M.I.Kopych*), R.Basiura**), A.K. Prykarpatsky***)

The symplectic study of motions in a perturbed van-der-pol dynamical system
Симплектичне вивчення поведiнки збуреної динамiчної системи ван-дер-поля
симплектическое изучение поведения возмущенной динамической системы ван-дер-поля

*),***) Dept.of Nonlinear Mathematical Analesis at the IAPPMM of the NAS, Lviv,Ukraina.
**) Dept. of Applied Mathematics at the AGH, Krakow, Poland.

      Abstract There is studied a weakly perturbed Van-der-Pol dynamical system and the structure of its trajectory behavior via the modern symplectic theory. Based on a Samoilenko-Prykarpatsky method of treating integral submanifolds of weakly perturbed completely integrable Hamiltonian systems one proves the regularity of deformations of the Lagrangian asymptotic submanifolds in a vicinity of the hyperbolic periodic orbit.

     Дослiджується слабко збурена динамiчна система Ван-Дер-Поля та структура поведiнки її траєкторiй на основi сучасної симплектичної теорiї. Використовується метод Самойленка-Прикарпатського дослiдження iнтегральних пiдмноговидiв слабко збурених повнiстю iнтегровних Гамiльтонових систем, встановлюється регулярнiсть деформацiй Лагранжевих асимптотичних пiдмноговидiв в околi гiперболiчної перiодичної орбiти.

     Исследуется слабо возмущенная динамическая система Ван-Дер-Поля и структура поведения ее траекторий на основе современной симплектической теории. Используется метод Самойленко-Прикарпатского исследования интегральных подмногообразий слабо возмущенных полностью интерированых Гамильтоновых систем, устанавливается регулярность деформаций Лагранжевых ассимптотических подмногообразий в окресности гиперболической периодической орбиты.


    1. Introduction We shall deal with the following Van-der-Pol dynamical system

\begin{displaymath} dx/dt=y,\ \ \ dy/dt=-\omega_0^2x-\alpha y(x^2-\beta^2)+ \varepsilon y \cos\omega t, \eqno(1.1) \end{displaymath}

where $(x,y)\in T^*(\bf R)$, $\omega_0,\omega\in {\bf R}_+$ -- the frequencies, $\alpha$, $\beta$, $\varepsilon\in {\bf R}_+$ -- some given constans and $t\in {\bf R}$ -- the evolution parameter. There is assumed too that parameters $\alpha,\varepsilon\in{\bf R}_+$ are small enough. It is evident that at $\alpha,\varepsilon=0$ dynamical system (1.1) doesn't possess hyperbolic peculiar points or periodic curves. This fact prompts us to devise a generalization of the theory developed in [1,2,3] for treating weakly perturbed Hamiltonian systems and their irregular motions caused by possible transversal separatrix splitting via the Birkhoff-Smale scenario. Herewith we suggest a new approach based on Samoilenko-Prykarpatsky imbedding submanifolds theory [6], to studying such a class of problems having no hyperbolic invariant submanifolds at $\alpha,\varepsilon=0$. In the case of the dynamical system (1.1) we prove at $\varepsilon=0$, $\alpha\in{\bf R}_+$ small enough the existence of the stable hyperbolic periodic invariant curve whose stable and unstable Lagrangean submanifolds are regular at $\varepsilon\in {\bf R}_+$ small enough in a vicinity of this attractive invariant curve.

      2. Integral submanifold analysis It is well known due to [5] that dynamical system (1.1) can be rewritten down as a Hamiltonian system on the extended phase space $T^*({\bf R}^2)$

\begin{displaymath} dq_j/dt=\partial H_\alpha^{(\varepsilon)}/\partial p_j,\ \ ... ...=-\partial H_\alpha^{(\varepsilon)}/\partial q_j, \eqno(2.1) \end{displaymath}


where $(q,p)\in T^*({\bf R}^2)$ and

\begin{displaymath} H_\alpha^{(\varepsilon)}=(q_2p_1-\omega_0^2q_1p_2)-\alpha(q_1^2-\beta^2)p_2q_2+ \varepsilon q_2p_2\cos\omega t \eqno(2.2) \end{displaymath}


is the corresponding Hamiltonian function. It is easy to verify that on $T^*({\bf R}^2)$ for $t\in {\bf R}$ the equations

