V. Melnikov

ON THE DETERMINATION OF THE DOMAIN OF CAPTURE FOR A SYSTEM CLOSE TO HAMILTONIAN

(Presented by Academician P. S. Aleksandrov on 21 II 1961)

The investigation of certain physical processes \((^{1,2})\) leads to the following problem. Let a system of differential equations with small parameter \(\varepsilon\) be given:

\[ \dot{x}=-\frac{\partial H}{\partial y}+\varepsilon f(x,y,t,\varepsilon),\qquad \dot{y}=\frac{\partial H}{\partial x}+\varepsilon g(x,y,t,\varepsilon), \tag{I_\varepsilon} \]

where \(H=H(x,y)\) is an analytic function of \(x\) and \(y\), and \(f(x,y,t,\varepsilon)\) and \(g(x,y,t,\varepsilon)\) are analytic functions of \(x,y,\varepsilon\), continuous together with the first derivative with respect to \(t\), and periodic in \(t\) with period \(2\pi\). Suppose further that the point \((x_c,y_c)\) is an equilibrium position of center type of the system \((I_0)\) (the system \((I_0)\) is obtained from the system \((I_\varepsilon)\) for \(\varepsilon=0\)). This means (see \((^3)\), p. 301) that there exists a neighborhood of the point \((x_c,y_c)\), containing no other equilibrium positions, such that the trajectory issuing from any point of this neighborhood distinct from \((x_c,y_c)\) is closed. We shall be interested in the case when the closed trajectories of the system \((I_0)\) do not fill the whole \((x,y)\)-plane. Moreover, we shall assume that the maximal neighborhood of the point \((x_c,y_c)\) filled with closed trajectories of the system \((I_0)\) lies in a bounded part of the plane. Then, as is known \((^4)\), there exists a boundary separating on the \((x,y)\)-plane the closed trajectories from the nonclosed ones. This boundary consists of a finite number of trajectories, each of which as \(t\to\pm\infty\) tends to some equilibrium position of saddle type. (Here, as everywhere below, it is assumed that all equilibrium positions of the system \((I_0)\) are simple, i.e. if at the point \((x_r,y_r)\)

\[ \frac{\partial H}{\partial x}=\frac{\partial H}{\partial y}=0, \]

then

\[ \frac{\partial^2 H}{\partial x^2}\frac{\partial^2 H}{\partial y^2} -\left(\frac{\partial^2 H}{\partial x\,\partial y}\right)^2\ne 0). \]

The problem is, for an arbitrarily chosen instant of time \(t_0\), to find in the \((x_0,y_0)\)-plane the set of points from which, at \(t=t_0\), oscillatory solutions issue.

Theorem 1. Let \((x_s,y_s)\) be an arbitrary equilibrium position of saddle type of the system \((I_0)\). Then there exists a change of variables of the form

\[ x=x_s+a\,(u+\varepsilon p(t,\varepsilon))\cos\varphi -\frac{1}{a}\,(v+\varepsilon q(t,\varepsilon))\sin\varphi, \]

\[ y=y_s+a\,(u+\varepsilon p(t,\varepsilon))\sin\varphi +\frac{1}{a}\,(v+\varepsilon q(t,\varepsilon))\cos\varphi, \]

where \(a\) and \(\varphi\) are constants, and the functions \(p(t,\varepsilon)\) and \(q(t,\varepsilon)\) are analytic in \(\varepsilon\) in a neighborhood of \(\varepsilon=0\), periodic in \(t\) with period \(2\pi\), and possess continuous partial derivatives with respect to \(t\) up to the second order inclusive, such that in the new variables the system \((I_\varepsilon)\) takes the form:

\[ \dot{u}=\lambda v-\frac{\partial \widetilde{H}}{\partial v} +\varepsilon \widetilde{f}(u,v,t,\varepsilon),\qquad \dot{v}=\lambda u+\frac{\partial \widetilde{H}}{\partial u} +\varepsilon \widetilde{g}(u,v,t,\varepsilon), \tag{II_\varepsilon} \]

where \(\lambda > 0,\ f(0,0,t,\varepsilon)\equiv g(0,0,t,\varepsilon)\equiv 0\), and the expansion of the function \(\bar H=\bar H(u,v)\) in a neighborhood of the point \((0,0)\) begins with terms of no lower than third order.

