数理解析研究所講究録
第 1786 巻 2012 年 93-115

93

Survey of Periodic Solutions of the Nonlinear Ordinary Differential
Equations and Study of Periodic Solutions of the Duffing Type Equation
with the Square Wave External Force
Nohara, B. T. and Arimoto, A.

Tokyo City University, Tokyo, Japan
FAX 03-5707-2147, E-mail: drben@tcu.ac.jp

1

Introduction

In the cource of studying on nonlinear ordinary differential equations, the existence of periodic
solutions has been forcused on as one of main subjects. The nontrivial periodic solutions haveimportant meanings in a variety of field such as engineering, medical, economic area and so on.
The question of whether a natural or social phenomenon has a certain periodicity is importantand interesting for us. For example, in the climatology, the past periodicity of global climatechange has been researched well but its future periodicity is a significant issure. Another example
is the terrestrial magnetism which turned the other way by the time rate of 1.5 times per a million
years [10],[14]. These are earth-scale examples but using a familiar one, there exists a rise or
fall in the exchange rate and stock market.
Moreover some phenomena in nature have a multiple-time periodicity, which means thatmultiple-time oscillations, like double-time, triple-time, quadruple-time oscillations and so on,
exist in a period. The electrocardiogram of human beings is a good example. The healthy heart,roughly speaking, beats triple-time oscillations in a period. The normal heart beating consistsof a wave, QRS complex and a wave[7] in a period. However man with heart defect does
not always beat triple-time oscillations.
The objective of this paper is to survey studies of periodic solutions of the nonlinear ordinary
differential equations and present the explicit form for periodic solutions of a nonlinear ordinary
differential equation(Eq.(2.12)) with the external force. Also we show the nontrivial periodic
solutions for Eq.(2.12) create multiple-time oscillations in a period depending on the period of
the external force.
The paper is organized as follows: first, we survey periodic solutions of the nonlinear ordinary
differential equations in the literature. In the following section, we treat the linear case $q=0$ in
Eq.(2.12). There exist such types of periodic solutions as the -periodic(Definition 3.1), hidden
periodic(Definition 3.2) and quasi-periodic one(Remark 3.5) even in the linear case. The explicit
forms of such solutions are shown as well as the periodic conditions. Then in the first half ofSection 4, the solution when $e(t)=$ const. is obtained and in the latter half we construct the
nontrivial periodic solution in the Farkas sense using the result of the first half. The ‘Farkas
sense’[5] means that it is periodic with a periodof the external force which can be chosen
appropriately. The numerical simulations are also presented at important positions.
$P$

$T$

$\omega$$\omega$2

Survey of periodic solutions of the nonlinear ordinary differential equations

In this section we survey studies of periodic solutions of the nonlinear ordinary differential
equations in the literature [3],[5],[6],[8],[9],[13],[15],[17],[19],[20]. First we state the following
proposition: Let.
$\dot{u}=\frac{du}{dt},\ddot{u}=\frac{d^{2}}{dt}\tau u$

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Proposition 2.1.
(2.1)$\ddot{u}+\varphi(u)\dot{u}+\psi(u)=0$

has an ‘essentially unique periodic solution under the following four $conditions[6J.\cdot(a)\varphi$ :and :are continuous and satisfies the Lipschitz condition. $(b)\varphi(u)=\varphi(-u)$ . $(c)$$\psi(u)=-\psi(-u)$ and $\psi(u)>0$ for $u>0$ . $(d) \int_{0}^{u}\varphi(s)ds<0$ for $0<u<u_{0}$ .
for$u>u_{0}$ .
is a monotone increasing function and
oo.
as
‘

$\mathbb{R}$$\psi$$\mathbb{R}arrow$$\mathbb{R}arrow \mathbb{R}$$\psi$$\int_{0}^{u}\varphi(s)ds>0$$\int_{0}^{u}\varphi(s)ds$$\int_{0}^{u}\varphi(s)dsarrow\infty$$uarrow$

Remark 2.1. The ‘essentially unique’ means that if $u=\xi(t)$ is a nontriivial periodic solutionof Eq. (2.1), then all other nontrivial periodic solutions of Eq. (2.1) are of the $fomu=\xi(t-\tau)$ ,where is a real number.
$\tau$

Proposition 2.1 was firstly proven by Levinson and Smith[8]. The important fact of this
proposition is that there exists only one periodic solution in Eq.(2.1). Moreover the periodic
orbit created by such an ‘essentially unique’ periodic solution becomes a unique limit cycle ofEq.(2.1), which is globally orbital stable. The fact that there exist a lot of periodic solutions if
the conditon (d) is not satisfed is known. Under weaker hypotheses, some improvements of this
proposition have been accomplished[5].
We call the special case in Eq.(2.1): $\psi(u)=u$ , i.e.,(2.2)$\ddot{u}+\varphi(u)\dot{u}+u=0$

the Li\’enard equation[9]. Moreover letting $\varphi(u)=\epsilon(u^{2}-1)$ yields$\ddot{u}+\epsilon(u^{2}-1)\dot{u}+u=0$

,

(2.3)

which is the van der Pol equation[17].
Example 2.1. The van der Pol equation (2.3) satisfies the conditions of Proposition 2.1. Therefore the van der Pol equation has an essentially unique nontrivial periodic solution.

Also letting $\varphi(u)=-(a-b\dot{u}^{2})$ leads to$\ddot{u}-(a-b\dot{u}^{2})\dot{u}+u=0$

(2.4)

,

which is the Rayleigh equation[13]. This equation also satisfies the conditions of Proposition
2.1.
Example 2.2. The Rayleigh equation (2.4) has an essentially unique nontrivial periodic solution.

Remark 2.2. By differentiating $Eq.(2.4)w.r.t$ .
identical with the van der Pol equation.

$t$and letting$\frac{du}{dt}=v$, the equation of

$v$

is

Eq.(2.5), which is the generalized Li\’enard equation and is based on a more realistic modelling,has been studied so far[20]. However we can only state that Eq.(2.5) has a periodic solution

which is not essentially unique.

Proposition 2.2.
(2.5)$\ddot{u}+\varphi(u,\dot{u})\dot{u}+\psi(u)=0$

has at least a periodic solution under the following four conditions::are continuous and satisfy the Lipschitz condition. $(b)u\psi(u)>0$ for $u\neq 0$ .
monotone increasing function and $|\psi(u)|arrow\infty$ as $|u|arrow\infty$ for $|u|\geq u_{0}$ . Moreover$(a)\varphi$$\mathbb{R}arrow \mathbb{R}$$\frac{\psi(u)}{\int_{0}^{u}\psi(s)ds}=\mathcal{O}(\frac{1}{|u|})$$(c)\exists u_{0}>0$for$\forall u,$$v$.

and $v_{0}>0,$ $s.t$ . $\varphi(u, v)\geq M>0$ for $|u|\geq u_{0},$. $(d)\varphi(0,0)<0$ .

$\mathbb{R}^{2}arrow \mathbb{R}$and$\psi(u)$:is a$\psi$(2.6)$|v|\geq v_{0}$and $\varphi(u, v)\geq-m,$ $(m>0)$95
Next we follow up periodic solutions which synchronize with a period of an external force.
Proposition 2.3.
(2.7)$\ddot{u}+\varphi(u,\dot{u})\dot{u}+\psi(u)=e(t)$has at least a periodic solution(period ) under the following four conditions::and :are continuous and satisfy the Lipschitz condition. $(b)u\psi(u)>0$ forislarge.
is a monotone increasing function for.
is large and and $|\psi(u)|arrow\infty$ as
Moreover.
(2.8)$(a)\varphi$$\omega$$\mathbb{R}arrow \mathbb{R}$$\psi$$\mathbb{R}^{2}arrow \mathbb{R}$$|u|$$|\psi(u)|$$|u|arrow\infty$$|u|$$\frac{\psi(u)}{\int_{0}^{u}\psi(s)ds}=\mathcal{O}(\frac{1}{|u|})$$(c)\exists u0>0$

for$\forall u,$$v$.

and $v_{0}>0,$ $s.t$ . $\varphi(u, v)\geq M>0$ foris continuous and $e(t)=e(t+\omega)$ .

$|u|\geq u_{0},$ $|v|\geq v_{0}$and $\varphi(u, v)\geq-m,$ $(m>0)$$(d)e:\mathbb{R}arrow \mathbb{R}$Proposition 2.3 was also proven by Levinson and Smith. The following proposition was provenby Yamaguti [19].

