数理解析研究所講究録 1264 巻 2002 年 159-172

159
PERIODIC SOLUTIONS FOR
FORCED VAN DER POL TYPE EQUATIONSCHIKAHIRO EGAMIANDNORIMICHI HIRANO

ABSTRACT. In the present paper, we will see that aVan der Pol type equations has aperiodic
solution when the forcing term is periodic. By showing the results of some simulations, we
illustrate periodicity of the solutions. We also prove that aperiodic solution found near theorigin is arepellor.

Key Words
Van der Pol type equation, Forcing term, Periodic solution, Banach space, Topological(Leray-Schauder) degree, Repellor, Attractor, Subharmonic solution

1INTRODUCTION
Let , ,
$f$$g$

$e:\mathrm{R}$$arrow \mathbb{R}$be continuous functions. The Lienard type equation with forcing term(L)$u_{tt}+f(u)u_{t}+g(u)=e(t)$$t\in \mathrm{R}$has been studied by many authors due to its adoption to awide variety of mechanical,
electrical, biological and economical systems. The equation (L) is usually called Duffing
type when $f(u)=\langle$ , or Van der Pol type when $g(u)=\eta u$ , where and are constants.
In economics, there have been anumber of elegant mathematical treatments of someof the traditional business cycle theories e.g. the treatment of Kaldor’s model by Changand Smyth (1971) and Schinasi (1982) and of acomplete Keynesian system by Torre (1977).
Because of one dimensional relaxation oscillatorjust like Van der Pol type without an forcing
term, their treatments concentrated on the question of the existence of limit cycles andconsequently made use of the planar properties such as Poincare-Bendixson theory andJordan curve (cf. C. Chiarella [14, \S 2, \S 3, \S 7], K. Kawamata [15, pp.131-148]).
Recently, $N$-dimensional extension of the equation (L) has been studied by several authat However it is not easy for $N\geq 2$ to obtain similar results as one dimensional cases.
$N$ -dimensional existence results for periodic solutions of the forced Van der Pol type are notyet established until now in comparison with those of the forced Duffing type (e.g., see [1]by J. Mawhin).
a continuous
Throughout this paper, unless otherwise explicitly stated, $F$,
functions, andis aperiodic function with period $T$ . We consider the existence
problem for periodic solutions of the forced $N$-dimensional Van der Pol type equation of the$\langle$$\eta$$g:\mathrm{R}^{N}arrow \mathrm{R}^{N}$$\mathrm{e}$$e\in L^{2}(\mathbb{R};\mathbb{R}^{N})$

form
(V)

$u_{tt}+ \frac{d}{dt}F(u)+g(u)=e(t)$where $u(t)=(u_{1}(t),u_{2}(t)$ ,$i\in[1, N]$.

$\ldots$,$\mathrm{u}(\mathrm{t})$, and$u:(t)\in \mathbb{R}$$t\in \mathrm{R}$,for each$t\in \mathrm{R}$and each integer

In the following section, we shall prove our main result for (V). We make use of theLeray-Schauder degree theory (cf. N. G. Lloyd [12, \S 4], K. Masuda [21, \S 23, \S 24]). In section

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3, we give the results of the simulations for some concrete models. In section 4, we provethat the periodic solution found near the origin by simulations in section 3is repellor.

2EXISTANCE OF PERIODIC SOLUTIONSIn this section, we establish an existence result for periodic solutions of (V). To state ourresult, we need some preliminaries.
, let us define that
For $x,y$$\in L^{2}([0,\eta j\mathrm{R}^{N})$$|x|$$=( \sum_{=1}^{N}x_{}^{2)^{1/2}},$$||x||=( \int_{0}^{T}|x(t)|^{2}dt)^{1/2}$,$(x,y)$ $= \int_{0}^{T}.\sum_{\Leftarrow 0}^{N}x:(t)y_{\dot{*}}(t)dt$.

In the folowing, we put$H=\{u\in L^{2}([0,T]j\mathrm{R}^{N}):"(0)=u(T),u_{t}\in L^{2}([0,T];\mathrm{R}^{N})\}$,

with the norm$||u||_{H}=(||u||^{2}+11\mathrm{u}11^{2})^{1/2}$

We also put$\tilde{H}=\{u\in H:\int_{0}^{T}u(t)dt$ $=0\}$

Further,$\dot{g}\mathrm{v}\mathrm{e}\mathrm{n}$aset$\Omega$, its closure is written$\overline{\Omega}$, its boundary.
$\partial\Omega$.

