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\begin{document}

\title{Title of Article}%%title

\author{Author 1\corrauth{email@xx.com}, Author 2}%%author

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\address{Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville,
Edwardsville,\addressNewline IL62026-1805, USA}%%address
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\Abstract{Each article should be preceded by
an abstract (8-11 lines long) that summarizes the content. The
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unrestricted access. This allows unregistered users to read the
abstract as a teaser for the complete article.Each article should be preceded by
an abstract (8-11 lines long) that summarizes the content. The
abstract will appear online at www.****.com and be available with
unrestricted access. This allows unregistered users to read the
abstract as a teaser for the complete article.}%%abstract
[\hfill keyword1 \par][][][][][2022 L\&H Scientific Publishing, LLC. All rights reserved.]%%

\maketitle
%\thispagestyle{fancy}
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\normalsize

\linenumbers

\section{Section Heading}

  Please note that the first line of text that follows a heading is not indented,
whereas the first lines of all subsequent paragraphs are.

Use the standard \verb|equation| environment to typeset your equations, e.g.
\begin{equation}
\ddot{x}+\delta\dot{x}-\alpha x+\beta x^3=Q_0\cos \Omega t
\end{equation}
where $\dot{{x}}=dx/dt$ is velocity. $Q_{0} $ and $\Omega $ are excitation
amplitude and frequency, respectively. $\delta $is damping coefficient.
$\alpha $ and $\beta$ are linear and nonlinear stiffness
coefficients of the Duffing oscillator.

however, for multiline equations we recommend to use the \verb|eqnarray|
environment\footnote{In physics texts please activate the class option \texttt{vecphys} to depict your vectors in \textbf{\itshape boldface-italic} type - as is customary for a wide range of physical subjects.}.
\begin{eqnarray}
a \times b = c \nonumber\\
\vec{a} \cdot \vec{b}=\vec{c}
\label{eq:01}
\end{eqnarray}


\begin{theorem}
theorem
\end{theorem}



\begin{proposition}
theorem
\end{proposition}

\begin{lemma}
theorem
\end{lemma}

\begin{coro}
theorem
\end{coro}

\begin{conj}
theorem
\end{conj}

In mechanical engineering, in 1918, Duffing \cite{1} presented the hardening
spring model to describe the vibration of electro-magnetized vibrating beam.
Since then, the Duffing oscillator has been extensively used to describe
nonlinear structural vibrations in structural dynamics. In 1964, Hayashi \cite{2}
discussed the approximate periodic solutions and the corresponding stability
by the averaging method and harmonic balance method. In 1973, Nayfeh \cite{3}
used the perturbation method to approximate periodic motion of the Duffing
oscillators (also see, Nayfeh and Mook \cite{4}). In 1979, Holmes \cite{5} showed the
strange attractors of chaotic motions in nonlinear oscillators via the
Duffing oscillator with a twin-well potential. In 1980, Ueda \cite{6} used
numerical simulations to show chaotic motion via period-doubling of periodic
motions of Duffing oscillators. In 1997, Luo and Han \cite{7} analytically
presented the stability and bifurcation conditions of periodic motions of
the Duffing oscillator. The constant term of the analytical solution for the
steady-state motion of the Duffing oscillator was not considered. In 1996,
Luo and Han \cite{8} presented an improved solution of the Duffing oscillator
with a twin-well potential. For analytical prediction of chaos, in 1999, Luo
and Han \cite{9} investigated chaotic motions in nonlinear rod through the
Duffing oscillator. For the periodically forced Duffing oscillator with
damping, the analytical prediction of periodic solutions is still very
difficult. In this paper, the analytical solutions of periodic motions will
be investigated and the analytical route of periodic motions to chaos will
be of great interest.