\begin{displaymath} dq_1/dt=q_2,\ \ \ dq_2/dt=-\omega_0^2q_1+\alpha q_2(\beta^2-q_1^2)+ \varepsilon q_2\cos\omega t, \end{displaymath}


\begin{displaymath} dp_1/dt=\omega_0^2p_2+2\alpha q_1q_2p_2,\ \ \ dp_2/dt=-p_1... ...ha(\beta^2-q_1^2)p_2-\varepsilon p_2\cos\omega t, \eqno(2.3) \end{displaymath}


hold that is the dynamical system (1.1) is enclosed in (2.3) as an invariant part. The corresponding to (2.1) canonical symplectic structure $\Omega^{(2)}\in \Lambda^2(T^*({\bf R}^2))$ is given as follows: $\displaystyle{ \Omega^{(2)}=\sum\limits_{j=1}^2dp_j\wedge dq_j.}$ Now we first shall study the dynamical system (2.3) at $\varepsilon=0$. Especially we are interested in finding the conditions under which there exists a second first integral in involution with the Hamiltonian (2.2) at $\varepsilon=0$. Put now $H_\alpha:=H_\alpha^{(\varepsilon)}\Big\vert _{\varepsilon=0}$ and

\begin{displaymath} F_\alpha^{(s)}:=\frac{1}{2}(q_2^2+\omega_0^2q_1^2)+\frac{s}... ...mega_0^2p_2^2)-\alpha F_1^{(s)}(q_1;c_1,c_2,c_3), \eqno(2.4) \end{displaymath}


where, by definition $c_1^2=q_2^2+\omega_0^2q_1^2,\ \ \ c_2^2=p_1^2+\omega_0^2p_2^2$, $c_3=\hbox{arcsin}(\omega_0q_1/c_1)+\hbox{arcsin}(\omega_0p_2/c_2)$, and, due to the condition $\{H_\alpha,F_\alpha^{(s)}\}=O(\alpha^2)$, $\alpha\to 0^+$,

\begin{displaymath} q_2\partial F_1^{(s)}/\partial q_1=2sq_1q_2p_1p_2+(q_1^2-\beta^2) (s\omega_0^2p_2^2-q_2^2). \eqno(2.5) \end{displaymath}


The following lemma is true.
     Lemma 2.1. The function $F_\alpha^{(s)}\in D(T^*({\bf R}^2))$ is a bounded one-valued $O(\alpha^2)$ -- conservation law of the dynamical system (2.1) at $\varepsilon=0$ if and only if

\begin{displaymath} \frac{s}{4}\left(\frac{c_1c_2}{\omega_0}\right)^2\left(1-\f... ...{8\omega_0^2}+ \frac{1}{2}\beta^2c_1^2=O(\alpha) \eqno(2.6) \end{displaymath}


holds for any $s\in{\bf R}$, $\alpha\to 0^+$.


      P r o o f. Defining angle variables $\varphi$ and $\psi\in [0,2\pi)$ via $\displaystyle{q_1=\frac{c_1}{\omega_0}\sin\varphi }$, $\displaystyle{q_2=c_1\cos\varphi }$ and $\displaystyle{p_1=c_2\cos\psi }$, $\displaystyle{p_2=\frac{c_2}{\omega_0}\sin\psi }$ one easily obtains that equation (2.5) is equivalent to the equation

\begin{eqnarray*} \omega_0\frac{\partial F_1^{(s)}}{\partial\varphi} &\!\!\!\!... ... \beta^2(sc_2^2\sin^2\psi-c_1^2\cos^2\varphi). \hskip4cm (2.7) \end{eqnarray*}


Since the function $F_1^{(s)}\in C^1([0,2\pi);{\bf R}^1)$ must be $2\pi$-periodic with respect to $\varphi\in[0,2\pi)$, from the evident condition $\displaystyle{\int\limits_0^{2\pi}\frac{\partial F_1^{(s)}}{\partial\varphi}d\varphi\equiv0 }$ and (2.7) one gets the necessary condition (2.6) for the expression (2.4) to be a $O(\alpha^2)$ -- conservation law of the flow (2.3) at $\varepsilon=0$. This proves the lemma. $\Delta$ As the condition (2.6) contains an arbitrary real parameter $s\in{\bf R}$, one derives right away that

\begin{displaymath} c_1^2(4\beta^2\omega_0^2-c_1^2)=O(\alpha),\ \ \ \frac{1}{4... ...\cos2c_3\right)+\frac{\beta^2}{2}c_2^2=O(\alpha). \eqno(2.8) \end{displaymath}