Our main result is the following,
Theorem 2.1 Let e $\in H$ . If $F(u)$ and $g(u)$ satisfy the follow .ng conditions (Fl), (F2) and(Gl), then theequation (V) has at least one periodic solution with period T.
$d|.ffe|\mathrm{e}nt|.al$

(F1)

where

$F_{}\in C^{1}(\mathrm{R};\mathrm{R})$

(F2)

where
(G1)

,

$F(u_{1},u_{2}, \ldots,u_{N})=(\begin{array}{l}F_{1}(u_{1})F_{2}(u_{2})\vdots F_{N}(u_{N})\end{array})$

for each integer: $\in[1, N]j$and$f_{}(0)<0$$f_{}$

denotes$\lim_{[\cdot 1arrow}\inf_{\infty}\frac{f_{}(s)}{s^{2}}>0$,$F_{j}’$$g(u)=Au$

and$\det A\neq 0$,$whm$ $A$ is a $N\mathrm{x}N$ constant matrix.

To prove Theorem 2.1 we need some lemmata. However, we omit those proofs here.
Lemma 2.2 Suppose that (Fl), (F2) and (Gl). Let e $\in H$ . If $u(t)$ is a T periodic solution
andof the differential equation (V), then u$\in\tilde{H}$$||u||\leq 2T||u_{t}||$.

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Remark. For the inequality in Lemma 2.2, which is called Poincare’s inequality, we can havestill stricter evaluation (cf. C. P. Gupta, J. J. Nieto and L. Sanchez [3]). However, thepresent evaluation is enough for lemmata in this paper.
Lemma 2.3 Let$\lambda_{0}\in(0, \min(_{\mathrm{T}}^{2\pi}, 1))$. Then the set

for some A$S_{1}= \{u\in H:u_{tt}+\frac{d}{dt}F(u)+\lambda Au=e(t)$$\in[\lambda_{0},1]\}$is bounded in $H$ .
Lemma 2.4 The set$S_{2}=\{u\in H$ : $u_{tt}+ \delta\frac{d}{dt}F(u)+\mathrm{X}$for some

Au $=6e(t)$$\delta$$\in[0,1]\}$is bounded in $H$ .
Here, let us introduce the topological (Leray-Schauder) degree and its properties.
Definition 2.5 LetX be Banach space, $D$ be a bounded open subset $ofX$ and be a compact, then we define the Leray-Schaudermapping from
into X. If
for any.
degree of at $p\in X$ relative to $D$ to be$\mathcal{L}$$\overline{D}$$x\in\partial D$$\mathcal{L}x\neq p$$\deg(\mathcal{L}, D,p)$$\mathcal{L}$The Leray-Schauder degree is known to have the following three properties.
(i) $\deg(I, D,p)=1$ for every $p\in D$ , where I denotes identity mapping,(ii) $\deg(\mathcal{L}, D,p)\neq 0$ implies(iii) If$\mathcal{H}(\xi)$$\mathcal{L}x=p$for some $x\in D$ .

is ahomotopy of compact mapping with, thenis independent of .

$\mathcal{H}(\xi)x\neq p$$\deg(\dot{\mathcal{H}}(\xi), D,p)$$\xi\in[0,1]$

for any$x\in\partial D$

and any$\langle$

Proof of Theorem 2.1 Let $B_{r}(0)$ be the open ball in
For each $\lambda\in[0,1]$ and $\delta\in[0,1]$ , we define aoperator$\tilde{H}$centered at 0with radius
by
:

$r$

$>0$

.

$\tilde{H}arrow\tilde{H}$$\mathcal{T}(\lambda, \delta)$,$\mathcal{T}(\lambda,\delta)u=-\delta\int_{0}^{t}F(u(s))ds-\lambda\int_{0}^{t}\int_{0}.Au(\tau)d\tau ds+\delta\int_{0}^{t}\int_{0}.e(\tau)d\tau ds+C$

where$C= \delta\int_{0}^{T}F(u(s))ds+\lambda\int_{0}^{T}\int_{0}^{\epsilon}$Au(r)$d \tau ds-\delta\int_{0}^{T}\int_{0}^{e}e(\tau)d\tau ds$.

is equicontinuous for any bounded subset
is aintegral operator,
Becauseis a
is relatively compact by Ascoli-Arzela theorem. Henceof . Thencompact operator. Furthermore, we put$\mathcal{T}(\lambda, \delta)U$$\mathcal{T}(\lambda, \delta)$$U$$\tilde{H}$$\mathcal{T}(\lambda, \delta)$$\mathrm{T}(\mathrm{A}, \delta)U$$\mathcal{T}(1-3(1-\lambda_{0})\xi, 1)u$$\mathcal{H}(\xi)u=\{$ $\mathcal{T}(\lambda_{0},2-3\xi)u$$\mathrm{T}(3\mathrm{A}\mathrm{O}(1-\xi), \mathrm{O})u$

for
for
for

is ahomotopy of compact mapping on
if and only ifthat is afixed points of$?t(\xi)$thatdefinition ofThen$\mathcal{H}(\xi)$$\mathcal{H}(0)$$S_{1}=\{u\in\tilde{H}$: $u=\mathcal{H}(\xi)u$$\xi\in[\frac{1}{3}, \frac{2}{3}]$$\xi\in[\frac{2}{3},1]$$\tilde{H}$$\mathcal{H}(\xi)$$u$$\xi\in[0, \frac{1}{3}]$

$u$

and
and
and

$u\in\tilde{H}$,,$u\in\tilde{H}$$u\in\tilde{H}$.