\begin{figure}[t!]
\centering
\includegraphics{fig1.eps}
%\vspace*{-30mm}
\caption{The analytical prediction of periodic solutions based on two
harmonic terms (HB3): (a) constant term $a_0$; (b)-(d)
harmonic amplitudes $A_k (k=1,2,3)$; and (e)-(f)
harmonic phases $\varphi_k$ ($k=1,2)$ for right potential well.
($\delta=0.5,\alpha=-10.0,\beta=10.0,Q_0=10.0)$.}
\label{fig1}
\end{figure}

To look for approximate analytical solution of nonlinear oscillator, such an
issue started from Lagrange \cite{10} to investigate the three-body problem as a
perturbation of the two-body problem by the method of averaging. In the end
of the 19th century, Poincare \cite{11} further developed the perturbation theory
to investigate the motions of celestial bodies. In 1920, van der Pol \cite{12}
used the method of averaging to determine the periodic solutions of
oscillation systems in circuits. Until 1928, the asymptotic validity of the
method of averaging was not proved. In 1928, Fatou \cite{13} gave the proof of
the asymptotic validity through the solution existence theorems of
differential equations. In 1935, Krylov, Bogoliubov and Mitropolsky \cite{14}
further developed the method of averaging, and the detailed presentation was
given. In 1964, Hayashi \cite{2} presented the perturbation methods including
averaging method and principle of harmonic balance. In 1969, Barkham and
Soudack \cite{15} extended the Krylov-Bogoliubov method for the approximate
solutions of nonlinear autonomous second order differential equations (also
see, Barkham and Soudack \cite{16}). In 1987, a generalized harmonic balance
approach was used by Garcia-Margallo and Bejarano \cite{17} to determine
approximate solutions of nonlinear oscillations with strong nonlinearity. In
the same year, Rand and Armbruster \cite{18} used the perturbation method and
bifurcation theory to determine the stability of periodic solutions. In
1989, Yuste and Bejarano \cite{19} used the elliptic functions rather than
trigonometric functions to improve the Krylov-Bogoliubov method. In 1990,
Coppola and Rand \cite{20} used the averaging method with elliptic functions to
determine approximation of limit cycle. In 1997, Luo and Han \cite{7}
analytically studied the stability and bifurcations of periodic solutions of
Duffing oscillators through the first order harmonic balance method, and
provided the analytical conditions for the Hopf and saddle-node
bifurcations. To obtain accurate results of periodic solutions in nonlinear
vibration, many harmonic terms are included in the harmonic balance method.
In 2008, Peng et al \cite{21} presented the approximate period-1 solution for the
Duffing oscillator by the HB3 method compared with the fourth-order
Runge-Kutta method. In 2011, Luo and Huang \cite{22} further discussed a
generalized harmonic balance method to obtain the analytical solution of
period-1 motion. Luo and Huang \cite{23} also presented a generalized harmonic
balance method to determine period-$m$ solutions in nonlinear oscillators.



In this paper, the generalized harmonic balance method will be used to
investigate analytical periodic motions in the periodically forced Duffing
oscillator with a twin-well potential. The bifurcation tree from period-1
motions to chaos will be presented with varying parameters. The
corresponding unstable periodic motions in the Duffing oscillator will be
presented for a better understanding of nonlinear dynamics in such a Duffing
oscillator. Numerical illustrations of stable and unstable periodic motions
will be carried out.

\section{Section Heading}

From Eq.(1), the standard form is
\setcounter{equation}{2}
\begin{equation}
\ddot{x}+f(x,\dot{x},t)=0
\end{equation}