We shall call the system \((\Pi_\varepsilon)\) a standard form of the system \((I_\varepsilon)\) in a neighborhood of the saddle. The theorem just stated makes it possible to formulate the following basic definition.

A solution \((u_\varepsilon(t),v_\varepsilon(t))\) of the system \((\Pi_\varepsilon)\) will be called a boundary solution if it is defined for all \(t\) greater than some \(t_0\), \(|u_\varepsilon(t)|+|v_\varepsilon(t)|\to 0\) as \(t\to\infty\), and there exists a time \(t_1\) such that for all \(t>t_1\) the conditions
\[ \frac{d}{dt}|u_\varepsilon(t)|<0 \quad\text{and}\quad \frac{d}{dt}|v_\varepsilon(t)|<0 \]
are satisfied. This definition, obviously, also has meaning for complex values of the parameter \(\varepsilon\). Using Theorem 1, it is not difficult to establish what is to be understood by a boundary trajectory of the system \((I_\varepsilon)\).

For an arbitrarily prescribed time \(t_0\), denote by \(\Gamma_\varepsilon(t_0)\) the set of points in the \((x_0,y_0)\)-plane such that, for \(t=t_0\), boundary trajectories of the system \((I_\varepsilon)\) issue from them.

Theorem 2. Let \(\Delta=\widetilde E^{\,2}\setminus\Gamma_\varepsilon(t_0)\), where \(\widetilde E^{\,2}\) is an arbitrary bounded part of the \((x_0,y_0)\)-plane from which the isolated points of the set \(\overline{\Gamma}_\varepsilon(t_0)\setminus\Gamma_\varepsilon(t_0)\) have been removed, and let \(\Delta'\subset\Delta\) be an arbitrary linearly connected set. Suppose that in the set \(\Delta'\) there exists a point \((x_0,y_0)\) from which, at \(t=t_0\), an oscillatory solution of the system \((I_\varepsilon)\) issues. Then, for sufficiently small \(\varepsilon\), any other solution of the system \((I_\varepsilon)\) issuing at \(t=t_0\) from an arbitrary point belonging to \(\Delta'\) will also be oscillatory.

It follows from Theorem 2 that, in order to find the oscillatory solutions of the system \((I_\varepsilon)\), it is necessary to find \(\Gamma_\varepsilon(t_0)\). With the aid of a certain development of the ideas of paper \([5]\), the following two theorems can be proved.

Theorem 3. There exist \(\varepsilon_0>0\) and \(\delta_0>0\) such that, for any complex \(\varepsilon\) and \(u_0\) satisfying the conditions \(|\varepsilon|<\varepsilon_0,\ |u_0|<\delta_0\), and for any \(t_0\), there exists a unique solution \((u_\varepsilon(t),v_\varepsilon(t))\) of the system \((\Pi_\varepsilon)\) such that
\[ u_\varepsilon(t_0)=u_0,\quad \frac{d}{dt}|u_\varepsilon(t)|<0 \quad\text{and}\quad \frac{d}{dt}|v_\varepsilon(t)|<0 \]
for all \(t\ge t_0\), and
\[ |u_\varepsilon(t)|+|v_\varepsilon(t)|\to 0 \quad\text{as } t\to\infty . \]

Theorem 4. There exist \(\varepsilon_1>0\) and \(\delta_1>0\) such that, for any complex \(\varepsilon\) and \(u_0\) satisfying the conditions \(|\varepsilon|<\varepsilon_1,\ |u_0|<\delta_1\), and for any \(t_0\), a solution \((u_\varepsilon(t),v_\varepsilon(t))\) of the system \((\Pi_\varepsilon)\) satisfying the conditions
\[ u_\varepsilon(t_0)=u_0,\quad \frac{d}{dt}|u_\varepsilon(t)|<0 \quad\text{and}\quad \frac{d}{dt}|v_\varepsilon(t)|<0 \]
for all \(t\ge t_0\), and
\[ |u_\varepsilon(t)|+|v_\varepsilon(t)|\to 0 \quad\text{as } t\to\infty, \]
has continuous partial derivatives with respect to \(\mu=\operatorname{Re}\varepsilon\) and \(\nu=\operatorname{Im}\varepsilon\), which satisfy the Cauchy–Riemann conditions for all \(t\ge t_0\).

