v_0)^2-(1-K\alpha)(K+1)^2}{[12K\beta g(\eta_0v_0)^2+(1-K\alpha)(K+1)^2]^5}\\ &\quad +\frac{288\beta^2(K+1)^{12}\eta_0^4g(\eta_0v_0)^2}{[12K\beta g(\eta_0v_0)^2+(1-K\alpha)(K+1)^2]^6}. \end{align*} Combing Theorem \ref{DerivativeFormulas} (i) and the fact that $\frac{\partial f}{\partial v} =1$ for $\eta=\eta_0$, $v=v_0$, we have \[ 12\beta g(\eta_0 v_0)^2-(\alpha +1)(K+1)^2>0. \] Noting that $g(\eta v)$ and $v$ have opposite sign, so all the terms in the RHS of \eqref{A} are negative. Therefore $A<0$. By similar arguments we have $B>0$. This completes the proof. \end{proof} The \emph{ old} period-$2$ orbit $\{p(\eta), -p(\eta)\}$ becomes unstable after the second bifurcation, and a pair of stable period-$2$ orbits appear. Denote them by $\{p_1(\eta) ,q_1(\eta)\}$ and $\{p_2(\eta) ,q_2(\eta)\}$ respectively, where $p_1$ and $p_2$ are around $p$, $q_1$ and $q_2$ are around $-p$. Let \[ p_1>p,\quad p_2-p,\quad q_2<-p, \] since $f$ is increasing around $p$ and $-p$. This pair of stable period-$2$ orbits can be illustrated by Figure \ref{Fig.4.1} and Figure \ref{Fig.4.2} (curves over $\eta\in (2.12,2.55)$). Let us look at the period-$4$ bifurcation. By period doubling bifurcation theorems, it occurs where $\frac{\partial }{\partial v}(f\circ f)|_{v=p_i}=f'(p_i)f'(q_i)=-1$, $i=1,2$. On the other hand, \[ \frac{\partial }{\partial v}(f\circ f)=f'(p)f'(-p)=1 \] at the pitchfork bifurcation point of period-$2$. Since $\frac{\partial }{\partial v}(f\circ f)$ varies continuously with respect to parameters and arguments, so $\frac{\partial }{\partial v}(f\circ f)$ must vanish at some period-$2$ point before period-$4$ bifurcation. Since $\frac{\partial }{\partial v}(f\circ f)=0$ if and only if the period-$2$ cycle contains extremal point $v_c$ or $-v_c$, the extremal point $v_c$ or $-v_c$ must be contained in a period-$2$ orbit before period-$4$ bifurcation. This process can be illustrated by the following experiment results and figures: We take $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=1.8$, then Theorem \ref{periodDoublingBifurcation} (ii) tells that the unique symmetric period-$2$ orbit is $\{0.3079,-0.3079\}$. Figure \ref{Fig.4.3} illustrates this fact. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-3} \end{center} %symmetryPeriod2.eps \caption{Stable symmetric period-$2$ orbit of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=1.8$.} \label{Fig.4.3} \end{figure} Then we take larger $\eta=2.2$, the symmetric period-$2$ orbit bifurcates into two branches of stable period-$2$ orbits, illustrated by Figure \ref{Fig.4.4}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-4} \end{center} %twoPeriod2.eps \caption{Two branches of stable period-$2$ orbits of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=2.2$.}\label{Fig.4.4} \end{figure} Take $\eta=2.3$, the two branches of period-$2$ orbits go apart, and pass by the extremal points, illustrated by Figure \ref{Fig.4.5} \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-5} \end{center} % passVc.eps \caption{Two branches of period-$2$ orbits of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=2.3$, period-$2$ orbits pass by extremal points $\pm v_c$.}\label{Fig.4.5} \end{figure} Take $\eta=2.6$, each period-$2$ orbit bifurcates into a stable period-$4$ orbit, illustrated by fig. \ref{Fig.4.6}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig4-6} \end{center} % twoPeriod4.eps \caption{Two branches of stable period-$4$ orbits of $f$, where $\alpha=0.5$, $\beta=1$, $K=0.7$, $\eta=2.6$.