V. Melnikov

ON THE BEHAVIOR OF THE TRAJECTORIES OF A SYSTEM CLOSE TO AN AUTONOMOUS HAMILTONIAN SYSTEM

(Presented by Academician N. N. Bogolyubov, 4 IX 1961)

In the present note we shall consider the behavior of the trajectories of the system

\[ \dot{x}=\partial H/\partial y+\varepsilon f(x,y,t,\varepsilon),\qquad \dot{y}=-\partial H/\partial x+\varepsilon g(x,y,t,\varepsilon), \tag{1_\varepsilon} \]

with respect to which we shall assume that the following conditions are satisfied:

1. \(H=H(x,y)\) is an analytic function of \(x\) and \(y\) in some neighborhood of the origin.

2. \(f(x,y,t,\varepsilon)\) and \(g(x,y,t,\varepsilon)\) are functions analytic in \(x,y\), and \(\varepsilon\) in some neighborhood of the point \(x=y=\varepsilon=0\), continuous in \(t\) together with the first derivative with respect to \(t\), and periodic in \(t\) with period \(2\pi\).

3. For \(x=y=0\), \(\partial H/\partial x=\partial H/\partial y=0\),
   \[ \Delta=(\partial^2 H/\partial x^2)(\partial^2 H/\partial y^2)-(\partial^2 H/\partial x\partial y)^2>0. \]
   The last condition means that for the system \((1_0)\) (the system \((1_0)\) is obtained from \((1_\varepsilon)\) for \(\varepsilon=0\)) the origin is an equilibrium position of center type, i.e., all trajectories of the system \((1_0)\) beginning in a sufficiently small neighborhood of the origin will be closed.

Let us pass in the system \((1_\varepsilon)\) from the variables \((x,y)\) to the variables \((H,\varphi)\), where \(H=H(x,y)\) is the Hamiltonian of the system \((1_0)\), and the function \(\varphi=\varphi(x,y)\) is defined in the following way. Let \(G_0\) be the maximal simply connected neighborhood of the origin, entirely filled by closed trajectories of the system \((1_0)\) and containing no other equilibrium positions of this system except the origin. Let, further, \((x_0,y_0)\) be an arbitrary point of the domain \(G_0\), distinct from the origin. Define the curve \(L=\{x=\alpha(s),\ y=\beta(s)\}\) as a solution of the system

\[ d\alpha/ds=\partial H/\partial\alpha,\qquad d\beta/ds=\partial H/\partial\beta \]

with the following initial conditions: \(\alpha(0)=x_0,\ \beta(0)=y_0\). It follows from this definition that the angle of intersection of the curve \(L\) with any trajectory of the system \((1_0)\) is equal to \(\pi/2\). Hence, and from the definition of the domain \(G_0\), it follows that there exists an interval \((s_1,s_2)\) of variation of \(s\) such that, as \(s\) varies in this interval, the curve \(L\) intersects every trajectory of the system \((1_0)\) lying in the domain \(G_0\), and moreover only once. Finally, let \((x(t,s),y(t,s))\) be a solution of the system \((1_0)\) satisfying the conditions: \(x(0,s)=\alpha(s),\ y(0,s)=\beta(s)\) \((s_1<s<s_2)\). Put \(\varphi(x,y)\) equal to the interval of time during which the solution \((x(t,s),y(t,s))\), passing through the point \((x,y)\), having started at \(t=0\) from the corresponding point on the curve \(L\), reaches the point \((x,y)\). It is not difficult to see that in this way \(\varphi(x,y)\) can be defined only up to an integer multiple of the period of the solution \((x(t,s),y(t,s))\). However, as is readily verified, each branch of the function \(\varphi(x,y)\) will be an analytic function of \(x\) and \(y\) in a neighborhood of any point of the domain \(G_0\) distinct from the origin. Considering the increment of the function \(\varphi(x,y)\) under displacement of the point \((x,y)\) along a trajectory of the system \((1_0)\) during the time \(\Delta t\), it is not difficult to be convinced that

\(\varphi(x,y)\) satisfies the equation

\[ \frac{\partial H}{\partial y}\frac{\partial \varphi}{\partial x} - \frac{\partial H}{\partial x}\frac{\partial \varphi}{\partial y} =1 . \]