. It is obvious from the definition ofis asolution of (V). We have by thefor some$\xi\in[0, \frac{1}{3}]\}$

162$S_{2}=\{u\in\tilde{H}$:for some$u=\mathcal{H}(\xi)u$$\xi\in[\frac{1}{3}, \frac{2}{3}]\}$

Then by Lemma 2.3 and Lemma 2.4, we have that there exists large$u\neq \mathcal{H}(\xi)u$For$\langle$$\in[\frac{2}{3},1]$

for any$u\in\partial B_{M_{0}}(0)$, we can see that each fixed point(2.1)$u\in\tilde{H}$$u=-(3 \lambda_{0}(1-())\int_{0}^{t}\int_{0}.$and anyof$\mathrm{v}(\mathrm{t})$such that$M_{0}$$\in[0, \frac{2}{3}]$.

satisfies$\mathcal{H}(\xi)$$d\tau ds$

$\xi$

.

$+C$

.

Then we immediately have that $u=0$ is the unique solution of (2.1), we also have that$u\neq \mathcal{H}(\xi)u$

for any$u\in\partial B_{M_{0}}(0)$and any$\xi\in[\frac{2}{3},1]$.

While, we should just calculate $\deg(I-\mathcal{H}$(0) $, B_{M_{0}}(0),0)$ to consider the fixed point
problem $u=\mathrm{H}(0)$ . Since
for any
and any $\xi\in[0,1]$ , then by theproperty (iii) of degree,$u\neq \mathcal{H}(\xi)u$$u\in\partial B_{M_{\mathrm{Q}}}(0)$$\deg(I-\mathcal{H}(0),B_{M_{0}}(0),\mathrm{O})=\deg(I-\mathcal{H}(1),B_{M_{0}}(0),\mathrm{O})=\deg(I,B_{M_{0}}(0),0)$.

Using the property (i) of degree, $\deg(I,Bu_{0}(0),0)=1$ . Therefore by the property (\"u) ofdegree, we obtain that $H(0)$ has at least one fixed point in , and Theorem 2.1 is proved.
$\tilde{H}$$\mathrm{O}$

3VAN DER POL OSCILLATOR
In section 3, We showed existence of periodic solution of (V). Then we would like to find out
where aperiodic solution with period exits. In this section, we introduce three concrete
examples and illustrate the results of simulations for each model.
$T$

3.1

PRELIMINARY

We define three kinds of periodic solutions with mutually different character.
Definition 3.1 Let u $\in L^{2}([0,T]jR^{N})$ be a periodic solution with period T of (V). If there
exists a neighborhood U of $\Gamma=\{(u(t),u_{t}(t));$ t $\in[0,T]\}$ such that$\lim_{tarrow+\infty}d((v(t),v_{t}(t))$, $\Gamma)=0$

for each$(v_{0},v_{t0})\in U$,

where $v(t)$ is a solution of (V) with initial value, denotes usual Euclid distance, then is said to be attractive or attractor.

$(\mathrm{v}(0),\mathrm{v}\mathrm{t}(\mathrm{O}))=(\mathrm{v}0\mathrm{i}\mathrm{v}\mathrm{t}0)\in R^{N}\mathrm{x}R^{N}$$d(\cdot$$\cdot$Definition 3.2 In contrast of Definition 31, if there$\lim_{tarrow-\infty}d((v(t),v_{t}(t))$

then

and

$u$

$)$

$u$

, $\Gamma)=0$$\dot{\varpi}stsU$

for eachof$\Gamma$such that$(v_{0},v_{t0})\in U$,

is said to be repellor.

Definition 3.3 If is a periodic solution with the integral multiple of , thenbe subharmonic solution
$u$

$T$

$u$

is said to

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3.2

NORMAL MODEL

We first introduce the most basic oscillating circuit with anegative resistor devised bystands for inductance, capacitance and resistance,andVan der Pol. On Fig.3.1, ,
$L$

$R$

$C$

$L$

Fig. 3.1: Van der Pol Oscillator
respectively. This $R$ , is called negative resistor, has the nonlinear property for current such
that
$i$

(3.1)

$R(i)=-\prime 0+r_{1}i$ % $r_{2}i^{2}$,

is nonnegative. Let be time. Then the circuit equation corresponding
where , andto Fig.3.1 is formalized by the following$r_{0}$$\tau$$r_{2}$$r_{1}$$Li_{\tau}+R(i)i+ \frac{1}{C}\int i(\tau)d\tau=0$.