     From (2.8) one follows that dynamical system (2.3) conservation law (2.5) for any $s\in{\bf R}$ if and only if a) $c_1=2\omega_0\beta+O(\alpha),\ \ \ c_2=O(\alpha^{1/2})$; (2.9) b) $c_1=O(\alpha^{1/2})$, $c_2=O(\alpha^{1/2})$, since the case b) seems to be more trivial, we further shall cope with the case a). The latter case qives that $p_1,p_2=O(\alpha^{1/2})$ as $\alpha\to 0^+$. Moreover, the first condition in (2.9 a)) evidently means that there exists an orbit $\sigma_{\alpha}\subset T^*({\bf R}^2)$ lying in compact submanifold $ M_{n,\alpha}^2 =\bigg\{(q,p)\in T^*({\bf R}^2):\ \ H_\alpha=h\in{\bf R},\ ... ...; 2\omega_0\beta;0;\varphi_0+\psi_0\right),\varphi_0,\psi_0\in[0,2\pi)\bigg\}$       Now our task will be centered on the imbedding mapping $\pi_h:\ M_{h,\alpha}^2\to T^*({\bf R}^2)$, in particular on finding the corresponding coordinates on $M_{h,\alpha}^2$ in which the motion on it is separable. This situation is allowed by the Liouville-Arnold theorem [9] on the complete integrability of Hamiltonian systems and the corresponding Hamilton-Jacobi separation variable procedure devised recently in [6,7]. For this to be made more effectively, let us consider now the same problem for the dynamical system (2.3) at $\alpha,\varepsilon=0$:

\begin{displaymath} dq_1/dt=q_2,\ \ \ dq_2/dt=-\omega_0^2q_1,\ \ \ dp_1/dt=\omega^2_0p_2,\ \ \ dp_2/dt=-p_1, \eqno(2.10) \end{displaymath}


having the following two involutive invariants:

\begin{displaymath} H_0:=H_\alpha\Big\vert_{\alpha=0}=q_2p_1-\omega_0^2q_1p_2,\... ...t_{\alpha=0}=\frac{1}{2}(q_2^2+\omega_0^2q_1^2). \eqno(2.11) \end{displaymath}


     It is useful to notice here that on $T^*({\bf R}^2)$ there exists another almost everywhere independent first integral $F_0'\in D(T^*({\bf R}^2))$, where $F_0'=dF_\alpha^{(s)}/ds\bigg\vert _{{s=0}\atop{\alpha=0}}= 1/2(p_1^2+\omega_0^2p_2^2)$. Since obviously, the commutator {F0,F0'} is a new first integral of (2.10), one gets easily that the function

\begin{displaymath} H_0':=p_1q_1+p_2q_2 \eqno(2.12) \end{displaymath}


commutes with the Hamiltonian $H_0\in D(T^*({\bf R}^2))$. Following now the method of [6], define two one-formes $\bar h^{(1)},\bar h'^{(1)}\in \Lambda^1(T^*(M_{h,h'}^2))$ satisfying the condition

\begin{displaymath} dH_0\wedge\bar h^{(1)}+dH_0'\wedge h'^{(1)}=\Omega^{(2)}, \eqno(2.13) \end{displaymath}


where, by definition, $\bar h^{(1)}:=\sum\limits_{j=1}^2\bar h_jdq_j$, $\bar h'^{(1)}:=\sum\limits_{j=1}^2\bar h_j'dq_j$. From (2.11) and (2.12) one gets on Mh,h'2 such a useful relationship:

\begin{displaymath} \left[\matrix{ \ \ q_2 & -\omega_0^2q_1\cr q_1 & \ \ q_2\... ...1 & -p_2\cr}\right]\pmatrix{ dq_1\cr dq_2\cr}. \eqno(2.14) \end{displaymath}