Proposition 2.4.
(2.9)$\ddot{u}+\varphi(u)\dot{u}+\psi(u, t)=e(t)$has at least a periodic solution(period ) under the following six conditions::and:are continuous and satisfy the Lipschitz condition $w.r.t$ . . $(b)\psi(x, t)$ has a partial
derivative coefficient $g_{t}(x, t)$ , which is continuous $w.r.t$ . . $(c)\psi(x, t)=\psi(x, t+\omega),$ $e(t)=e(t+\omega)$
and $\int_{0}^{\omega}e(t)dt=0$ .
sgn $uarrow\infty$ as as $|u|arrow\infty$ and $| \int_{0}^{t}e(s)ds|<E_{0}$ . $(e)$$(a)\varphi$$\omega$$\mathbb{R}arrow \mathbb{R}$$\mathbb{R}^{2}arrow \mathbb{R}$$\psi$$u$$t$$(d) \int_{0}^{u}\varphi(s)ds$$\psi(u, t)$$k_{0},$$u_{0},$

sgnu $\geq k_{0}>0$ for $|u|>u_{0}$ .
$k_{1},$$u_{1}$$(f)| \int_{0}^{u}\varphi(s)ds|>\frac{1}{k_{1}}|\frac{\partial t\int_{0}^{u}\psi(s,t)ds}{\psi(u,t)}|$are positive definite and $0<k_{1}<1$ .

for $|u|>u_{1}$ , where

Moreover, the perturbed Li\’enard equation$\ddot{u}+\varphi(u)\dot{u}+\psi(u)=\epsilon f(\frac{t}{\omega}, u,\dot{u})$(2.10)

has been studied recently and the existence of a nontrivial periodic solution of Eq.(2.10) is
proven under the mild conditions[3].
Finally, in this section, we shall introduce the study of Taam[16], which stimulates the authors’
motivation. The equation is based on the Duffing equation with a periodic external force.
Proposition 2.5. Let$p,$$q>0$ .

The equation$\ddot{u}+pu+2qu^{3}=e(t)$,(2.11)

where $e(t)=e(t+\omega),$ $e(t)=-e(-t)$ and $e(t)>0$ for $0<t< \frac{\omega}{2},e(0)=e(\omega)=0$ , has a periodic
solution of period such as. Here $M$ is a constant number which is obtained by$\omega$$\omega\leq\frac{2\pi}{\sqrt{2qM^{2}+p}}$

solving an algebmic equation induced by coefficients of $Eq.(2.11)$ .

Also Taam derived such a condition that Eq.(2.11) has-of-phase solutions comparing
with an external force.
In this paper, our objective is to present concrete solutions in order to understand the solution
stracture of Eq.(2.11). To do so, we let an external force a definite function. Therefore, ourtarget equation is the following Duffing type equation[4],[11]:$\frac{\omega}{2}-out$$\ddot{u}+pu+2qu^{3}=f(t)$ ,(2.12)

96
where

$p,$

$q>0$ . Also we impose the external force as follows: let $\nu=0,1,2,$$\ldots$(2.13)$f(t)=\{\begin{array}{l}\frac{e}{2}, \nu\omega\leq t<(\nu+\frac{1}{2})\omega-\frac{e}{2}, (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega\end{array}$where $e>0$ and $\omega>0$ , which indicate the amplitude and period of the external force, respectively. If the external force $f=0$ , Eq.(2.12) is the standard Duffing equation[4], which is anonlinear oscillator with a cubic stiffness term to describe the hardening spring effect observedin many mechanical problems. We have the following fact concerning the standard Duffing
equation.

Fact 2.1. [12] The standard Duffing equation$\ddot{u}+pu+qu^{3}=0$,(2.14)

where $p>0,$ $q>0$ , has the following essential unique periodic solution for any initial condition:$u(t)=\sqrt{\frac{\sqrt{p^{2}+qE}-p}{q}}$cn$((\sqrt{p^{2}+qE})^{\frac{1}{4}}t,$$\sqrt{\frac{\sqrt{p^{2}+qE}-p}{2\sqrt{p^{2}+qE}}})$

Here $E>0$ is determined by the initial condition. The period$\tau=\frac{4K(\sqrt{\frac{\sqrt{p^{2}+qE}-p}{2\sqrt{p^{2}+qE}}})}{(\sqrt{p^{2}+qE})^{\frac{1}{4}}}$$\tau$.

(2.15)of the solution is presented as.

(2.16)

Many studies concerning the Duffing equation have been carried out, in particular, the chaos
related researches[18] have been used to study after the discovery of chaos phenomenon in
Eq.(2.12) with a damping factor(u) and a sinesoidal fuction for the external force. However the
problem of whether Eq.(2.12) itself has a nontrivial periodic solution or not had been almostly
forgotten except such a few researches as Taam stated before.

3

The linear case $(q=0)$ : harmonic oscillation with the external
force

First we study the linear case in Eq.(2.12). That is, in the differential equation:$\ddot{u}+I^{\gamma u}=F,$$t\geq 0$,

where we suppose that $F=F(t)=F(t+\omega),$ $\omega>0,p>0$ . We categorize the relations of
to clarify periodic solutions as follows:(1) $\omega\sqrt{p}\neq 0(mod 2\pi)$ , moreover more precicely,$h\in Z^{+},$ $h\neq 1,$(1-1)
and are irreducible.

$p$$\frac{\omega}{2\pi/\sqrt{p}}=\frac{g}{h},$(1-2)$\frac{\omega}{2\pi/\sqrt{p}}=$

$g,$

$g$

$h$

irrational number.

(2) $\omega\sqrt{p}=0(mod 2\pi)$ .
Some definitions are presented before describing theorems.

(3.1)$\omega$and97
Definition 3.1. Letand. Also let $u=u(t)$ , which is times$u^{(n)}(t)$, andbe the n-th order derivative of $u(t)$ . In the$g:\mathbb{R}^{n+1}arrow \mathbb{R}(n\geq 1)$

differentiable function defined infollowing, differential equation:$t\in \mathbb{R}$$F:\mathbb{R}arrow \mathbb{R}$$g(u(t),\dot{u}(t),\ddot{u}(t),$where $F(t)=F(t+\omega)$ , we call the solution$\ldots,$$u^{*}(t)$$n$$u^{(n)}(t))=F(t)$ ,the$\omega$(3.2)-periodic solution if $u(t)=u(t+\omega)$ .

Remark 3.1. We consider is a polynominal of $u(t),\dot{u}(t),$, in this paper. We distinguishthe -periodic solution from another solution by $indicating*likeu^{*}$ if necessary.
$g$$\ldots$$\omega$

Remark 3.2. If $u(t)$ has period , then the solution has also period $2T,$ $3T,$
the smallest period, then we call this smallest the period of $u(t)$ .
$T$$\ldots$

Suppose

$T$

is

$T$

Definition 3.2. In the differential equation (3.2), we call the solution $u\#(t)$ the hidden periodic
solution if. We call the hidden period of the solution.
$u(t)=u(t+\hat{\omega}),\hat{\omega}\neq\omega$$\hat{\omega}$

Remark 3.3. We distinguish the hidden periodic solution from another solution by indicating
like
if necessary.

$\#$

$u\#$

Theorem 3.1. Suppose that$\frac{\omega}{2\pi/\sqrt{p}}=\frac{g}{h}$. Here$g,$$h\in Z^{+},$$h\neq 1$

, and

$g$

and

are irreducible.

$h$(1) The differential equation (3.1) has the -periodic solution, that is, $u^{*}(t)=u^{*}(t+\omega)$ iff theinitial condition $sat\iota sfies(u(O),\dot{u}(0))=(u^{*}(O),\dot{u}^{*}(0))$ .
(2) The w-periiodic solution is presented by$\omega$$u^{*}(t)= \frac{1}{2\sqrt{p}\tan(\sqrt{p}\omega/2)}l^{t+\omega}\cos\sqrt{p}(t-s)F(s)ds-\frac{1}{2\sqrt{p}}\int^{t+\omega}\sin\sqrt{p}(t-s)F(s)ds$(3) Suppose$\omega isn’ t$

.

(3.3)

the period of the solution of Eq. (3.1). The differential equation (3.1) has thehidden periodic solution, that is, $u\#(t)=u\#(t+\hat{\omega})$ where
period which is the least common multiple of$\int_{0}^{\omega h}\sin\sqrt{p}sF(s)ds=0$$\omega$andand$\frac{2\pi}{\sqrt{p}}$$\hat{\omega}(=\omega h or \frac{2\pi g}{\sqrt{p}})$indicates the hidden

iff$\int_{0}^{\omega h}\cos\sqrt{p}sF(s)ds=0$

.

(3.4)

(4) The hidden periodic solution is presented by$u \#(t)=u(O)\cos\sqrt{p}t+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}t+\frac{1}{\sqrt{p}}\int_{0}^{t}\sin\sqrt{p}(t-s)F(s)ds$

.

(3.5)

Proof. (1) Let$u(t)=(\begin{array}{l}u(t)\dot{u}(t)\end{array}),$$A=(\begin{array}{ll}0 1-p 0\end{array}),$ $F(t)=(\begin{array}{l}0F(t)\end{array})$

,

(3.6)

then Eq.(3.1) can be rewritten as$\dot{u}(t)=$Au $(t)+F(t)$ .

(3.7)

We have the solution of Eq.(3.7) as the following form:$u(t)=e^{tA}u(0)+e^{tA}\int_{0}^{t}e^{-sA}F(s)ds$

.

(3.8)

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If this solution is -periodic, then $u^{*}(t)=u^{*}(t+\omega)$ . Using this fact, we easily obtain$\omega$.