Theorem 4 together with Theorem 1 make it possible to compute the boundary trajectories of the system \((I_\varepsilon)\) by expanding them in a series in powers of the parameter \(\varepsilon\) in a certain, generally speaking, “small” neighborhood of the saddle-type equilibrium position under study. However, using Poincaré’s theorem (see \([6]\), pp. 153–160), one can show that the boundary trajectories of the system \((I_\varepsilon)\) will be analytic functions of the parameter \(\varepsilon\) also in a “large” neighborhood of the saddle-type equilibrium position, since the expansions obtained for the boundary trajectories in powers of the parameter \(\varepsilon\) will converge also for \(t<t_0\) on any finite interval \([t_0-T,t_0]\) \((T>0)\), provided \(\varepsilon\) is sufficiently small. This turns out to be sufficient for proving the following theorem.

Theorem 5. Let \(G_c\) be the maximal neighborhood of the point \((x_c,y_c)\) filled with closed trajectories of the system \((I_0)\). Number by the integers from one to \(n\) all saddle-type equilibrium positions lying on the boundary of the region \(G_c\), in the order in which they are situated on the boundary when traversing along the boundary of the region \(G_c\) in the clockwise direction, and suppose that the motion along the closed trajectories of the system \((I_0)\),

lying in the region \(G_c\), also proceeds clockwise (if this is not so, then this can always be achieved by replacing \(t\) by \(-t\)). Denote by \((x_i(t), y_i(t))\) \((i=1,2,\ldots,n)\) the solution of system \((I_0)\) which, as \(t\to -\infty\), tends to the \(i\)-th equilibrium position of saddle type and, as \(t\to +\infty\), tends to the \((i+1)\)-st equilibrium position of saddle type (for \(i=n\), the \((i+1)\)-st equilibrium position is to be understood as the 1st equilibrium position), and let

\[ I_i(t_0)=\int_{-\infty}^{\infty} \{f(x_i(t),y_i(t),t-t_0,0)\dot y_i(t) -g(x_i(t),y_i(t),t-t_0,0)\dot x_i(t)\}\,dt \]

be such that

\[ \sum_{i=1}^{n}\int_{0}^{2\pi} I_i(t_0)\,dt_0>0. \]

Let, further, for \(\delta>0\), \(\overline{G}_{c,\delta}^{-}\subset G_c\) be the set of points of the region \(G_c\) lying at a distance greater than \(\delta\) from the boundary of the region \(G_c\), and let \(G_{c,\delta}^{+}\supset G_c\) be the \(\delta\)-neighborhood of the set \(\overline{G}_c\). Then, for sufficiently small \(\varepsilon>0\), there exists \(\delta(\varepsilon)>0\) such that all solutions of system \((I_\varepsilon)\) issuing, at \(t=t_0\), from the region \(G_{c,\delta(\varepsilon)}^{-}\) will be oscillatory, and their trajectories for \(t\ge t_0\) do not leave the region \(G_{c,\delta(\varepsilon)}^{+}\), with \(\delta(\varepsilon)\to 0\) as \(\varepsilon\to +0\).

In conclusion, I take this opportunity to express my deep gratitude to S. V. Fomin for his assistance in carrying out this work.

Joint Institute
for Nuclear Research

Received
1 II 1961

REFERENCES

1. G. Sansone, Rend. Acc. Naz. Dei. XI, Ser. IV, 8, 1 (1957).
2. Yu. S. Sayasov, V. K. Melnikov, ZhTF, 30, no. 6, 656 (1960).
3. A. Poincaré, On curves defined by differential equations, Moscow—Leningrad, 1947.
4. N. A. Sakharnikov, Prikl. matem. i mekh., 15, no. 3, 349 (1951).
5. V. K. Melnikov, Matem. sbornik, 49 (91), no. 4, 353 (1959).
6. V. V. Golubev, Lectures on the analytic theory of differential equations, Moscow—Leningrad, 1950.