     Solving now the determining equation (2.13), one obtains

\begin{displaymath} \bar h_1=\frac{q_2}{q_2^2+\omega_0^2q_1^2},\ \ \bar h_2=\... ...\ \bar h_2'=\frac{q_2}{q_2^2+\omega_0^2q_1^2}. \eqno(2.15) \end{displaymath}


     Since on the manifold Mh,h'2 there are fulfilled the relationships

\begin{displaymath} \bar h^{(1)}\Big\vert _{M_{h,h'}^2}=dt,\ \ \ \bar h'^{(1)}\Big\vert _{M_{h,h'}^2}=dt', \eqno(2.16) \end{displaymath}


where the parameters $(t,t'): M_{h,h'}^2\to{\bf R}^2$ are globally parametrizing coordinates on Mh,h'2, one can define the following imbedding mapping: $\overline{\pi}: M_{h,h'}^2\to{\bf R}^2\subset T^*({\bf R}^2)$ not depending everywhere on $(h,h')\in{\bf R}^2$: $q_j=q_j(\mu_1,\mu_2),\ \ \ j=\overline{1,2}$, and the associated cotangent imbedding mappings $p_j=p_j(\mu_1,\mu_2;h,h')$, $j=\overline{1,2}$,where $\mu_j\in{\bf S}^1$, $j=\overline{1,2}$ -- separable coordinates in case when the integral submanifold Mh,h'2, is compact and diffeomorphic to the torus ${\bf T}^2\simeq{\bf S}^1\times{\bf S}^1$, what will be assumed in the sequel. Making use of the equality $\alpha^{(1)}:=\sum\limits^2_{j=1}p_jdq_j=\sum\limits_{j=1}^2w_jd\mu_j$, where $\Omega^{(2)}:=d\alpha^{(1)}$, one gets for $j=\overline{1,2}$

\begin{displaymath} p_j=\sum\limits_{k=1}^2w_k\partial\mu_k/\partial q_j. \eqno(2.17) \end{displaymath}


next one needs to assume that one transformation $T^*({\bf R}^2)\ni(\mu,w): \to (t,t';h,h')\in T^*({\bf R}^2)$ is canonical in a vicinity of the cotangent vector bundle T*(Mh,h'2), that is

\begin{displaymath} \sum\limits_{j=1}^2w_jd\mu_j=-tdh-t'dh'+dS_0(\mu;h,h'), \eqno(2.18) \end{displaymath}


where $S_0: ({\bf S}^1\times {\bf S}^1)\times{\bf R}^2\to{\bf R}$ -- the corresponding generating function. From the relationships (2.18) and (2.16) one obtains easily that on T*(Mh,h'2)

\begin{displaymath} S_0(\mu;h,h')=\sum\limits_{j=1}^2\int\limits_{\mu_j^0}^{\mu_j}w_j(\tau;h,h') d\tau, \eqno(2.19) \end{displaymath}


where for $j=\overline{1,2}$

\begin{displaymath} \frac{\partial w_j(\mu_j;h,h')}{\partial h}=\frac{q_2}{q_2^... ...}{q_2^2+\omega_0^2q_1^2} \frac{\partial q_2}{\partial\mu_j}, \end{displaymath}


\begin{displaymath} \frac{\partial w_j(\mu_j;h,h')}{\partial h'}= \frac{\omega... ...a_0^2q_1^2} \frac{\partial q_2}{\partial\mu_j}. \eqno(2.20) \end{displaymath}


     From (2.20) one deduces right away that

\begin{displaymath} \frac{\partial w_j(\mu_j;h,h')}{\partial h'}=\frac{1}{2} \... ...artial\ln(q_2^2+\omega_0^2q_1^2)}{\partial\mu_j} \eqno(2.21) \end{displaymath}


for $j=\overline{1,2}$. Since the right hand side of the equality (2.21) doesn't depend on w-variables; one can for the definiteness put for instance

\begin{displaymath} \partial w_1(\mu_1;h,h')/\partial h':=\mu_1^{-1}, \eqno(2.22) \end{displaymath}


where $\mu_1\in{\bf S}^1$. The latter gives rise to the following one-half imbedding mapping of the integral submanifold $M_{h,h'}^2\subset T^*({\bf R}^2)$: $q_2^2+\omega_0^2q_1^2=\mu_1^2$. Considering similarly the next equation from (2.21) for j=2, one obtains easily that