Differentiating above equation by , we have$\tau$$i_{\tau\tau}- \frac{1}{L}(r_{0}-2\mathrm{r}\mathrm{x}\mathrm{i}-3r_{2}i^{2})i_{\tau}+\frac{1}{LC}i=0$

Here, transforming$\tau$

(3.2)

Put

into

$\sqrt{LC}t$and putting$i=\sqrt{\overline{3}r_{2}\prime[perp]}x$.

, we have$x_{tt}-r_{0} \sqrt{\frac{C}{L}}(1-\frac{2r_{1}}{\sqrt{3r_{0}r_{2}}}x-x^{2})xt+x=0$$\epsilon=\prime_{0}\sqrt{\frac{c}{L}}$.

now. If we suppose $r_{1}=0$ , then we obtain the Van der Pol type equation(M1)$x_{tt}+\epsilon(x^{2}-1)\mathrm{x}\mathrm{t}+x=0$,.
represents the nonlinearity of the system (M1) and (3.1) holds thewhereessence of Self-induced oscillation. In case that is equal to zero, (M1) is actually just a
linear oscillator. It is known as arelaxation oscillation that the solutions of the autonomous
plane by Poincare-Bendixson
system (M1) have the unique limit cycle for each on-limitset of the orbits. Where
the
Theorem (cf. F. Verhulst [10, \S 4.3]). Fig.3.2 gives
axis indicates $x(t)$verticalthechange from 0to 20 by 2step. The horizontal axis andcycle
becomes large, respectively. We can see in Fig.3.2 the amplitude of limit
andand $=15$ is about
as grows. For example, the period of limit cycle for
7.63, 13.79 and 26.80$x(t)\in \mathrm{R}$$\epsilon$$\epsilon$$\epsilon$$(x,x_{t})$$\omega$$\epsilon$$x_{t}(t)$$\epsilon$$\epsilon=2,\epsilon=6.5$$\epsilon$

164$\mathrm{x}’$$\mathrm{x}$Fig. 3.2: Limit Cycle

3.3

FORCED MODEL

We next introduce the Van der Pol oscillator with external power source. On Fig.3.3, $E$ isthe voltage of external power source. Using transformation similar to (3.2), we can formulate
the circuit equation corresponding to Fig.3.3 as follows(3.3)$x_{tt}-r_{0} \sqrt{\frac{C}{L}}(1-\frac{2r_{1}}{\sqrt{3r_{0}r_{2}}}x-x^{2})x_{t}+x=\sqrt{\frac{3r_{2}}{r_{0}}}\sqrt{\frac{C}{L}}E\mathrm{c}\mathrm{o}\mathrm{e}t$

We putand. We call forcing coefficient. If we assume $r_{1}=0$,
then we have the forced Van der Pol type equation$\epsilon=r_{0}\sqrt{\tau c}$

(M2)

$B$

$B=\sqrt{\underline{3}\mathrm{a}\mathrm{o}},\sqrt{\tau c}E$$x_{tt}+\epsilon(x^{2}-1)x_{\mathrm{C}}+x=B$

coe .
$t$

We show three results of simulations of the model (M2). Then we have to set a
initial value on $[0, 2\pi]$. Let us set four initial values $(t_{0},x(t_{0}),x_{t}(t_{0}))$ at(0,0,0), (0,0,3), $(0,$ $-1, 1)$ and (0, 1,5). Let be fixed at 10.0 and $B$ be set at 2.0, 6.5$\mathrm{x}\mathrm{R}^{N}\mathrm{x}\mathrm{R}^{N}$$\epsilon$

Fig. 3.3: Van der Pol Oscillator with External power source

165
and 15.0. It is rather more important for us than the relation between initial values andorbits whether aperiodic solution exists or where aperiodic solution was observed or its
period. Fig.3.4, Fig.3.5 and Fig.3.6 are 3-dimensional plots and projections onto $t=0$ oftime series of and of (M2) in case that the amplitude of external force $B$ is weak 2.0,
strong 15.0 and middle 6.5, respectively. In Fig.3.4, we can find a -periodic repellor in
the region of $t<0$ which oscillate near the origin with small amplitude, and a
periodicsubharmonic solution in the region of $t>0$ in which all four orbits reach. In Fig.3.5, we canobserve a -periodic attractor in the region of $t>0$ in which all four orbits was attracted
as time progresses. In the region of $t<0$ , we can hardly calculate because the orbits diverge
instantly to infinity for all four initial values. Fig.3.6 gives acomplicated state that we havearepellor in the region of $t<0$ and both an attractor and asubharmonic solution in theregion of $t>0$ .
$x$$x_{t}$$2\pi$$6\pi$$2\pi$$\mathrm{x}$Fig. 3.5: Case of $B=2$$\mathrm{x}$Fig. 3.5: Case of $B=15$$\mathrm{x}$Fig. 3.6: Case of $B=6.5$