(3.9)$u^{*}(t)=e^{\omega A}(1-e^{\omega A})^{-1}e^{tA}l^{t+\omega}e^{-sA}F(s)ds$

Eqs.(3.8) and (3.9) leads to$u(t)-u^{*}(t)=e^{tA}(u(O)-u^{*}(O))$ .

(3.10)

This relation follows the theorem.
(2) By simple calculations we have$e^{\omega A}=$$(\cos\sqrt{p}\omega$$\frac{\sin\sqrt{p}\omega}{\cos\sqrt{p}w\sqrt{p}}),$$(1-e^{\omega A})^{-1}= \frac{1}{2(1-\cos\sqrt{p}\omega)}(1-\cos\sqrt{p}\omega$$1 \sqrt{p}\omega\frac{\sin\sqrt{p}\omega}{-cos,(3.11)\sqrt{p}})$Using these equations, we obtain Eq.(3.3) easily.
(3) Eq.(3.8) directly yields.

(3. 12)$u(t)=u(0) \cos\sqrt{p}t+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}t+\frac{1}{\sqrt{p}}\int_{0}^{t}\sin\sqrt{p}(t-s)F(s)ds$

So we obtain$u(t+ \hat{\omega})=u(0)\cos\sqrt{p}(t+\hat{\omega})+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}(t+\hat{\omega})+\frac{1}{\sqrt{p}}\int_{0}^{t+\hat{\omega}}\sin\sqrt{p}(t+\hat{\omega}-s)F(s)ds$

From the assumption$\hat{\omega}\sqrt{p}=2\pi g$$\frac{\omega}{2\pi/\sqrt{p}}=\frac{g}{h}$, we have the hidden period$\hat{\omega}=\omega h=\frac{2\pi g}{\sqrt{p}}$. $(3.13)$. Using the relation, then we have$u(t+ \hat{\omega})=u(0)\cos\sqrt{p}t+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}t+\frac{1}{\sqrt{p}}\int_{0}^{t+\hat{\omega}}\sin\sqrt{p}(t-s)F(s)ds=$.

$=u(0) \cos\sqrt{p}t+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}t+\frac{1}{\sqrt{p}}\int_{0}^{t}\sin\sqrt{p}(t-s)F(s)ds+\frac{1}{\sqrt{p}}\int^{t+\hat{\omega}}\sin\sqrt{p}(t-s)F(s)ds$(3.14)The most right term can be rewritten as. $(3.15)$$\int^{t+\hat{\omega}}\sin\sqrt{p}(t-s)F(s)ds=\sin\sqrt{p}t\int^{t+\hat{\omega}}\cos\sqrt{p}sF(s)ds-\cos\sqrt{p}t\int^{t+\hat{\omega}}\sin\sqrt{p}sF(s)ds$

Here let$t\in[(m-1)\hat{\omega}, m\hat{\omega}),$ $m\inN$

without loss of generality then we have$\int^{t+\hat{\omega}}\cos\sqrt{p}sF(s)ds=l^{m\hat{\omega}}\cos\sqrt{p}sF(s)ds+\int_{m\hat{\omega}}^{t+\hat{\omega}}\cos\sqrt{p}sF(s)ds=$$= \int^{m\hat{\omega}}\cos\sqrt{p}sF(s)ds+\int_{(m-1)\hat{\omega}}^{t}\cos\sqrt{p}sF(s)ds=$$= \int_{(m-1)\hat{\omega}}^{m\hat{\omega}}\cos\sqrt{p}sF(s)ds=$$= \int_{0}^{\hat{\omega}}\cos\sqrt{p}sF(s)ds=const$.

(3. 16)

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Similarly,$l^{t+\hat{\omega}} \sin\sqrt{p}sF(s)ds=\int_{0}^{\hat{\omega}}\sin\sqrt{p}sF(s)ds=const$.

(3.17)

Consequently, the relations Eqs.(3.14), (3.15), (3.16) and (3.17) imply that
(3.18)$u(t+\hat{w})=u(t)$iff Eq. (3.4) holds. This equation means that $u(t)$ is the hidden periodic solution.
(4) This is obvious in the proof of (3).
$\blacksquare$

Remark 3.4. The fact that the hidden periodic solution depends on the initial condition is clear
from the expression of $u\#(t)$ . The hidden periodic solution has the period and $isn’ t$ uniquedepending on the initial condition.
$\hat{\omega}$

Theorem 3.2. Suppose thatthe$\omega$$\frac{\omega}{2\pi/\sqrt{p}}=irmtional$

number. The differential equation (3.1) has-periodic solution presented by $Eq.(3.3)$ if the initial condition satisfies $(u(O),\dot{u}(0))=$$(u^{*}(0),\dot{u}^{*}(0))$.

There doesn’t exist the hidden periodic solution.

Proof. The first half of the statement can be proven by the same manner of the proof of Theorem
3.1(1). See [18](pl47-l49) for the proof of the latter half.
Remark 3.5. The solution except the -periodic one in Theorem 3.2 is called the quasi-periodic
one.
$\omega$

Theorem 3.3. Suppose thatare$\omega$(mod ). All solutions in the differential equation (3.1)-periodic and don’t depend on the initial condition iff$\omega\sqrt{p}=0$$\int_{0}^{\omega}\sin\sqrt{p}sF(s)ds=0$$2\pi$

and$\int_{0}^{\omega}\cos\sqrt{p}sF(s)ds=0$.

(3.19)The solutions formula is presented by.

(3.20)$u^{*}(t)=u(0) \cos\sqrt{p}t+\frac{\dot{u}(0)}{\sqrt{p}}\sin\sqrt{p}t+\frac{1}{\sqrt{p}}\int_{0}^{t}\sin\sqrt{p}(t-s)F(s)ds$

Proof. We can prove this simply by letting $h=1$ in Theorem 3.1(2).
We give some examples of the theorems of this section. Letthat is, we consider the following linear differential equation:$f$of Eq.(2.13) be$F$in Eq.(3.1),$\ddot{u}+pu=\{\begin{array}{l}\frac{e}{2}, \nu\omega\leq t<(\nu+\frac{1}{2})\omega-\frac{e}{2}, (\nu+\frac{1}{2})\omega\leq t<(l\text{ノ}+1)\omega\end{array}$

Example 3.1. Let $p=1,$
We obtain the following,$e=1$ in Eq. (3.21), then this case corresponds to Theorem 3.1.
-periodic solution by computing $Eq.(3.3)$ concretely:$\omega=\pi,$$\omega(=\pi)$(3.21)$u^{*}(t)=\{\begin{array}{ll}\frac{1}{\not\in}(\sin(t-(\nu+\frac{1}{2})\pi)-\sin(t-\nu\pi)+1), \nu\pi\leq t<(\nu+\frac{1}{2})\pi\overline{2}(\sin(t-(\nu+\frac{1}{2})\pi)-\sin(t-(\nu+1)\pi)-1), (\nu+\frac{1}{2})\pi\leq t<(\nu+1)\pi\end{array}$

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Also the hidden periodic solution, in which period is$2\pi$, is computed by $Eq.(3.5)$ as follows:$u^{\#}(t)=\{\begin{array}{ll}(u(0)-\frac{1}{\int})\cos t+\dot{u}(0)\sin t+\frac{1}{2}, \nu\pi\leq t<(\nu+\frac{1}{2})\pi(u(0)-)\cos t\overline{\not\in}+(\dot{u}(0)+1)\sin t-\frac{1}{\not\in}, (\nu+\frac{1}{2})\pi\leq t<(\nu+1)\pi(u(0)+\overline{\not\in})\cos t+(\dot{u}(0)+1)\sin t+\overline{2}’ (\nu+1)\pi\leq t<(\nu+\frac{3}{2})\pi(u(0)+\overline{2}) \cos t+\dot{u}(0)\sin t-\frac{1}{2}, (\text{ノ}+\frac{3}{2})\pi\leq t<(\nu+2)\pi\end{array}$Figures 3.1 and 3.2 show the numerecal results directly computed from the differental equation.
-periodic solution in the initial condition: $u(O)=$In Figure 3.1, the phase portmit of the$\omega(=\pi)$and the time histories up to four periods are shown. On the other hand, Figure
3.2 shows the hidden periodic one. Note that the hidden periodic solution depends on the initialcondition, so the orbit of the hidden periodic one varies with the initial condition.