\begin{displaymath} \partial w_2(\mu_2;h,h')/\partial h'=1/2\partial\mu_1^2/ \partial\mu_2\equiv0 \eqno(2.23) \end{displaymath}


as variables $\mu_j$, $j=\overline{1,2}$, are mutually independent. Thus, from (2.20), (2.22) and (2.23) one follows that on T*(Mh,h'2)

\begin{displaymath} w_1(\mu_1;h,h')=\frac{1}{\mu_1}h'+\overline{C}_1(\mu_1)+C_1... ..._2(\mu_2;h,h')=\overline{C}_2(\mu_2)+C_2(\mu_2)h \eqno(2.24) \end{displaymath}


for some mappings $C_j,\overline{C}_j: {\bf S}^1\to{\bf R}$, $j=\overline{1,2}$. The result (2.24) must evidently satisfy exactly the first pair of equations (2.20) provided the basic conditions (2.11), (2.12) and (2.17) are fulfilled identically, that is on T*(Mh,h'2)

\begin{displaymath} h=q_2p_1-\omega_0^2q_1p_2,\ \ \ h'=p_1q_1+p_2q_2. \eqno(2.25) \end{displaymath}


     Before verifying the condition mentioned above, let us analyze the first pair of equations (2.20) provided the (2.24). One obtains hence easily for j=1 that $\partial q_1/\partial\mu_1-q_1/\mu_1=C_1(\mu_1)q_2$, and similarly, for j=2: $\partial q_1/\partial \mu_2=C_2(\mu_2)q_2$.      For the mappings $C_j:{\bf S}^1\to {\bf R}$, $j=\overline{1,2}$ should be determined exactly, let us now proceed whith verifying the conditions (2.24) and (2.17) on T*(M2h,h'). As a result of simple calculations we get such relationships: $q_1=\mu_1\omega_0^{-1}\sin\mu_2$, $q_2=\mu_1\cos\mu_2$, $C_2=\omega^{-1}_0$, $\bar C_1=0=\bar C_2$, C1=0, solving completely the imbedding problem for the integral submanifold $M^2_{h,h'}\subset T^*({\bf R}^2)$. Based now on the expression (2.19), we can find the generating function $S_0:({\bf S}^1\times{\bf S}^1)\times {\bf R}^2\to {\bf R}:$

\begin{displaymath} S_0(\mu;h,h') =\int_{\mu^0_1}^{\mu_1}ds(h's^{-1})+\int_{\m... ...^{-1}_0)= h'ln(\mu_1/\mu^0_1)+h\omega^{-1}_0(\mu_2-\mu^0_2), \end{displaymath}


where $(\mu^0_1,\mu^0_2)\in {\bf S}^1\times {\bf S}^1$ - arbitrary points on M2h,h'. The corresponding canonical symplectic structure on the submanifold T*(M2h,h') becomes due to (2.17): $\Omega^{(2)}=\sum^2_{j=1} dw_j\wedge d\mu_j$. The result obtained makes it possible to rewrite our dynamical system (2.10) in canonical variables $(\mu,w)\in T^*(M^2_{h,h'})$. Especially, from (2.25) one sees that $H_0=\omega_0w_2$, $H_0'=\mu_1w_1,\ \ F_0'=\mu^2_1/2$. In case when the parameter $\alpha\ne 0$, one obtains also that

\begin{eqnarray*} H^{(\varepsilon)}_{\alpha} &\!\!\!\!\!=&\!\!\!\!\!\omega_0 w... ...u_2-w_2\sin\mu_2\mu_1^{-1}\bigr)\cos\omega t , \hskip4cm(2.26) \end{eqnarray*}


where me made use of such easily deducable relationships, valid on T*(M2h,h'):

\begin{displaymath} w_1(\mu_1;h,h')=h'/\mu_1, \ \ \ \ \ \ w_2(\mu_2;h,h')=h/\omega_0, \end{displaymath}


\begin{displaymath} p_1=w_1\omega_0\sin\mu_2+w_2\frac{\omega_0}{\mu_1}\cos\mu_2, \ \ \ p_2=w_1\cos\mu_2-w_2\frac{\sin\mu_2}{\mu_1} \end{displaymath}


\begin{displaymath} \frac{\partial\mu_1}{\partial q_2}=\cos\mu_2, \ \ \frac{\p... ...ial\mu_2}{\partial q_2}=-\frac{\sin\mu_2}{\mu_1} \eqno(2.27) \end{displaymath}