Hence, taking \((1_\varepsilon)\) into account, we obtain

\[ \begin{aligned} \dot H&=\varepsilon\left(H'_x f+H'_y g\right)=\varepsilon F(H,\varphi,t,\varepsilon),\\ \dot\varphi&=1+\varepsilon\left(\varphi'_x f+\varphi'_y g\right) =1+\varepsilon G(H,\varphi,t,\varepsilon), \end{aligned} \tag{2_\varepsilon} \]

where \(F(H,\varphi,t,\varepsilon)\) and \(G(H,\varphi,t,\varepsilon)\) are analytic functions of \(H,\varphi\), and \(\varepsilon\), except at the point \(H=H_0=H(0,0)\), continuous in \(t\) together with the first derivative with respect to \(t\), and periodic in \(t\) with period \(2\pi\); in the variable \(\varphi\) these functions will also be periodic with period \(T(H)\), depending on \(H\).

Let now \(H_{m,n}\ne H_0\) be such that \(T(H_{m,n})=2\pi m/n\), \(T'(H_{m,n})\ne 0\) (\(m\) and \(n\) are relatively prime integers). In system \((2_\varepsilon)\) make the substitution:
\[ H=H_{m,n}+\mu h,\quad \varphi=\frac{nt}{2m\pi}T(H)+\psi,\quad \text{where } \mu=\sqrt{\varepsilon} \]
(here and in what follows, for definiteness, we shall assume that \(\varepsilon>0\)). After the substitution we obtain

\[ dh/dt=\mu A(\psi,t)+\mu^2 P(h,\psi,t,\mu), \]

\[ d\psi/dt=\mu a h+\mu^2 Q(h,\psi,t,\mu), \]

where
\[ a=-\frac{n}{2m\pi}T'(H_{m,n})\ne 0, \]
and \(A(\psi,t)\), \(P(h,\psi,t,\mu)\), and \(Q(h,\psi,t,\mu)\) are analytic functions in the variables \(h,\psi\), and \(\mu\), continuous in \(t\) together with the first derivative with respect to \(t\), and periodic in \(t\) with period \(2m\pi\). Finally, after the substitution

\[ h=u-i\mu m\sum_{p\ne 0}\frac{1}{p}A_p(v)e^{ipt/m},\quad \psi=v,\quad \text{where }\quad A_p(v)=\frac{1}{2m\pi}\int_0^{2m\pi} A(v,t)e^{-ipt/m}\,dt, \]

we obtain the system

\[ \dot u=\mu A_0(v)+\mu^2 R(u,v,t,\mu),\quad \dot v=\mu au+\mu^2 S(u,v,t,\mu), \tag{3_\mu} \]

where \(R(u,v,t,\mu)\) and \(S(u,v,t,\mu)\) are analytic functions of \(u,v\), and \(\mu\), continuous in \(t\) together with the first derivative with respect to \(t\), and periodic in \(t\) with period \(2m\pi\).

Let us make one essential remark about the function \(A_0(v)\). By definition,

\[ A_0(v)=\frac{1}{2m\pi}\int_0^{2m\pi} \{-f(x_0(t+v),y_0(t+v),t,0)\dot y_0(t+v) \]

\[ +\,g(x_0(t+v),y_0(t+v),t,0)\dot x_0(t+v)\}\,dt, \]

where \((x_0(t),y_0(t))\) is a solution of system \((1_0)\), periodic with period \(2\pi m/n\), satisfying the condition \(H(x_0(t),y_0(t))\equiv H_{m,n}\). It follows that \(A_0(v)\) is an analytic function of \(v\), periodic in \(v\) with period \(2\pi m/n\). On the other hand, replacing under the integral sign, with the help of which \(A_0(v)\) is defined, \(t+v\) by \(t\), we obtain that \(A_0(v)\) has period \(2\pi\). Hence it follows easily that \(A_0(v)\) has period \(2\pi/n\), i.e., on the interval \([0,2\pi m/n]\) exactly \(m\) periods of the function \(A_0(v)\) are contained. In what follows we shall assume of the function \(A_0(v)\) that it has only simple zeros and, in particular, is not identically equal to zero. The case when \(A_0(v)\) does not vanish at all is elementary. In this case the following theorem holds, characterizing the behavior of the trajectories of system \((1_\varepsilon)\) in a certain neighborhood of the curve \(H(x,y)=H_{m,n}\).