166
Fig.3.7 and Fig.3.8 are drawn in order to give clearly the orbits of three periodic solutions.
for large sufficiently, and Fig.3.8
Fig.3.7 is the 3-dimensional plot of$t=0$
is its projection onto
. In Fig.3.8, we can see that the oscillation with the smallest
amplitude is repellor, the orbit is attractor which oscillate by the inner side of the largest
swing, and the orbit is subharmonic solution which also has the small amplitude crossing
axis while oscillating with the largest amplitude.
$|t|$$(t\mathrm{m}\mathrm{o}\mathrm{d} 2\mathrm{n}\mathrm{l}\mathrm{x}(\mathrm{t})\mathrm{J}\mathrm{x}\mathrm{t}(\mathrm{t}))$$x$$\mathrm{x}$Fig. 3.7: Invariant Set for $|t|>>1$

Fig. 3.8: Periodic Solutions

In Fig.3.9, Fig.3.10 and Fig.3.11,
are drawn for $\in[1W,20]$ , $t\in$$[100,200]$ and $t\in[-2W, -1\mathrm{M}]$ to investigate the exact periods of periodic solutions, where$T=2\pi,2\pi$ and, respectively. In order to get the exact period, we have to choose
may be overlapped at one.
appropriately so that the curves$t$$(t\mathrm{m}\mathrm{o}\mathrm{d} T,x(t))$$T$$6\pi$$(t \mathrm{m}\mathrm{o}\mathrm{d} T,x(t))$$\mathrm{t}$$\mathrm{t}$Fig. 3.9: Repellor

Fig. 3.10: Attractor$\mathrm{x}$$\mathrm{t}$Fig. 3.11: Subharmonic Solution

167
3.4

COUPLED MODEL WITH FORCING TERM

We introduce the coupling model connected two oscillating circuits with external powersource.

Fig. 3.12: Coupled Van der Pol Oscillator with External power source

On the Fig.3.12, and represents the small capacitance added artificially and theminute resistance included in the lead portion of the circuit, respectively. The characteristicof negative resistance is described by
$c$

$r$

for $n=1,2$ .

$R_{n}(i)=-\mathrm{r}\mathrm{n}\mathrm{O}+rnli+r_{n2}i^{2}$

Now assume
and put $i_{1}=\sqrt{3r_{12}r}" x$ and. By using sametransformation as (3.3), we have the following circuit equation corresponding to Fig.3.12$\sqrt{L_{1}C_{1}}=\sqrt{L_{2}C_{2}}$$i_{2}=\sqrt{\frac{r}{3}r_{22}2\mathrm{L}}y$$x_{tt}-r_{10} \sqrt{\frac{C_{1}}{L_{1}}}(1-\frac{2r_{11}}{\sqrt{3r_{10}r_{12}}}x-x^{2})x_{t}+x=\sqrt{\frac{3r_{12}}{r_{10}}}\sqrt{\frac{C_{1}}{L_{1}}}E_{1}\cos t$$+ \frac{C_{1}}{c}(y-x)+r\sqrt{\frac{C_{1}}{L_{1}}}(y_{\mathrm{C}}-x_{t})$$y_{tt}-r_{20} \sqrt{\frac{C_{2}}{L_{2}}}(1-\frac{2r_{21}}{\sqrt{3r_{20}r_{22}}}y-y^{2})y_{t}+y=\sqrt{\frac{3r_{22}}{r_{20}}}\sqrt{\frac{C_{2}}{L_{2}}}$a$\cos t$$+ \frac{C_{2}}{c}(x-y)+r\sqrt{\frac{C_{2}}{L_{2}}}(x_{\ell}-y_{t})$Here we put,
and, then we can build up our main model$k_{n}=\mathrm{C}\mathrm{n}/\mathrm{c}$$\epsilon_{n}=r_{n0}\sqrt{\frac{c}{L_{\mathrm{n}}}}$$B_{n}=\sqrt{\frac{3}{r}\underline{r_{\mathrm{B}0}}l}\sqrt{\frac{c}{L_{n}}}E_{n}$,.