     Similarly based on the equation (2.7) one obtains the expression for the second invariant $F^{(0)}_{\alpha}\in D(T^*(M^2_{h,h'}))$ for the compact case of the integral submanifold M2h,h':

\begin{displaymath} F^{(0)}_{\alpha}=\mu^2_1/2-\alpha\bigl[ \frac{\mu^2_1\sin^... ...{\mu^2_1}{4\omega^2_0} \cos^2\mu_2\bigr)\bigr], \eqno(2.28) \end{displaymath}


where $F^{(0)}_{\alpha}\vert_{M^2_{h,h'}}=2\beta^2\omega^2_0\in {\bf R}_{+}$, that is needed according to the conditions of Lemma 2.1. Thus, on the submanifold $M^2_{h,\alpha}\subset T^*({\bf R}^2)$ in the canonical coordinates $(\mu,w)\in T^*(M^2_{h,h'})$ at $\varepsilon\ne 0$ our dynamical system (2.3) is written down as

\begin{displaymath} d\mu_1/dt=\partial H^{(\varepsilon)}_{\alpha}/\partial w_1 ... ...u_2+ \varepsilon\mu_1\cos^2\mu_2, \hskip3cm\phantom{tttttt} \end{displaymath}


\begin{displaymath} d\mu_2/dt=\partial H^{(\varepsilon)}_{\alpha}/\partial w_2=... ...epsilon}{2}\sin2\mu_2\cos\omega t, \hskip3cm\phantom{tttttt} \end{displaymath}


\begin{eqnarray*} d w_1/dt &\!\!\!\!\!=&\!\!\!\!\!-\partial H^{(\varepsilon)... ...igr)- \varepsilon w_1\cos^2\mu_2\cos\omega t, \hskip1cm(2.29) \end{eqnarray*}


\begin{eqnarray*} d w_2/dt &\!\!\!\!\!=&\!\!\!\!\!-\partial H^{(\varepsilon)}... ...2+ w_2\cos 2\mu_2\bigr) \cos\omega t. \hskip6cm\phantom{FFFFF} \end{eqnarray*}


      The system (2.29) can be now analyzed further making use of the time independent canonical transformation of the variables $(\mu, w)\in T^*(M^2_{h,\alpha})$ to variables $(\bar\mu,\bar w)\in T^*(M^2_{\bar h,\alpha})$ where by definition,

\begin{displaymath} \sum^2_{j=1} w_jd\mu_j-H^{(\varepsilon)}_{\alpha}dt=- \sum... ...pha}dt+ dS^{(\varepsilon)}_{\alpha}(\mu;\bar w) \eqno(2.30) \end{displaymath}


with $\overline S^{(\varepsilon)}_{\alpha}:({\bf R}^1\times {\bf S}^1)\times {\bf R}^2 \longrightarrow {\bf R}$ - the corresponding generating function, chosen from the following conditions:

\begin{displaymath} \bar\mu_1=\mu_1, \ \ \ \bar w_1=w_1, \ \ \ \ \partial \bar... ...silon)}_{\alpha}/\partial \bar\mu_2 =O(\alpha^2) \eqno(2.31) \end{displaymath}


as $\alpha\to 0^+$. As a result of (2.30) one obtains easily that $w_2=\bar w_2-\alpha\partial \overline S^{(\varepsilon)}_1/\partial\mu_2+O(\alpha^2)$, $\mu_2=\bar\mu_2-\alpha\partial \overline S^{(\varepsilon)}_1 /\partial\bar w_2 + O(\alpha^2)$, where $\overline S^{(\varepsilon)}_{\alpha}=\sum^2_{j=1}\bar w_j\mu_j+ \alpha \overline S^{(\varepsilon)}_1 (\mu_2;\bar w_2)+O(\alpha^2)$. From (2.30) and (2.31) one gets now that $\bar H^{(\varepsilon)}_{\alpha}=\bar H^{(\varepsilon)}_0+ \alpha\bar H^{(\varepsilon)}_1+O(\alpha^2)$ where