$0, \dot{u}(0)=-\frac{1}{2}$Figure 3.1: (left):The phase portrait of the $\omega(=\pi)$ -periodic solution in the initial condition:
in Example 3.1. (right): The time history of the $\omega(=\pi)$ -periodic solution.
The time is shown up to four periods.
$u( O)=0,\dot{u}(0)=-\frac{1}{2}$Figure 3.2: (left):The phase portrait of the hidden $(2\pi)$ periodic solution in the initial condition:

in Example 3.1. (right): The time history of the hidden $(2\pi)$ periodic
solution. The time is shown up to two periods.
$u( O)=\frac{1}{2},\dot{u}(0)=\frac{1}{2}$Example 3.2. Let $p=1,$ $\omega=1,$ $e=1$ in $Eq.(3.21)$ , then we find$\frac{\omega}{2\pi/\sqrt{p}}=\frac{1}{2\pi}$so thatthis example corresponds to Theorem 3.2. We obtain the following, $\omega(=1)$ -periodic solution by
Eq. (3.3):$u^{*}(t)=\{\begin{array}{l}\frac{l}{2\sin\frac{1}{2}}(\sin(t-(\nu+\frac{1}{2}))-\sin(t-\nu)+\sin\frac{1}{2}), \nu\leq t<\nu+\frac{1}{2}\frac{l}{2\sin\frac{1}{2}}(\sin(t-(\nu+\frac{1}{2}))-\sin(t-(\nu+1))-\sin\frac{1}{2}), \nu+\frac{1}{2}\leq t<\nu+1\end{array}$

101
The initial conditions except $u( O)=0,\dot{u}(0)=\frac{\cos\frac{l}{2}-1}{2\sin\frac{1}{2}}=-0.1276709606\ldots$ create quasiperiodic solutions. Fig.3.3 shows the $\omega(=1)$ -periodic solution’s orbit with the initial condition:$u( O)=0,\dot{u}(0)=\frac{\cos\frac{l}{2}-1}{2\sin\frac{1}{2}}$and the time histories shown up to four periods. Fig.3.4 shows thequasi-periodic orbit with the initial condition: $u(O)=0,\dot{u}(0)=-0.12$ .

Figure 3.3:

(left):The $\omega(=1)$ -periodic orbit with the initial condition:$\frac{\cos\frac{l}{2}-1}{2\sin\frac{1}{2}}=-0.1276709606\ldots$$u(O)=0,\dot{u}(0)=$in Example 3.2. (right):The time histories shown up to four

periods.

Figure 3.4: The quasi-periodic orbit with the initial condition:
Example 3.2. The orbit is shown up to $t=30$ .

$u(O)=0,\dot{u}(0)=-0.12$ in

Example 3.3. Let $p=4,$ $\omega=2\pi,$ $e=1$ in $Eq.(3.21)$ , then we find $\omega\sqrt{p}=0(mod2\pi)$ so that
this example corresponds to Theorem 3.3. We obtain the following, $\omega(=2\pi)$ -periodic solution
by Eq. (3.20):$u^{*}(t)=\{(u(0)-\frac{1}{})cos(2t)+\frac sin(2t)+\frac{1}{8,l}(u(0)+\frac{81}{8})cos(2t)+\frac{\dot{u}(0)\dot{u}f_{o)}}{2}sin(2t)-\frac{}{8’},\nu\pi\leq t<(\nu+1)\pi(\nu+1)\pi\leq t<(\nu+2)\pi$

All solutions become the $\omega(=2\pi)$ -periodic solution the initial condition. Fig.3.5 shows the$\omega(=2\pi)$ -periodic orbit with the initialcondition:and the time $histor^{J}ies$shown up to two periods.
$u( O)=\frac{3}{10},\dot{u}(0)=\frac{3}{10}$102$\frac{Fi3}{10},\dot{u}(0)=\frac{3}{10}inExample33.(right):Thetimehistoriesshownuptotwoperiodsgure3.5:(1eft):The\omega(=.2\pi)-periodicsolution’ sorbitwiththeinitialcondition:u(0)=$4Periodic solutions of the Duffing equation with the squarewave external force

In this section, we obtain the solution of the nonlinear differential equation,$\ddot{u}+pu+2qu^{3}=\{$$\nu\omega\leq t<(\nu+\frac{1}{2})\omega,$$\nu=0,1,2,$$\ldots$$- \frac{e}{2’}\frac{e}{2}$$( \nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega$First, we treat the following nonlinear differential equation modified from the above equaiton:$\ddot{u}+\mu\iota+2qu^{3}=\frac{e}{2},$with the initial condition:
we let$u(O)=u0,\dot{u}(0)=u_{00}$$0\leq t,$$p,$ $q,$$e>0$ ,

(4.1)

. From the first integral of motion in Eq.(4.1),$f(u)=c+eu-pu^{2}-qu^{4}$ ,(4.2)

where $c=u_{00}^{2}-eu_{0}+pu_{0}^{2}+qu_{0}^{4}$ . Here let $c>0$ .
Now let $\alpha_{i}(i=1,2,3,4)$ be the roots of $f(u)=0$ and we define$p_{1}$

and$p_{2}$

as
(4.3)
(4.4)
(4.5)

$f(u)=-qp_{1}(u)p_{2}(u)$ ,$p_{1}(u)=(u-\alpha_{1})(u-\alpha_{2})$,$p_{2}(u)=(u-\alpha_{3})(u-\alpha_{4})$.

, and two real ones, one is positive
We easily find two complex roots, which are named
and the other negative, named. The
for $f(u)=0$ . So we haveandelementary symmetric polynomials of Eq.(4.3) are$\alpha_{1},$$\alpha_{3},$$\alpha_{2}$$\alpha_{4}<0<\alpha_{3}$$\overline{\alpha}_{1}=\alpha_{2}$$\alpha_{4}$$\sigma_{1}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}$,$\sigma_{2}=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{3}+\alpha_{2}\alpha_{4}+\alpha_{3}\alpha_{4}$,,$\sigma_{3}=\alpha_{1}\alpha_{2}\alpha_{3}+\alpha_{1}\alpha_{2}\alpha_{4}+\alpha_{1}\alpha_{3}\alpha_{4}+\alpha_{2}\alpha_{3}\alpha_{4}$$\sigma_{4}=\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}$,then we have the following relations using Eq.(4.2):$\sigma_{1}=0,$$\sigma_{2}=\frac{p}{q},$$\sigma_{3}=\frac{e}{q},$$\sigma_{4}=-\frac{c}{q}$

.

(4.6)

We obtain$\alpha_{1}+\alpha_{2}=-(\alpha_{3}+\alpha_{4})$from$\sigma_{1}=0$

.

(4.7)

103
Lemma 4.1.

$\alpha_{3}+\alpha_{4}>0$.

Proof. Using Eq.(4.7), we have$\sigma_{3}=\alpha_{1}\alpha_{2}\alpha_{3}+\alpha_{1}\alpha_{2}\alpha_{4}+\alpha_{1}\alpha_{3}\alpha_{4}+\alpha_{2}\alpha_{3}\alpha_{4}=$$=\alpha_{1}\alpha_{2}(\alpha_{3}+\alpha_{4})+(\alpha_{1}+\alpha_{2})\alpha_{3}\alpha_{4}=$$=(\alpha_{1}\alpha_{2}-\alpha_{3}\alpha_{4})(\alpha_{3}+\alpha_{4})$

,

(4.8)

namely$\frac{e}{q}=(\alpha_{1}\alpha_{2}-\alpha_{3}\alpha_{4})(\alpha_{3}+\alpha_{4})$In Eq.(4.9), the facts that$\frac{e}{q}>0,$$\alpha_{1}\alpha_{2}=|\alpha_{1}|^{2}>0$

and.

$\alpha_{3}\alpha_{4}<0$

(4.9)

imply

$\alpha_{3}+\alpha_{4}>0$.

$\blacksquare$

By the variable transform$v=u-\alpha_{4}$$p_{1}(u),p_{2}(u)$,(4.10)

are transformed to$p_{1}^{*}(v)=(v-N)(v-\overline{N})=v^{2}-(N+\overline{N})v+|N|^{2}$,(4.11)(4.12)$p_{2}^{*}(v)=v(v-M)=v^{2}-Mv$ ,where $M=\alpha_{3}-\alpha_{4},$ $N=\alpha_{1}-\alpha_{4}$ . Here we construct the quadratic equation[2] using thecoefficients ofas follows:$p_{1}^{*}(v),p_{2}^{*}(v)$.

(4.13)$(M-(N+\overline{N}))x^{2}+2|N|^{2}x-M|N|^{2}=0$

Let $m,$$n$

be the roots of Eq.(4.13), then we have
$m= \frac{\sqrt{|N|^{4}+M|N|^{2}(M-(N+\overline{N}))}-|N|^{2}}{M-(N+\overline{N})}$,$n= \frac{-\sqrt{|N|^{4}+M|N|^{2}(M-(N+\overline{N}))}-|N|^{2}}{M-(N+\overline{N})}$(4.14).

(4.15)

Since $M-(N+\overline{N})=2(\alpha_{3}+\alpha_{4})>0$ from Lemma 4.1 and $M=\alpha_{3}-\alpha_{4}>0$ , we find $m>0$

and $n<0$ .
Here we check the signs of$p_{1}^{*}(m),p_{1}^{*}(n),p_{2}^{*}(m)$and$p_{2}^{*}(n)$.

Lemma 4.2.
$p_{1}^{*}(m)>0,p_{1}^{*}(n)>0,p_{2}^{*}(m)<0,p_{2}^{*}(n)>0$

are clear since
Proof. $p_{1}^{*}(m)>0$ andsince $n<0,$ $M>0$ . On the other hand, since$p_{1}^{*}(n)>0$$p_{1}^{*}$.