Theorem 1. Let the function \(A_0(v)\), occurring in the system \((3_\mu)\), not vanish. Then there exist constants \(\mu_0>0\), \(\delta_0>0\), \(c_0>0\) such that, for any positive \(\varepsilon<\mu_0^2\), every solution of the system \((1_\varepsilon)\) which at \(t=0\) starts from an arbitrary point of the domain
\[ H_{m,n}-\mu\delta_0 < H(x,y) < H_{m,n}+\mu\delta_0, \]
for all \(t>c_0/\mu\) will lie in the domain
\[ H(x,y)<H_{m,n}-\mu\delta_0, \]
if
\[ T'(H_{m,n})\operatorname{sign} A_0(v)>0, \]
and in the domain
\[ H(x,y)>H_{m,n}+\mu\delta_0, \]
if
\[ T'(H_{m,n})\operatorname{sign} A_0(v)<0. \]

To investigate the case where the function \(A_0(v)\) has simple zeros, we shall try to find an analogy between the system \((3_\mu)\) and the system considered in note [1]. To this end, in the system \((3_\mu)\) we replace the independent variable \(t\) by \(\tau/\mu\). After the substitution we obtain the system
\[ du/d\tau=A_0(v)+\mu R(u,v,\tau/\mu,\mu),\qquad dv/d\tau=au+\mu S(u,v,\tau/\mu,\mu), \tag{4_\mu} \]
which is similar to the system considered in note [1]; however, the presence, in the arguments of the functions \(R(u,v,\tau/\mu,\mu)\) and \(S(u,v,\tau/\mu,\mu)\), of the singular factor \(1/\mu\) does not allow the results of note [1] to be applied directly to the system \((4_\mu)\). Nevertheless, we shall try to obtain for the system \((4_\mu)\) results analogous to the results of note [1], and with their help to investigate the disposition of the trajectories of the system \((1_\varepsilon)\). For this purpose, instead of the system \((4_\mu)\), we consider the system
\[ \dot u=A_0(v)+\mu R(u,v,\omega t,\mu),\qquad \dot v=au+\mu S(u,v,\omega t,\mu), \tag{5_\mu} \]
in which the frequency \(\omega\) does not depend on \(\mu\). However, we shall greatly shorten the path leading to the solution of the problem if, before passing from the system \((3_\mu)\) to the system \((5_\mu)\), we subject the system \((3_\mu)\) to two transformations.

The meaning of the first transformation is contained in the following lemma.

Lemma 1. Let \(v=v_r\) be a simple root of the equation \(A_0(v)=0\). Then there exist two functions \(p_r(t,\mu)\) and \(q_r(t,\mu)\), analytic in a certain disk \(|\mu|<\mu_0\), continuous in \(t\) together with their partial derivatives with respect to \(t\) up to and including the second order, and periodic in \(t\) with period \(2m\pi\), such that, after making in the system \((3_\mu)\) the substitution
\[ u\to u+\mu p_r(t,\mu),\qquad v\to v+\mu q_r(t,\mu), \]
we obtain a system of the same form as the system \((3_\mu)\), with the additional property that, for \(u=0\), \(v=v_r\), the right-hand sides of the resulting system vanish identically.

The second transformation is given by the following formulas:
\[ u=\xi-i\mu^2 m\sum_{p\ne 0}\frac{1}{p}R_p(\xi,\eta)e^{ipt/m}, \qquad v=\eta-i\mu^2 m\sum_{p\ne 0}\frac{1}{p}S_p(\xi,\eta)e^{ipt/m}, \]
where
\[ R_p(\xi,\eta)=\frac{1}{2m\pi}\int_0^{2m\pi} R(\xi,\eta,t,0)e^{-ipt/m}\,dt, \]
\[ S_p(\xi,\eta)=\frac{1}{2m\pi}\int_0^{2m\pi} S(\xi,\eta,t,0)e^{-ipt/m}\,dt. \]