. If we suppose $r_{n1}=0$ ,$r\langle(1$(M3)$x_{tt}-\epsilon_{1}(1-x^{2})x_{t}+x=B_{1}\cos t+k_{1}(y-x)$,$y_{tt}-\epsilon_{2}(1-y^{2})y_{t}+y=B_{2}\cos t+k_{2}(x-y)$,

where we call
coupling coefficient.
We show two results of simulations of the model (M3). Then we have to set ainitialvalue on $[0, 2\pi]$. Let , and , be fixed at 2.0, 10.0 and 0.5, 0.3,
respectively. We suppose $B_{1}=B_{2}$ .
$k_{n}$$\mathrm{x}\mathrm{R}^{N}\mathrm{x}\mathrm{R}^{N}\mathrm{x}\mathrm{R}^{N}\mathrm{x}\mathrm{R}^{N}$$\epsilon_{1}$$\epsilon_{2}$$k_{1}$$k_{2}$

168
We first show case of weak external force $B_{1}=B_{2}=1.5$ . Let us set four initial valat (0, 0, 0, 0, 0), $(0, 0,$ $-1, 0,$ $-1)$ , $(0,$ $-2,$ $-5,$ $-1,$ $-10)$ and$(0,$ -2,
-2.5, -10 .
Fig.3.13 and Fig.3.14 are time series plotting $(t,x(t),x_{t}(t))$ and $(t,y(t),y_{t}(t))$ , respectively. In both Fig.3.13 and Fig.3.14, we can observe a -periodic repellor in the regionof $t<0$ and a -periodic subharmonic solution in the region of $t>0$ . Then $x(t)$ of thissubharmonic solution in Fig.3.13 appears like -periodic. But watching $y(t)$ in Fig.3.14,
we have its period is .
Furthermore, we expect that, even if amodel was multidimensionalize, the amplitude ofeach variable depend on :of each dimension. In this case, the amplitude of $x(t)$ and $y(t)$depend on
and , respectively. However, if we suppose that coupling coefficients 4islarger, then we may have amore complicated situation of the orbits.

ues$(t_{0},x(t_{0}),x_{\ell}(t_{0}),y(t_{0}),y_{(}t_{0}))$
$5,$

$)$

$2\pi$$6\pi$$2\pi$$6\pi$$\mathcal{E}$$\epsilon_{1}$$\epsilon_{2}$$\mathrm{x}$Fig. 3.13: $(t,x(t),x_{l}(t))$ in case of $B_{1}=B_{2}=1.5$$\mathrm{y}$Fig. 3.14: $(t,y(t),y_{t}(t))$ in case of $B_{1}=B_{2}=1.5$
Fig.3.15, Fig.3.16 and Fig.3.17 Fig.3.18 give the period of repellor and subharmonic solution,
respectively. We can check that each period isand .
We next show case of strong external force $B_{1}=B_{2}=15$ . Let us set four initial values$(t_{0},x(t_{0}),x_{t}(t_{0}),y(t_{0})$ ,at (0, 0, 0, 0, 0), $(0,$ $-2,$ $-5,$ $-1,$ $-10)$ , $(0,$ -2, -2.5, -10 and$2\pi$$y_{t}(t_{0}))$$(0, 3,$ $-1, 3, 1)$$6\pi$

$5,$

.

$)$

In Fig.3.19 and Fig.3.20, we can observe a -periodic attractor in the region of $t>0$
like $B=15$ of case in the last section.In the region of $t<0$ , the orbits diverge instantly toinfinity for all four initial values.Fig.3.21 and Fig.3.22 give the period of attractor is .
In the end, by the results of some simulations, we can expect that forced Van der Pol
system has an attractor in case that the external force is strong or arepellor in case that theexternal force is weak. But it is not easy to investigate completely setting of parameters for$2\pi$$2\pi$

169$\mathrm{t}$$\mathrm{t}$Fig. 3.15: Repellor$(t\mathrm{m}\mathrm{o}\mathrm{d} 2\pi,x(t))$$\mathrm{x}$Fig. 3.16: Repellor$(t\mathrm{m}\mathrm{o}\mathrm{d} 2\pi, y(t))$$\mathrm{y}$tFig. 3.17: Subharmonic(t$\mathrm{m}\mathrm{o}\mathrm{d} 6\pi,x(t)$$\mathrm{t}$Fig. 3.18: Subharmonic(t$\mathrm{m}\mathrm{o}\mathrm{d} 6\pi,y(t)$$\mathrm{x}$Fig. 3.19: $(x(t),x_{t}(t))$ in case of $B_{1}=B_{2}=15$$\mathrm{y}$Fig. 3.20:$(y(t),y_{\mathrm{C}}(t))$

in case of $B_{1}=B_{2}=15$)170$\mathrm{y}$$\mathrm{t}$$\mathrm{t}$Fig. 3.21: Attractor$(t\mathrm{m}\mathrm{o}\mathrm{d} 2\pi,x(t))$