(4.16)

has no real roots. Also $p_{2}^{*}(n)=n(n-M)>0$$\sqrt{|N|^{4}+M|N|^{2}(M-(N+\overline{N}))}<|N|^{2}+M(M-(N+\overline{N}))$ ,(4.17)

then$m-M= \frac{\sqrt{|N|^{4}+M|N|^{2}(M-(N+\overline{N}))}-|N|^{2}}{M-(N+\overline{N})}-M<$$< \frac{|N|^{2}+M(M-(N+\overline{N}))-|N|^{2}}{M-(N+\overline{N})}-M=0$

So we have $p_{2}^{*}(m)=m(m-M)<0$ .

$\blacksquare$.

(4.18)

104
Theorem 4.1. Let the initial condition be $u(O)=u_{0},\dot{u}(0)=u00$ and suppose $c=u_{00}^{2}-eu0+$. The nonlinear differential equation$pu_{0}^{2}+qu_{0}^{4}>0$$\ddot{u}+\psi u+2qu^{3}=\frac{e}{2},0\leq t,$ $p,$ $q,$

$e>0$ ,

has the solution$u(t)= \frac{m-n}{1+|B|cn(\Omega(t-t_{0}),k)}+n+\alpha_{4}$where$t_{0}$(4.19),

is determined by the initial condition and$\Omega=\frac{\sqrt{A^{2}+B^{2}}\sqrt{qp_{1}^{*}(n)p_{2}^{*}(n)}}{m-n},$Proof. The first integral of Eq.(4.1) is$A^{2}= \frac{p_{1}^{*}(m)}{p_{1}^{*}(n)},$ $B^{2}=- \frac{p_{2}^{*}(m)}{p_{2}^{*}(n)},$$\dot{u}^{2}=f(u)$$k= \frac{|B|}{\sqrt{A^{2}+B^{2}}}$.

(4.20), that is,$\dot{u}^{2}=c+eu-pu^{2}-qu^{4}$.

(4.21)

Using the variable transform Eq.(4.10), Eq.(4.21) becomes$\dot{v}^{2}=-qpi(v)p_{2}^{*}(v)$.

(4.22)

Here let Eq.(4.22) rewrite using the linear transform$v= \frac{m+nw}{1+w}$.

(4.23)

First$p_{1}^{*}(v)=p_{1}^{*}( \frac{m+nw}{1+w})=(\frac{m+nw}{1+w})^{2}-(N+\overline{N})\frac{m+nw}{1+w}+|N|^{2}=$$= \frac{1}{(1+w)^{2}}((m^{2}-m(N+\overline{N})+|N|^{2})+(2mn-(N+\overline{N})(m+n)+2|N|^{2})w+$$+(n^{2}-n(N+\overline{N})+|N|^{2})w^{2})$Here $m$ and$n$

are the roots of Eq.(4.13) so we have $m+n=- \frac{2|N|^{2}}{M-(N+\overline{N})},$(4.24)$mn=- \frac{M|N|^{2}}{M-(N+\overline{N})}$

Then we easily find the second term in Eq.(4.24) is zero, that is,$2mn-(N+\overline{N})(m+n)+2|N|^{2}=0$ .

Therefore$p_{1}^{*}(v)$(4.25)

is rewritten as$p_{1}^{*}(v)= \frac{1}{(1+w)^{2}}(p_{1}^{*}(m)+p_{1}^{*}(n)w^{2})$.

(4.26)$p_{2}^{*}(v)= \frac{1}{(1+w)^{2}}(p_{2}^{*}(m)+p_{2}^{*}(n)w^{2})$.

(4.27)

Similarly,

Consequently, the linear transform Eq.(4.23) to Eq.(4.22) yields$\dot{w}^{2}=\frac{-q}{(m-n)^{2}}(p_{1}^{*}(m)+p_{1}^{*}(n)w^{2})(p_{2}^{*}(m)+p_{2}^{*}(n)w^{2})$.

(4.28).

105
Taking account Lemma 4.2 in Eq.(4.28), we have.

(4.29)$\frac{dw}{\sqrt{(\frac{p_{1}^{*}(m)}{p_{1}^{*}(n)}+w^{2})(-\frac{p_{2}^{*}(m)}{p_{2}^{*}(n)}-w^{2})}}=\pm\frac{\sqrt{qp_{1}^{*}(n)p_{2}^{*}(n)}}{m-n}dt$Let$A^{2}= \frac{p_{1}^{*}(m)}{p_{1}^{*}(n)},$$B^{2}=- \frac{p_{2}^{*}(m)}{p_{2}^{*}(n)}$$\frac{1}{\sqrt{A^{2}+B^{2}}}$cn, then we obtain.

$-1( \frac{w}{|B|})\frac{|B|}{\sqrt{A^{2}+B^{2}}}I=\pm\frac{\sqrt{qp_{1}^{*}(n)p_{2}^{*}(n)}}{m-n}(t-t_{0})$(4.30)

this, we directly have

$F\}om$$w(t)=|B|$ cn $( \frac{\sqrt{A^{2}+B^{2}}\sqrt{qp_{1}^{*}(n)p_{2}^{*}(n)}}{m-n}(t-t_{0}),$$\frac{|B|}{\sqrt{A^{2}+B^{2}}})$(4.31).

Inversing the linear transform Eq.(4.23) and variable transform Eq.(4.10) leads to Eq.(4.19).
$k)\neq$
Note that the denominator of Eq.(4.19) does not become zero, that is, $1+|B|$ cn. Because $0<m<-n$ and $0<M-m<M-n$ lead to $0<m(M-m)<n(n-M)$ . Then$\blacksquare$$(\Omega(t-t_{0}),$$0$$B^{2}= \frac{m(M-m)}{n(n-M)}<1$. So we have $|B|<1$ .

Corollary 4.1. In Theorem 4.1, we let $u(O)=0,\dot{u}(0)=\sqrt{c}$ .
$cn(\Omega t_{0})=-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$$sn(\Omega t_{0})<0$

in Eq.

$( \Omega t_{0})=-\frac{m+\alpha}{|B|(n+\alpha 4)}$,in this Corollary. Flromin Eq.(4.19). Also we obtain$u_{00}=\sqrt{c}$$\dot{u}(t)=\frac{(m-n)|B|\Omega sn(\Omega(t-t_{0}),k)dn(\Omega(t-t_{0}),k)}{(1+|B|cn(\Omega(t-t_{0}),k))^{2}}$Since $m-n>0$ and the assumption: $\dot{u}(0)=\sqrt{c}>0$ , sn
Lemma 4.3. Let the right-hand side of Eq.

$(4\cdot 19)$$\ddot{u}+pu+2qu^{3}=-\frac{e}{2},$Pmof.

$Substituing-h_{1}(t-\frac{\omega}{2})$must satisfy$(4\cdot 19)$.

Proof. First, if $u(O)=0$ , then $u_{00}^{2}=c(>0)$ . We treat$u(O)=0$ , we directly have cn$t_{0}$

be$(\Omega t_{0})<0$$h_{1}(t)$$\frac{\omega}{2}\leq t,$.

.

(4.32)

must be satisfied.

satisfies$Then-h_{1}(t- \frac{\omega}{2})$$p,$ $q,$$\blacksquare$(4.33)$e>0$ .

to the equation: the (left side)-(right side) of Eq. (4.33) yields$- \ddot{h}_{1}(t-\frac{\omega}{2})-ph_{1}(t-\frac{\omega}{2})-2qh_{1}^{3}(t-\frac{\omega}{2})+\frac{e}{2}=$$=-( \ddot{h}_{1}(t-\frac{\omega}{2})+ph_{1}(t-\frac{\omega}{2})+2qh_{1}^{3}(t-\frac{\omega}{2})-\frac{e}{2})=$$=-( \ddot{h}_{1}(\tau)+ph_{1}(\tau)+2qh_{1}^{3}(\tau)-\frac{e}{2})=$

$=0$

since

$h_{1}(t)$satisfies Eq.(4.1).

$\blacksquare$

$($

for

$\frac{\omega}{2}\leq t)$

for$0\leq\tau)$

$($

106
Lemma 4.4. $|B|=- \frac{m}{n}$ .

Proof. We remember that

$m$

and

$n$

are the roots of the quadratic equation (4.13). Then we have$m+n=- \frac{-2|N|^{2}}{M-(N+\overline{N})},$$mn=- \frac{-M|N|^{2}}{M-(N+\overline{N})}$.

Using these relations, we find that the following trivial equality$m(M-n)+n(M-m)=(m+n)M-2nm$
equals zero. Therefore, $- \frac{m}{n}=\frac{M-m}{M-n}$ . This follows the lemma.

Lemma 4.5. Let $H( \tau)=\frac{m-n}{1+|B|cn(\tau,k)}+n+\alpha_{4}$ . Then

(4.34)$\blacksquare$$\max_{\tau}H(\tau)=\alpha_{3},$$\min_{\tau}H(\tau)=\alpha_{4}$.