}\label{Fig.4.6} \end{figure} It is well known that a discrete dynamical system is chaotic if it has a homoclinic orbit. According to Theorem \ref{HomoclinicOrbits}, $f$ has homoclinic orbits and chaos when $\eta \geq \frac{3\sqrt{3}(K+1)}{2(1+\alpha)}$. For $\alpha =0.5$ and $K=0.7$, the condition is as $\eta \geq 2.9445$, which agrees with bifurcation diagrams Figure \ref{Fig.4.1} and Figure \ref{Fig.4.2}. In addition to homoclinic orbits, period three is a classical criteria for chaos. \begin{theorem}\label{periodThree} Given $\alpha$, $\beta$ and $K$, there exists $\eta_3$ such that $F(\eta_3\cdot)$ has period $3$. \end{theorem} \begin{proof} It is easy to verify that $F(\bar{\eta}\cdot)$ has period three, where $\bar{\eta}$ is the critical value of $\eta$ such that the local maximum equals to the positive intercept with line $u=v$. Let $d=M$, $c=-v_c$, $b\in (0,v_c)$ such that \begin{equation} F(\bar{\eta}b)=-v_c, \end{equation} and $a\in (v_c,M]$ such that \begin{equation} F(\bar{\eta}a)=b. \end{equation} It is easy to see that $c0$. \end{lemma} Then the dynamics of $u$ and $v$ are determined by the iterative compositions of $F_{\alpha,\beta,K}(G_\lambda(e^{-(1+\frac{1}{K})cL}\cdot))$ and $G_\lambda(e^{\int_0^Ld(\eta)d\eta-c\frac{L}{K}} F_{\alpha,\beta}(e^{-cL-\int_0^Ld(\eta)d\eta}\cdot))$. \subsection*{Acknowledgements} This research was supported by grant \#NPRP 4-1162-1-181 from the from the Qatar National Research Fund (a member of Qatar Foundation). \begin{thebibliography}{10} \bibitem{ChHs1} G. Chen, S. B. Hsu, Y. Huang, M. Roque-Sol; The spectrum of chaotic time series (I): Fourier analysis, \emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.} \textbf{5} (2011), 1439--1456. \bibitem{ChHs2} G. Chen, S. B. Hsu, J. Zhou; Chaotic vibrations of the one-dimensional wave equation subject to a self-excitation boundary condition, Part I; \emph{Trans. Amer. Math. Soc.} \textbf{350} (1998), 4265--4311. \bibitem{ChHs3} G. Chen, S.B. Hsu, J. Zhou; Chaotic vibrations of the one-dimensional wave equation subject to a self-excitation boundary condition, Part II, Energy injection, period doubling and homoclinic orbits, \emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.} \textbf{8} (1998), 423--445. \bibitem{ChHs4} G. Chen, S. B. Hsu, J. Zhou, Chaotic vibrations of the one-dimensional wave equation subject to a self-excitation boundary condition, Part III, Natural hysteresis memory effects, \emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.} \textbf{8} (1998), 447--470. \bibitem{ChHs5} G. Chen, S. B. Hsu, J. Zhou; Snapback repellers as a cause of chaotic vibration of the wave equation due to a van der Pol boundary condition and energy injection in the middle of the span, \emph{J. Math. Phys.} \textbf{39} (1998), 6459--6489. \bibitem{ChSb} G. Chen, B. Sun, T. Huang; Chaotic oscillations of solutions of the Klein-Gordon equation due to imbalance of distributed and boundary energy flows, \emph{Internat. J. Bifur. Chaos.} \textbf{24} (2014), 1430021. \bibitem{Devaney} Robert L. Devaney; An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, Inc. New York, 1989. \bibitem{Saber} Saber N. Elaydi; Discrete Chaos, Chapman Hall/CRC, New-York, 1999. \end{thebibliography} \end{document}