\begin{displaymath} \bar H^{(\varepsilon)}_0=<H^{(\varepsilon)}_0>^{{\bf S}^1},... ...ilon)}_0+\alpha H^{(\varepsilon)}_1+O(\alpha^2) \eqno(2.32) \end{displaymath}


with the operation $< \cdot >^{{\bf S}^1}$ acting as the usual ${\bf S}^1$ - averaging with respect to the variable $\mu_2\in {\bf S}^1$. Whence based on (2.26) one obtains that

\begin{displaymath} \bar H^{(\varepsilon)}_0(\mu_1, w_1;\bar w_2)=\omega\bar w_... ...igl( \frac{\mu^2_1}{4\omega^2_0}-\beta^2\bigr). \eqno(2.33) \end{displaymath}


for all $\varepsilon\in {\bf R}$ as $\alpha\to 0^{+}$. Thus the resulting dynamical system on $T^*(M^2_{\bar h,\alpha})$ can be written as

\begin{displaymath} d\mu_1/dt=\partial\bar H^{(\varepsilon)}_{\alpha}/\partial ... ...mu_1}{2}\cos\omega t+O(\alpha^2), \hskip0.03cm\phantom{tttt} \end{displaymath}


\begin{displaymath} d w_1/dt=-\partial\bar H^{(\varepsilon)}_{\alpha}/\partial\... ...w_1}{2}\cos\omega t -\alpha\frac{w_1\beta^2}{2}+O(\alpha^2), \end{displaymath}


\begin{displaymath} d\bar\mu_2/dt=\partial\bar H^{(\varepsilon)}_{\alpha}/\part... ...ilon)}_{\alpha}/\partial\bar\mu_2 =O(\alpha^2). \eqno(2.34) \end{displaymath}



     Let us analyze the dynamical system (2.34) as $\alpha\to 0^{+}$. From the first equation in (2.34) one follows right away that $d\mu^2_1/dt<0$ for all $t\in {\bf R}_{+}$ if $\mu^2_1\ge4\beta^2\omega^2_0+k_1+O(\alpha)$ as $\alpha\to 0^{+}$ for any k1>0, and conversly, $d\mu^2_1/dt>0$ for all $t\in {\bf R}_{+}$ if $0<\mu^2_1\le4\beta^2\omega^2_0-k_1 +O(\alpha)$ as $\alpha\to 0^{+}$ for any k1>0. Thus, based on the well known Poincare theorem [8] , the folllowing assertion is true.
     Theorem 2.2. The dynamical system (2.34) at $\varepsilon=0$ for $\alpha\in{\bf R}_+$ is small enough possesses an invariant limiting periodic cycle of a radius $r_{\beta}=2\beta\omega_0+O(\alpha)$ $\sigma_{\beta}\subset T^*(M^2_{\bar h,\alpha})$ lying on the plane $\bar w_1=O(\alpha^2)$, $\bar w_2=O(\alpha^2)$.
     P r o o f. From the consideration presented above and equations (2.34) one follows that $\vert w_1\vert\to O(\alpha^2)$ and $\bar w_2\simeq O(\alpha^2)$ as $t\to\infty$. Since there exists no peculiar point for all $(\mu_1,w_1)\in U_{\alpha}(\sigma_{\beta})=\{(\mu_1,w_1)\in {\bf R}^1\times{\bf R}^1:\bigl\vert\mu^2_1/4\omega^2_0-\beta^2 \bigr\vert<k^2_1+O(\alpha)$ as $\alpha\to 0^{+}$, where k1>0, |w1|<k2, k2>0}, based on the mentioned above Poincare theorem one derives that in the neighbourhood $U_{\alpha}(\sigma_{\beta})$ there exists an invariant attracting limiting cycle of the radius $r_{\beta}=2\beta\omega_0+O(\alpha)$ as is stated in the theorem. The latter proves the theorem.