. The first integral of Eq.(4.1) is $\dot{u}^{2}=c+eu-pu^{2}-qu^{4}$ andProof. First, we show
, which have the relation
we find that $f(u)=c+eu-pu^{2}-qu^{4}=0$ has two real roots:, and the others are complex ones. So we have
. Also we obtain Eq.(4.19),$\alpha_{4}\leq H(\tau)\leq\alpha_{3}$$\alpha_{3},$$0<\alpha_{3}$$\alpha_{4}<$$\alpha_{4}$$\alpha_{4}\leq u(t)\leq\alpha_{3}$therefore it follows$\alpha_{4}\leq H(\tau)\leq\alpha 3$. Using Lemma 4.4, $H( \tau)=\frac{n(m-n)}{n-mcn(\tau,k)}+n+\alpha_{4}$ .

Therefore,$\max_{\tau}H(\tau)=H(\tau)|_{cn(\tau,k)=-1}=\frac{n(m-n)}{m+n}+n+\alpha_{4}=$$= \alpha_{4}+\frac{2mn}{m+n}=\alpha_{4}+M=\alpha_{4}+(\alpha_{3}-\alpha_{4})=\alpha_{3}$$\min_{\tau}H(\tau)=H(\tau)|_{cn(\tau,k)=1}=\frac{n(m-n)}{n-m}+n+\alpha_{4}=\alpha_{4}$,.

$\blacksquare$

From now on, we pay for Eq.(4.35) in the sence of Farkas[5]. That is, we let a period of theexternal force be able to be chosen appropriately. Also we fix the initial condition $(u(O),\dot{u}(0))=$$c>0$ in order to show the existence of the w-periodic solutions.
$\omega$$(0, \sqrt{c}),$Theorem 4.2. Suppose that $c>0$ . Let $T= cn^{-1}(-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}),$ $0<T<2K(k)$ and$\omega=\frac{4}{\Omega}(2(1+2l)K(k)-T)$

for some non-negative integers . The nonlinear differential equation$l$,$\ddot{u}+pxu+2qu^{3}=\{$$\nu\omega\leq t<(\nu+\frac{1}{2})\omega,$$\nu=0,1,2,$$\ldots$$- \frac{e}{2’}\frac{e}{2}$(4.35)$( \nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega$with the initial condition $(u(O),\dot{u}(0))=(0, \sqrt{c})$ has $C^{1}\omega-per^{J}iodic$ solutions$u^{*}(t)=\{\begin{array}{ll}h_{+}^{o}(t-\nu\omega), \nu\omega\leq t<(\nu+\frac{1}{2})\omega,-h_{+}^{o}(t-(\nu+\frac{1}{2})\omega), (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega,\end{array}$(4.36)

where$h_{+}^{o}(t)= \frac{m-n}{1-|B|cn(\Omega(t-\frac{\omega}{4}),k)}+n+\alpha_{4}$.

(4.37)

107
We use the suffix ” in Eq.(4.36). This means “odd“-time oscillations, that is, the solution has
the odd number of oscillations in a period like single-time, triple-time, quintic-time oscillations
and so on. Also the suffix ” means “even”-time oscillations, that is, the solution has theeven number of oscillations in a period like double-time, quadruple-time, sextic-time oscillations
and so on. We will claim in Theorem 4.3, the solution given in Thereom 4.2 has “odd”-time
oscillations.
$0$

$(e$

Proof. We formally construct the
Lemma 4.3 as follows:$\omega$-periodic solution for Eq.(4.35) using Theorem 4.1 and(4.38)$u^{*}(t)=\{\begin{array}{ll}h_{1}(t-\nu\omega), \nu\omega\leq t<(\nu+\frac{1}{2})\omega,-h_{1}(t-(\nu+\frac{1}{2})\omega), (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega,\end{array}$where$h_{1}(t)= \frac{m-n}{1+|B|cn(\Omega(t-t_{0}),k)}+n+\alpha_{4}$.

(4.39)

Basically, the form of Eq. (4.38) is -periodic so that the following conditions must hold inorder that the solution is guaranteed asin $t\geq 0$ :$\omega$$C^{1}$$\lim_{\epsilonarrow 0}u^{*}((\nu+\frac{1}{2})\omega-\epsilon)=\lim_{\epsilonarrow 0}u^{*}((\nu+\frac{1}{2})\omega+\epsilon),$(4.40)$\forall\nu$(4.41)$\forall\nu$$\lim_{\epsilonarrow 0}u^{*}(\nu\omega-\epsilon)=\lim_{\epsilonarrow 0}u^{*}(\nu\omega+\epsilon),$$\lim_{\epsilonarrow 0}\dot{u}^{*}((\nu+\frac{1}{2})\omega-\epsilon)=\lim_{\epsilonarrow 0}\dot{u}^{*}((\nu+\frac{1}{2})\omega+\epsilon),$(4.42)$\forall\nu$(4.43)$\forall\nu$$\lim_{\epsilonarrow 0}\dot{u}^{*}(\nu\omega-\epsilon)=\lim_{\epsilonarrow 0}\dot{u}^{*}(\nu\omega+\epsilon),$From Eqs.(4.40) and (4.41), we obtain. Also from Eqs.(4.42) and (4.43), we
obtain. Therefore Eqs. (4.40) to (4.43) are rewritten as$h_{1}( \frac{\omega}{2})=-h_{1}(0)$$\dot{h}_{1}(\frac{\omega}{2})=-\dot{h}_{1}(0)$(4.44),$\frac{m-n}{1+|B|cn(\Omega(\frac{\omega}{2}-t_{0}),k)}+n+\alpha_{4}=-(\frac{m-n}{1+|B|cn(\Omega t_{0},k)}+n+\alpha_{4})$$\frac{(m-n)|B|\Omega sn(\Omega(\frac{\omega}{2}-t_{0}),k)dn(\Omega(\frac{\omega}{2}-t_{0}),k)}{(1+|B|cn(\Omega(\frac{\omega}{2}-t_{0}),k))^{2}}=\frac{(m-n)|B|\Omega sn(\Omega t_{0},k)dn(\Omega t_{0},k)}{(1+|B|cn(\Omega t_{0},k))^{2}}$.

(4.45)On the other hand, since the initial condition: $u(O)=0$ , we directly havecn

(4.46)$( \Omega t_{0}, k)=-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$from Eq.(4.39). Applying Eq.(4.46) to Eq.(4.44) leads tocn$( \Omega(\frac{\omega}{2}-t_{0}), k)=-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$,and from these two equations we obtain the relationcn$( \Omega(\frac{\omega}{2}-t_{0}), k)=$cn$(\Omega t_{0)}k)$.

Therefore we obtain the following necessary condition:$\Omega(\frac{\omega}{2}-t_{0})=\Omega t_{0}+4\ell K(k),$ョ$\ell\in Z$ ,(4.47)

108
for the-periodic solution which satisfies $u(O)=0$ . Note that Eq.(4.45) is automaticallysatisfied by the condition Eq.(4.47). So can be written by$C^{1}\omega$$t_{0}$$t_{0}= \frac{\omega}{4}+2\ell\frac{K(k)}{\Omega}$.

(4.48)

Next we consider such condition that the solution also must satisfies the other initial condition:. Substituting Eq.(4.48) toyields$\dot{u}(0)=\sqrt{c}$$\dot{h}_{1}(0)$$\dot{h}_{1}(0)=-\frac{(m-n)|B|\Omega sn(\Omega t_{0},k)dn(\Omega t_{0},k)}{(1+|B|cn(\Omega t_{0},k))^{2}}=$$=- \frac{(m-n)|B|\Omega sn(\frac{\omega}{4}\Omega+2lK(k),k)dn(\frac{\omega}{4}\Omega+2\ell K(k),k)}{(1+|B|cn(\frac{\omega}{4}\Omega+2\ell K(k),k))^{2}}$.

(4.49)

From $\omega=\frac{4}{\Omega}(2(1+2l)K(k)-T)$ , it follows $4lK(k)< \frac{\omega}{4}\Omega=2(1+2l)K(k)-T<2(1+2l)K(k)$.

Then sn$( \frac{\omega}{4}\Omega, k)>0$. Therefore in order to $\dot{u}(0)=\dot{h}_{1}(0)=\sqrt{c}>0,$$m-n>0$ . Therefore substituting Eq.(4.48), that is,$t_{0}= \frac{\omega}{4}+2\ell\frac{K(k)}{\Omega}$yields Eq.(4.37).
The rest of the proof is to examine the suitability of $0<T<2K(k)$ .

cn$\ell$$(T, k)=- \frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$$T$

must be odd since( :odd) to Eq.(4.39)$\ell$

is defined by.

(4.50)

Since $H(T)= \frac{m-n}{1+|B|cn(T,k)}+n+\alpha_{4}$ has a period $4K(k)$ and $H(O)=\alpha 4<0,$ $H(2K(k))=$(from Lemma 4.5), we can selecttheorem.
$\alpha_{3}>0$$T$

as $0<T<2K(k)$ from the intermediate value$\blacksquare$

Theorem 4.3. The
in a period.