Fig. 3.22: Attractor$(t\mathrm{m}\mathrm{o}\mathrm{d} 2\mathrm{n}\mathrm{t}\mathrm{y}\{\mathrm{t}))$all cases that asubharmonic solution exists (e.g. regarding the results for one dimensional
system, see J. E. Flaherty and F. C. Hoppensteadt [2] .
$)$4REPELLOR
In this section , we prove that the periodic solution found near the origin by simulations in
section 3is repellor.
Proposition 4.1 Let be a small constant, $u\in L^{2}(R;R^{N})$ be a -periodic solution of (V)utsth $||u||\leq\rho$ and is a Hermite conjugate matrix. Then is a repellor.
$T$$\rho$$A$$u$

Proof. Using condition (F2), we have that, there exists $\rho:>0$ for $:\in[1,N]$ such that$f\dot{.}(u:)<0$

then put$\rho=\mathrm{n}\cdot \mathrm{n}:\epsilon[1,N1$$\rho:$

for$|u:|<\rho:$,. Putting$f(u)=(\begin{array}{llll}f_{1}(u_{1}) 0 00 f_{2}(u_{2}) 0\vdots \vdots \ddots \vdots 0 0 f_{N}(u_{N})\end{array})$,

we can rewrite (V) to(4.1)$u_{tt}+f(u)u_{t}+Au=e(t)$.

Let $u\in L^{2}(R_{j}R^{N})$ be a periodic solution of (4.1) with $||u||\leq\rho$ . Further letconstant and $u+h\phi$ be asolution (4.1). Then we have
$T$

(4.2)

$(u+h\phi)_{tt}+f(u+h\phi)(u+h\phi)_{t}+A(u+h\phi)$ $=e(t)$

$h$

be a

.

By (4.1) and (4.2), we have$\phi_{tt}+\frac{f(u+h\phi)-f(u)}{h\phi}u_{t}\phi+f(u+h\phi)\phi_{t}+A\phi=0$

By letting
(4.3)

$h$

.

g0 to 0for the equation above, we get the linearized equation$\phi_{tt}+f(u)\phi_{t}+(f’(u)u_{t}+A)\phi=0$,

where $f’(u)$ is Hessian of $f(u)$ . Here we rewrite the equation above as asystem of first order
ordinary equations of the form
(4.4)$(\begin{array}{l}\phi_{l}\chi_{t}\end{array})=(\begin{array}{ll}0 1-f’(u)u_{\mathrm{C}}-A -f(u)\end{array})(\begin{array}{l}\phi\chi\end{array})$

171
We consider the initial value problem of autonomous system (4.4) with $(\phi(0), \chi(0))=$. We put $\Phi(t)=(\phi(t), \chi(t))$ for $t\geq 0$ . Then by Floquet’s theoremof(e.g., see [9, \S 1.4] by F. C. Hoppensteadt), we have that the fundamental solution(4.4) can be written in the form$(00, \chi_{0})\in \mathrm{R}^{N}\mathrm{x}\mathrm{R}^{N}$$\Phi(t)$$\Phi(t)=Q(t)\exp(\Lambda t)$,) is a -periodic function, and A
where $Q(t)=\{q_{j}\}$ is amatrix such that each element
is aJordan matrix. To prove that is arepellor, it is sufficient to see that each eigenvalueof Ais positive. Let is asolution of (4.3). Then we have$T$$\mathrm{q}\mathrm{i}\mathrm{j}\{\mathrm{t}$$u$$\lambda_{:}$$\varphi$$\varphi_{tt}+f(u)\varphi_{t}+(f’(u)u_{t}+A)\varphi=0$.

Integrating the equation above over $[0, t]$ , we have$\varphi_{t}(t)+(f(u)\varphi)(t)+A\int_{0}^{t}\varphi(s)ds=\varphi_{t}(0)+(f(u)\varphi)(0)$Here we put $\sigma=\varphi_{t}(0)+(f(u)\varphi)(0)$ . We also put$\phi(t)=\int_{0}^{t}\varphi(s)ds$$\psi_{tt}+f(u)\psi_{t}+A\psi=\sigma$

Then multiplying the equality above by(4.5)$\psi_{t}$for t$\geq 0$. Then we have.

and integrating over $[0, t]$ , we get$| \psi_{t}(t)|^{2}+(\langle A\psi(t),\psi(t)\rangle\rangle\geq|\psi_{t}(0)|^{2}+(\langle A\psi(0),\psi(0))\rangle+\frac{1}{2}\int_{0}^{t}|\psi_{t}(s)|^{2}ds+\langle\langle\sigma,\psi(t)-\psi(0)\rangle)$.