     3. The deformation of Lagrangian submanifoldds. It is now easy to verify that this attractive limiting cycle $\sigma_{\beta}\subset T^*(M^2_{\bar h,\alpha})$ is realized at $F^{(0)}_{\alpha}=2\beta^2\omega^2_0$, $H^{(0)}_{\alpha}=\bar h$ for any $\bar h\in {\bf R}$ being a hyperbolic periodic solution to the dynamical system (2.3) at $\varepsilon=0$. Therefore, one can try to study the corresponding to $\sigma_{\beta}$ stable and unstable Lagrangian submanifolds $\Lambda^{\pm}_{\alpha,0}(\sigma_{\beta})$ in $T^*(M^2_{\bar h,\alpha})$ and their deformations $\Lambda^{\pm}_{\alpha,\varepsilon} (\sigma_{\beta,\varepsilon})$ as $\varepsilon\to 0$, defined by means of the following expressions [9,10]:

\begin{displaymath} \Lambda^{\pm}_{\alpha,0}(\sigma_{\beta}) =\bigg\{(\mu_1, w... ...pha}): w_1=\partial\bar S^{(0)}_{\alpha,\pm}/ \partial\mu_1, \end{displaymath}


\begin{displaymath} \bar w_2=\partial\bar {S}^{(0)}_{\alpha,\pm}/\partial\bar\m... ...\ \ \lim_{t\to\pm\infty}\vert w_1\vert=0\bigg\}, \eqno(3.1) \end{displaymath}


\begin{displaymath} \Lambda^{\pm}_{\alpha,\varepsilon}(\sigma_{\beta,\varepsil... ...partial\bar {S}^{(\varepsilon)}_{\alpha,\pm}/ \partial\mu_1, \end{displaymath}


\begin{displaymath} \bar w_2=\partial\bar {S}^{(\varepsilon)}_{\alpha,\pm}/\par... ...2\omega^2_0,\ \ \lim_{t\to\pm\infty}\vert w_1\vert=0\bigg\}, \end{displaymath}


where, by definition, $\sigma_{\beta,\varepsilon}\subset T^* (M^2_{\bar h,\alpha})\times {\bf S}^1$ - the properly deformed limiting cycle $\sigma_{\beta}\subset T^*(M^2_{\bar h,\alpha})$ and

\begin{displaymath} \partial\bar{S}^{(\varepsilon)}_{\alpha,\pm}/\partial t+\ba... ...^{(\varepsilon)}_{\alpha,\pm}/\partial\bar w_2)=0 \eqno(3.2) \end{displaymath}



     Making use now of the explicit expression of the Hamiltonian function $\bar H^{(\varepsilon)}_{\alpha}$, from (3.2) one simply deduces the validity of the following Lemma.
     Lemma 3.1. The following asymptotic Lagrangian submanifolds $\Lambda^{-}_{\alpha,0}(\sigma_{\beta})$ and $\Lambda^{-}_{\alpha,\varepsilon}(\sigma_{\beta,\varepsilon})$ are empty. Based on Lemma 3.1. and standard considerations of the Poincare theory on the arrangement of Lagrangean submanifolds in phase space [11], one can formulate the next theorem.  
     Theorem 3.2. All motions of the Van-der-Pol dynamical system (1.1) as $\alpha\to 0^{+}$ and $\varepsilon=O(\alpha^{1+\delta})$, $\delta\in (0,1)$ is arbitrary, are regular in a vinicity of the attractive limiting cycle $\sigma_{\beta}\subset T^*(M^2_{\bar h,\alpha})$. This means that the $\varepsilon$-perturbation chosen in (1.1) doesn't give rise to the phenomenon of transversal splitting stable and unstable asymptotic Lagrangian submanifolds of the corresponding limiting cycle , what is , in general, the main scenario of arising chaotic motions via the Poincare- Birkhoff theory [11,12]. Thereby, the old problem of proving a possible existence of chaotic motions in a weakly perturbed Van-der-Pol dynamical system needs a further study using both analytical and numerical methods of investigation. The next study of the Van-der-Pol dynamical system by means of symplectic methods will be centered at problems of existing its adiabatic invariants and a possibility of strongly irregular motions under both slow and weak perturbations.

     Acknowledgements. The authors are cordially indebted to Profs. Anatoliy M. Samoilenko, Denis L.Blackmore and John Tavantzis for many stimulating discussions of the problem studied. Special thanks are to Dr. Valeriy H. Samoylenko, Mgrs. Yarema Prykarpatsky and Andrzej Szum for a help while analyzing the integral submanifold imbedding problem and its solution presented.

References

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