$C^{1}\omega$-periodic solution in Theorem 4.2 has $(1+4l)-oscillations(l=0,1,2, \ldots)$

Proof. The number of oscillations in a period can be found by accounting the number of zerosof$\dot{u}^{*}(t)=0,$ $\nu\omega\leq t<(\nu+1)\omega$. Let $N_{0}=\#\{t|\dot{u}^{*}(t).=0, \nu\omega\leq t<(\nu+1)\omega\}$ . From the structureof Eq.(4.36), it is sufficient to account $n_{0}= \#\{t|h_{+}^{o}(t)=0,0\leq t<\frac{\omega}{2}\}$ . That is,Consequently, the number of oscillations in a period equals . Now, we easily have$n_{0}=-N_{A}2^{\cdot}$$n_{0}$$\dot{h}_{+}^{o}(t)=\frac{(m-n)|B|\Omega sn(\Omega(t-\frac{\omega}{4}),k)dn(\Omega(t-\frac{\omega}{4}),k)}{(1-|B|cn(\Omega(t-\frac{\omega}{4}),k))^{2}}$(4.51)

then we only need to account the number of zeros, say , of sn. From this we find that sn $(-2(1+2l)K(k)+T)<0$ at $t=0$ since-2 $(1+2l)K(k)<$$-2(1+2l)K(k)+T<-4lK(k)(\cdot.\cdot 0<T<2K(k))$ . Also we have sn$(2(1+2l)K(k)-T)>0$
at. Therefore the argument of the sn function lies in $8lK(k)<8lK(k)+2(2K(k)-T)<$$4(1+2l)K(k)$ . From this fact
equals the number of zeros of the sn function during overperiods and less than 1 $+2l$ periods. Moreover, since sn $(-2(1+2l)K(k)+T)<0$ and
sn $(2(1+2l)K(k)-T)>0$ , we obtain $n_{0}^{s}=1+4l$ . Of cource,then the theorem isproven.
We only state the other theorems since we can prove them similarly.
$n_{0}^{s}$$( \Omega(t-\frac{\omega}{4}), k)|_{\omega=\frac{4}{\Omega}(2(1+2l)K(k)-T)}=$$0,0 \leq t<\frac{\omega}{2}$$t= \frac{\omega}{2}$$n_{0}^{s}$$2l$$n_{0}=n_{0}^{s}$$\blacksquare$

109
Theorem 4.4. Suppose that $c>0$ . Let $T= cn^{-1}(-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}),$ $0<T<2K(k)$ and$\omega=\frac{4}{\Omega}(4(1+l)K(k)-T)$

for some non-negative integers . The nonlinear differential equation$l$,$\ddot{u}+pu+2qu^{3}=\{$$\nu\omega\leq t<(\nu+\frac{1}{2})\omega,$$\nu=0,1,2,$$\ldots$$- \frac{e}{2’}\frac{e}{2}$$( \nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega$with the initial condition $(u(O),\dot{u}(0))=(0, \sqrt{c})$ has$C^{1}\omega$-periodic solutions(4.52)$u^{*}(t)=\{\begin{array}{ll}h^{\underline{o}}(t-\nu\omega), \nu\omega\leq t<(\nu+\frac{1}{2})\omega,-h^{\underline{o}}(t-(\nu+\frac{1}{2})\omega), (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega,\end{array}$where$h_{-}^{o}(t)= \frac{m-n}{1+|B|cn(\Omega(t-\frac{\omega}{4}),k)}+n+\alpha_{4}$Theorem 4.5. The
in a period.

$C^{1}\omega$(4.53).

-periodic solution in Theorem 4.4 has $(3+4l)-oscillations(l=0,1,2, \ldots)$

Next we state the theorem regarding “even”-time oscillations.
Theorem 4.6. Suppose that $c>0$ . Let $T=$ cn$\omega=8(1+l)\frac{K(k)}{\Omega}$$-1(- \frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}),$$0<T<2K(k)$ andfor some non-negative integers . The nonlinear differential equation$l$with the initial condition $(u(O),\dot{u}(0))=(0, \sqrt{c})$ has$C^{1}$$(4\cdot 35)$

w-periodic solutions(4.54)$u^{*}(t)=\{\begin{array}{ll}h_{+}^{e}(t-\nu\omega), \nu\omega\leq t<(\nu+\frac{1}{2})\omega,-h_{+}^{e}((\nu+1)\omega-t), (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega,\end{array}$where$h_{+}^{e}(t)= \frac{m-n}{1+|B|cn(\Omega t+T),k)}+n+\alpha_{4}$Proof. We construct the$\omega$

as follows:.

(4.55)-periodic solution for Eq.(4.35) as the same manner of Theorem 4.2(4.56)$u^{*}(t)=\{\begin{array}{ll}h_{1}^{e} (t- vw ), \nu\omega\leq t<(\nu+\frac{1}{2})\omega,-h_{1}^{e}((\nu+1)\omega-t), (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega,\end{array}$where$h_{1}^{e}(t)= \frac{m-n}{1+|B|cn(\Omega(t-t_{0}),k)}+n+\alpha_{4}$The form of Eq. (4.56) issolution is guaranteed as$\omega$$C^{1}$.

(4.57)-periodic so that Eqs.(4.40) to (4.43) must hold in order that thein $t\geq 0$ . Eqs.(4.40) and (4.41) are written as$\frac{m-n}{1+|B|cn(\Omega(\frac{\omega}{2}-t_{0}),k)}+n+\alpha_{4}=-\frac{m-n}{1+|B|cn(\Omega(\frac{\omega}{2}-t_{0}),k)}-(n+\alpha_{4})$$\frac{m-n}{1+|B|cn(\Omega t_{0},k)}+n+\alpha_{4}=-\frac{m-n}{1+|B|cn(\Omega t_{0},k)}-(n+\alpha_{4})$.

,(4.58)(4.59)

Note that Eqs.(4.42) and (4.43) are automatically satisfied. Eq.(4.59) is identical with $u(O)=0$ .
Also Eq.(4.58) becomes to be identical with Eq.(4.59) using $\omega=8(1+l)\frac{K(k)}{\Omega}$ . Therefore

110
Eq.(4.56) is guaranteed as the$C^{1}\omega$-periodic solution if $\omega=8(1+l)\frac{K(k)}{\Omega}$ and $u(O)=0$ . From

Eq.(4.56) the condition $u(O)=0$ indicates cn$( \Omega t_{0}, k)=-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$Moreover, we will check the another initial condition:$\dot{u}(0)=\sqrt{c}$.

, that is,.

$- \frac{(m-n)|B|\Omega sn(\Omega t_{0},k)dn(\Omega t_{0},k)}{(1+|B|cn(\Omega t_{0},k))^{2}}=\sqrt{c}$This leads to the condition sn $(\Omega t_{0}, k)<0$ must hold. From the relations sn $(\Omega t_{0}, k)<0$ andcn$( \Omega t_{0}, k)=-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}$, we can select such$\tau_{+}=\Omega t_{0}$as $2K(k)<\tau_{+}<4K(k)$ . Because

there exists a root of $H(T)= \frac{m-n}{1+|B|cn(T,k)}+n+\alpha_{4}=0$ in $(0,2K(k))$ and $(2K(k),4K(k))$ ,has a period $4K(k)$ and $H(O)=\alpha_{4}<0,$ $H(2K(k))=\alpha_{3}>0$ , from
Lemma 4.5). Here let cn$0<T<2K(k)$ , we find easily $\tau_{+}=4K(k)-T$ .
respectively.

$(\cdot.\cdot H(T)$$(T, k)=- \frac{m+\alpha_{4}}{|B|(n+\alpha_{4})},$

Substituting$t_{0}= \frac{\tau_{+}}{\Omega}=\frac{4K(k)-T}{\Omega}$

Theorem 4.7. The$0,1,2\ldots)$ in a period.

$C^{1}$

to Eq.(4.57) yields Eq.(4.55).

$\blacksquare$

w-periodic solutions in Theorem 4.6 create $2(1+l)$ -oscillations(lProof. The proof can be performed by the same manner in Theorem 4.3.

$=$$\blacksquare$

We can analize the case of $(u(O),\dot{u}(0))=(0, -\sqrt{c})$ similarly. We summarize the result of thissection including the case of $(u(O),\dot{u}(0))=(0, -\sqrt{c})$ in Tables 1 and 2. That is, the nonlinear
differential equation:$\ddot{u}+\psi u+2qu^{3}=\{\begin{array}{ll}\frac{e}{2}, \nu\omega\leq t<(\nu+\frac{1}{2})\omega, \nu=0,1,2, \ldots-\frac{e}{2}, (\nu+\frac{1}{2})\omega\leq t<(\nu+1)\omega\end{array}$has the$C^{1}\omega$-periodic solutions shown in Tables 1 and 2.

Finally, we give an example of Theorems.