. Suppose that there exists anegative
where we put (:for ,
appropriately, we haveof A. Then by choosing the initial value
eigenvaluethat for some $D>0$ ,$\langle x, y\rangle\rangle=\sum_{=1}^{N}.\cdot x:y$$x$$y\in \mathrm{R}^{N}$$(\varphi(0), \varphi_{t}(0))$$\lambda_{:}$$|\psi_{t}|\leq De^{\lambda:t},|\psi|\leq De^{\lambda:t}$for all t

This implies that
This contradicts to (4.5). This completes the proof.

$\lim_{tarrow\infty}|\psi_{t}(t)|^{2}+\langle\langle A\psi(t),\psi(t)\rangle\rangle=0$

and$\geq 0$.

$\lim_{tarrow\infty}\langle\langle\sigma,\psi(t)-\psi(0))\rangle=0$.

$\square$

REFERENCES[1] Jean Mawhin, An Extension of a Theorem of A.C. Lazer on Forced Nonlinear Oscillations, Journal of Mathematical Analysis and Applications, Vo1.40, pp.20-29, (1972).
[2] J.E.Flaherty, F.C.Hoppensteadt, Frequency Entrainment of a Forced van der Pol Oscillator, Studies in Applied Mathematics, Vo1.58, pP.5-15, (1978).
[3] Duane W.Storti, Per G.Reinhall, Stability of in-phave and out-Of-phase modes forpair of linearly coupled van der Pol oscillators, Nonlinear dynamics, PP.1-23, Series on
Stability, Vibration and Control Systems. Series B, Vo1.2, World Scientific Publishing,River Edge, NJ, (1997).

$a$[4] Nguyen Phuong C\’ac, Periodic Solutions of a Li\’enard equation with forcing term, Nonlinear Analysis, Vo1.43, pp.403415, (2001).
[5] Chaitan P.Gupta, Juan J.Nieto, Luis Sanchez, Periodic Solutions of Some Lienard andDuffing Equations, Journal of Mathematical Analysis and Applications, Vo1.140, PP.6782, (1989).

172[6] Norman Levinson, A Second Order Differential Equation with Singular Solutions, Annals of Mathematics, Vo1.50, No.1, pp.127-153, (1949).
[7] M.L.Cartwright, J.E.Littlewood, Some Fo.eed Point Theorems, Annals of Mathematics,
V01.54, NO.1, pp.1-37, (1951).

[8] John Guckenheimer, Philip Holmes, “Nonlinear Oscillations, Dynamical Systems, andBifurcations of Vector Fields”, Applied Mathematical Sciences, 42, Springer-Verlag,(1983).

[9] Frank C. Hoppensteadt, “Analysis and Simulation of Chaotic Systms” , Applied Mathematical Sciences, 94, Springer-Verlag, (2000).
[10] Ferdinand Verhulst “Nonlinear Differential Equations and Dynamical Systems”,
Springer-Verlag, (1990).
[11] Wolfgang Walter uOrdinary Differential Equations n, Springer-Verlag, (1998).
[12] N.G.Lloyd, “Degree Theory”, Cambridge University Press, (1978).

[13] G.Hardy, J.E.Littlewood, G.P61ya, “INEQUALITIES”, Cambridge University Press,(1934).

[14] Carl Chiarella, “The Elements of aNonlinear Theory of Economic Dynamics”, LectureNotae in Economics and Mathematical Systems, 343, Springer-Verlag, (1990).
[15] Toru Maruyama, Wataru Takahashi, uNoffiinear and Convex Analysis in Economic, LectureNotes in Economics and Mathematical Systems, 419, Springer-Verlag,(1995).
$\mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{y}"$[16] Gianni Ricci, Kumaraswamy Velupillai, “Growth Cycles and Multisectoral Economics:
the Goodwin Tradition”, LectureNotes in Economics and Mathematical Systems, 309,

Springer-Verlag, (1988).

[17] 宮寺功, u 関数解析” , 理工学社, (1972).
[18] 青木利夫, 高橋渉, 平野載倫, u 演習集合位相空間”, 培風館, (1985).
[東] 浅田統一郎, u 成長と循環のマクロ動学” , 日本経済評論社, (1990).

[20] 佐藤力, u 非線形振動論 , 基礎物理科学シリーズ, 2, 朝倉書店, (1970).
$n$[21] 増田久弥, u 非線形数学”, 新数学講座, 15, 朝倉書店, (1985).

CHIKAHIRO EGAMI\dagger AND NORIMICHI$\mathrm{H}\mathrm{I}\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{O}^{\mathit{1}}$Division of Information Media Environment Sciences,
Graduate School of Environment and Information Sciences,
Yokohama National University,
795 Tokiwadai, Hodogaya, Yokohama, Kanagawa, 2408501
E-mail address: \dagger chika@hiranolab .jks.ynu.ac.jp, hirano@math sci .ynu.ac.jp