\subsection{Subsection Heading}

The Fourier series expression of any periodic motion in nonlinear systems
needs infinite terms to give the exact solution of such a periodic motion.
In practice, it is impossible to do so. Thus, the truncated Fourier series
solutions will be used to give an approximate solution that can be close to
the exact solution. From such approximate, analytical solutions, the
equilibrium solution of coefficient dynamical system for the Fourier series
of the periodic motion can be obtained from Eq.(3) using Newton-Raphson
method, and the stability and bifurcation analysis of the such equilibrium
points can be completed through the eigenvalue analysis. The system
parameters are
\begin{equation}
\delta=0.5,\alpha=-10.0,\beta=10,Q_0=10.0
\end{equation}
The backbone curves of harmonic amplitude varying with excitation frequency
$\Omega$ are illustrated. The harmonic amplitude and phase are
defined by
\begin{equation}
A_{k/m}\equiv\sqrt{b^2_{k/m}+c^2_{k/m}},\varphi_{km}=\arctan\dfrac{c_{k/m}}{b_{k/m}}
\end{equation}
and the corresponding solution in Eq.(43) is
\begin{equation}
x^*(t)=a^{(m)}_0+\sum^N_{k=1}A_{k/m}\cos\left(\dfrac{k}{m}\Omega t-\varphi_{k/m}\right).
\end{equation}

In Luo and Han \cite{8}, one term harmonic term was considered for period-1
motions for the large and small orbit. In this paper, many harmonic terms
will be considered to achieve a more accurate prediction of the periodic
motions. For period-1 motion, the first three harmonic terms of the Fourier
series expansion (HB3) will be used to obtain the approximate periodic
solutions. The constant term $a^{(1)}_0\equiv a_0$ and the first three
harmonic amplitudes $A_k$ and phases $\varphi_k(k=1,2,3)$ versus excitation frequency are plotted in
Fig.1(a)-(g), respectively. A parameter map is presented in Fig.2.

The initial conditions for stable period-1 motion ($\Omega=2.75)$,
unstable period-1 motion and stable period-2 motion ($\Omega=2.753)$,
unstable period-1 motion, unstable period-2 motion and stable period-4
motion ($\Omega=2.7537)$ are listed in Table 1.

\begin{table}[t!]
\centering
\caption{\rm Input data for numerical simulations of periodic motions $(\delta=0.5,\alpha=-10.0,\beta=10.0,Q_0=10.0)$.}
\setlength{\tabcolsep}{20pt}
\footnotesize{
\begin{tabular*}{16.5cm}{cccccc}
\hline\hline\hline
&  \raisebox{-2ex}[0pt][0pt]{$\Omega$}  &  \multicolumn{2}{c}{Initial conditions $(t=0.0)$}  &  \raisebox{-2ex}[0pt][0pt]{Stability}  &  \raisebox{-2ex}[0pt][0pt]{Period-$m$}\\\cline{3-4}
& & $x_0$  &  $\dot{x}_0$\\\hline
Fig.10(a)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)\\
Fig.10(b)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)\\
Fig.10(c)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)\\
Fig.10(d)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)\\
Fig.10(e)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)\\
Fig.10(f)  &  2.75  &  $-0.724512$  &  0.251206  &  Stable(HB)  &  Period-1(HB5)
\\\hline\hline\hline\end{tabular*}}
\end{table}



\begin{figure}[t!]
\centering
\includegraphics{fig2.eps}
\caption{A parameter map from the analytical prediction of periodic
solutions based on three harmonic terms (HB3): (a) Global view and (b)
zoomed view. ($\delta=0.5,\alpha=-10.0,\beta=10,Q_0=10.0)$.}
\label{fig1}
\end{figure}



\section{Conclusions}
In this paper, analytical routines of period-1 motions to chaos in the
Duffing oscillator with a twin-potential well were discussed comprehensively
through the generalized harmonic balance method. The analytical solutions of
period-$m$ motions were developed by the Fourier series and the corresponding
Hopf bifurcations of periodic motions cause new periodic motions with
period-doubling. Three analytical routes of asymmetric period-1 motions to
chaos were developed. The approximate, analytical periodic solutions were
verified via numerical simulations, and the analytical, unstable periodic
motions were given as well. With exact unstable periodic motion, the
numerical simulations should stay with the analytical solution if without
any computational errors. The analytical routes with unstable periodic
motions can lead us to find unstable chaos.



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\end{document}