Example 4.1. We consider the following example:$\ddot{u}+3u+2u^{3}=\{\begin{array}{l}13, t\in(\nu\omega, (\nu+\frac{1}{2})\omega),-13, t\in((\nu+\frac{1}{2})\omega, (\nu+1)\omega).\end{array}$

We obtain the$(0, -\sqrt{30})$

as$C^{1}\omega$(4.60)-periodic solution for the “odd “-time oscillations with the initial condition$h_{-}^{o}(t)= \frac{30}{4+cn(\sqrt{15}(t-\frac{\omega}{4}),\frac{1}{\sqrt{5}})}-7$,

and with the initial condition $(0, \sqrt{30})$ as$h_{+}^{o}(t)= \frac{30}{4-cn(\sqrt{15}(t-\frac{\omega}{4}),\frac{1}{\sqrt{5}})}-7$

.

111

Table 1:The

$0,1,2,$$\nu=0,1,2,$$\ldots,$$C^{1}$

Table 2:

The

$0,1,2,$$\nu=0,1,2,$$\ldots,$$C^{1}$$\omega$-periodic solutions for “odd”-time oscillations.

$\ldots$$\omega$and $T=$ cn$-1(- \frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}),$

and$T= cn^{-1}(-\frac{m+\alpha_{4}}{|B|(n+\alpha_{4})}),$

$l$

$=$

In this table,

$l$

$=$

$0<T<2K(k)$ .

-periodic solutions for “even”-time oscillations.

$\ldots$

In this table,$0<T<2K(k)$ .

112
Also we have the$(0, -\sqrt{30})$ as$C^{1}\omega$-periodic solution for the “even”-time oscillations with the initial condition,$h_{-}^{e}(t)= \frac{30}{4+cn(\sqrt{15}t-T,\frac{1}{\sqrt{5}})}-7$

where $T=$. 1.337 and with the initial condition$(0, \sqrt{30})$

as.

$h_{+}^{e}(t)= \frac{30}{4+cn(\sqrt{15}t+T,\frac{1}{\sqrt{5}})}-7$

We find$\omega$concretely as follows: For the “odd “-time oscillations with the initial condition$(0, -\sqrt{30})$, 4.808, 8.236, . . .
and for the “odd “-time oscillations with the initial condition $(0, \sqrt{30})$, 5.475, 8.903, . . .
moreover the “even”-time oscillations with the initial condition $(0, \pm\sqrt{30})$$\omega\fallingdotseq 1.380$$\omega\fallingdotseq 2.047$, 6.856, 10.284, . . .
The numerical computations are shown in Figures 4.1 to 4.8. Figures 4.1 to 4.3 show “odd “time oscillations orbits with the initial condition: $u^{*}(O)=0,\dot{u}^{*}(0)=-\sqrt{30}$. Figures 4.4 and
4.5 also show “odd “-time oscillations orbits with the initial condition: $u^{*}(O)=0,\dot{u}^{*}(0)=\sqrt{30}$ .
Moreover Figures 4.6 to 4.8 show “even”-time oscillations with the initial condition: $u^{*}(O)=$$0,\dot{u}^{*}(0)=\sqrt{30}$or $\dot{u}^{*}(0)=-\sqrt{30}$ .
$\omega\fallingdotseq 3.428$

Figure 4.1: (left):The “single-time oscillation” orbit with the initial condition: $u(O)=0,\dot{u}(0)=$. (right): The time history of the “single-time oscillation”
in Example 4.1.
orbit. The time is shown up to two periods.
$-\sqrt{30}$$\omega\fallingdotseq 1.380$

Figure 4.2: (left):The “triple-time oscillations” orbit with the initial condition: $u(O)=0,\dot{u}(0)=$. (right): The time history of the “triple-time oscillations”
in Example 4.1.
orbit. The time is shown up to two periods.
$-\sqrt{30}$$\omega\fallingdotseq 4.808$

113

Figure 4.3: (left):The “quintic-time oscillations” orbit with the initial condition: $u(O)=0,\dot{u}(0)=$in Example 4.1.
. The orbit is the same of “quintic-time oscillations” one.
(right): The time history of the “quintic-time oscillations” orbit. The time is shown up to two$-\sqrt{30}$$\omega\fallingdotseq 8.236$

periods.

Figure 4.4: (left):The “single-time oscillation” orbit with the initial condition: $u(O)=0,\dot{u}(0)=$in Example 4.1.
. 2.047. (right): The time history of the ”single-time oscillation” orbit.
The time is shown up to two periods.
$\sqrt{30}$$\omega=$

Figure 4.5: (left):The time history of the “triple-time oscillations” orbit with the initial condition:$u(O)=0,\dot{u}(0)=\sqrt{30}$ . The time is shown up to two periods.
. 5.4755. (right): The time
history of the “quintic-time oscillations” orbit with the initial condition: $u(O)=0,\dot{u}(0)=\sqrt{30}$.
The time is also shown up to two periods.
. 8.9036. The orbits of both figures are the same$/or4.3$of Figure 4.2 and.
$\omega=$$\omega=$

Figure 4.6: (left):The time history of the “double-time oscillations” orbit with the initial condition: $u(O)=0,\dot{u}(0)=-\sqrt{30}$. The time is shown up to two periods.
. 3.428. (right): The time
history of the ”double-time oscillations” orbit with the initial condition: $u(O)=0,\dot{u}(0)=\sqrt{30}$ .
The time is also shown up to two periods. The orbits of both figures are the same of Figure 4.2
and $/or4.3$ .
$\omega=$

114

Figure 4.7: (left):The time history of the “quadruple-time oscillations” orbit with the initial. (right):
condition: $u(O)=0,\dot{u}(0)=-\sqrt{30}$ . The time is shown up to two periods.
The time history of the “quadruple-time oscillations” orbit with the initial condition: $u(O)=$$0,\dot{u}(0)=\sqrt{30}$. The time is also shown up to two periods. The orbits of both figures are thesame of Figure 4.2 and/or 4.3.
$\omega\fallingdotseq 6.856$

Figure 4.8: (left):The time history of the ”sextic-time oscillations” orbit with the initial con10.284.
dition: $u(O)=0,\dot{u}(0)=-\sqrt{30}$ . The time is shown up to two periods.
(right): The time history of the “sextic-time oscillations” orbit with the initial condition:$u(O)=0,\dot{u}(0)=\sqrt{30}$ . The time is also shown up to two periods. The orbits of both figures are
the same of Figure 4.2 and $/or4.3$ .
$\omega\fallingdotseq$

References[1] Alligood, K.T., Sauer, T.D. and Yorke, J.A., Chaos. An Introduction to Dynamical Systems,
Springer-Verlag, 1997.
[2] Ando, S., Introduction:Elliptic Integral and Elliptic Functions, Nisshinshuppan, 1970.(inJapanese)[3] Chouikha,A.R., ‘Periodic perturbation of non-conservative second order differential equations’, Electron.J.Qual.Theory.Differ.Equ., 49, 122/136, 2002.
[4] Duffing, G., Erzwungene Schwingungen bei Ver\"anderlicher Eigenfrequenz, F. Viewegu. Sohn:

Braunschweig, 1918.
[5] Farkas,M., Periodic motions, Springer-Verlag, 1994.
[6] Hsu, S.B., ‘Differential Equations with Applications’, World Scientific Publishing Co., 2006.
[7] Keener, J. and Sneyd, J., ‘Mathematical Physiology’, Springer-Verlag, 1998.

[8] Levinson,N. and Smith,O.K., ‘General equation for relaxation oscillations’, Duke Math, J.9.
382/403, 1942.
[9] Li\’enard,A., ‘\’Etude des Oscillations Entretenues’, Rev.G\’en.Electricit\’e 23, 901/912, 1928.

[10] Mathews, T. and W. Gardner, Field reversals of paleo magnetic type in coupled disk dynamos, U.S. Naval Res.Lab.Rep.5886, 1963.

[11] Nohara, B.T. and Arimoto, A., ‘Solutions of the Duffing Equation with a higher ordernonlinear term’, Theoretical and Applied Mechanics Japan, 59, 133/141, 2011. Proc. KoreaJapan Joint Workshop on Dynamical Systems and Related Topics, 25/33, 2010.

115[12] Nohara, B.T. and Arimoto, A., ‘Exact Solutions of Generalized Duffing Equations‘, Bulletinof Tokyo City University, 3, 47-62, 2010. (in Japanese)[13] Rayleigh, L., Theory of Sound, Vol.1, London, 1894.
[14] Rikitake, T., Oscillations of a system of disk dynamos, Proc.Cambr.Phil.Soc.,54, 89/105,

1958.
[15] 清水辰次郎，非線型振動論，培風館，昭和 4O 年．
[16] Taam, C.T., ‘The solution of Nonlinear Differential Equation III’, Duke Mathematical Journal, 24, 511-519, 1957.

[17] van der Pol, B., ‘On relaxation oscillations’, Philos. Mag., 2, pp.978-992, 1926.
[18] Wiggins, S., ‘Introduction to Applied Nonlinear Dynamical Systems and Chaos’, 2nd ed.,Springer, 2003.
[19] Yamaguti,M., ‘Some properties of non-linear differential equations of the parametric excitation’, Memoirs of the College of Science, University of Kyoto, Series A, 28,2,87/961953.

[20] Zhang,L.H. and Wang,Y., ‘A Note on Periodic Solutions of a Forced Li\’enard-Type Equation’, The ANZIAM Journal, 51, 350/368, 2010.