After carrying out the transformations described, the system \((3_\mu)\) will take the form
\[ \dot\xi=\mu A_0(\eta)+\mu^2 R_0(\xi,\eta)+\mu^3\widetilde R(\xi,\eta,t,\mu), \]
\[ \dot\eta=\mu a\xi+\mu^2 S_0(\xi,\eta)+\mu^3\widetilde S(\xi,\eta,t,\mu), \]
where
\[ A_0(v_r)=R_0(0,v_r)=S_0(0,v_r)=0,\qquad \widetilde R(0,v_r,t,\mu)=\widetilde S(0,v_r,t,\mu)=0. \]
Thus, we have arrived at the system
\[ \dot\xi=A_0(\eta)+\mu R_0(\xi,\eta)+\mu^2\widetilde R(\xi,\eta,\omega t,\mu), \]
\[ \dot\eta=a\xi+\mu S_0(\xi,\eta)+\mu^2\widetilde S(\xi,\eta,\omega t,\mu), \tag{6_\mu} \]

where the functions $\widetilde R(\xi,\eta,\omega t,\mu)$ and $\widetilde S(\xi,\eta,\omega t,\mu)$ are periodic in $\theta=\omega t$ with period $2m\pi$. For this system the following two theorems are valid.

Theorem 2. Let $\eta=v_r$ be a root of the equation $A_0(\eta)=0$ such that $aA_0'(v_r)>0$. Then there exist $\mu_0>0$, $\delta_0>0$, $c_0>0$ such that, for any complex $\mu$, $\xi_0$ satisfying the inequalities $|\mu|<\mu_0$, $|\xi_0|<\delta_0$, for any real $\omega$ satisfying the inequality $-c_0/|\mu|^2<\omega<c_0/|\mu|^2$, and for any $t_0$, there exists a unique solution $(\xi_\mu(t),\eta_\mu(t))$ of system $(6_\mu)$ such that $\xi_\mu(t_0)=\xi_0$; for all $t\geqslant t_0$

\[ \frac{d}{dt}|\xi_\mu(t)|<0,\qquad \frac{d}{dt}|\eta_\mu(t)-v_r|<0;\qquad |\xi_\mu(t)|+|\eta_\mu(t)-v_r|\to 0 \quad \text{as } t\to\infty . \]

Theorem 3. Let $\eta=v_r$ be a root of the equation $A_0(\eta)=0$ such that $aA_0'(v_r)>0$. Then there exist $\mu_1>0$, $\delta_1>0$ and $c_1>0$ ($\mu_1\leqslant\mu_0$, $\delta_1\leqslant\delta_0$, $c_1\leqslant c_0$) such that, for any complex $\mu$ and $\xi_0$ satisfying the inequalities $|\mu|<\mu_1$, $|\xi_0|<\delta_1$, for any real $\omega$ satisfying the inequality $-c_1/|\mu|^2<\omega<c_1/|\mu|^2$, and for any $t_0$, the solution $(\xi_\mu(t),\eta_\mu(t))$ of system $(6_\mu)$ satisfying the conditions: $\xi_\mu(t_0)=\xi_0$, for all $t\geqslant t_0$

\[ \frac{d}{dt}|\xi_\mu(t)|<0 \quad \text{and} \quad \frac{d}{dt}|\eta_\mu(t)-v_r|<0, \qquad |\xi_\mu(t)|+|\eta_\mu(t)-v_r|\to 0 \quad \text{as } t\to\infty , \]

possesses continuous partial derivatives with respect to $\sigma=\operatorname{Re}\mu$ and $\nu=\operatorname{Im}\mu$, which satisfy the Cauchy—Riemann conditions for all $t\geqslant t_0$, $\sigma^2+\nu^2<\mu_1^2$ and $-c_1/|\mu|^2<\omega<c_1/|\mu|^2$.

On the basis of Theorem 3 we can find the boundary trajectories of system $(6_\mu)$ in the form of series in powers of the parameter $\mu$, which will converge for all $|\mu|<\mu_1$ and $-c_1/|\mu|^2<\omega<c_1/|\mu|^2$. Setting $\omega=1/\mu$ ($\mu>0$) in the obtained series, we obtain series which will converge for all real $\mu$ satisfying the inequality $0<\mu<\min(\mu_1,c_1)$. Analysis of the series obtained makes it possible to reveal a number of interesting regularities concerning the behavior of the trajectories of system $(1_\varepsilon)$.

Thus, for example, it can be shown that, under certain conditions, in the domain $G_0$ there will exist families consisting of a finite number of pairwise nonintersecting domains $G_k$ $(k=1,\ldots,m)$ such that any solution of system $(1_\varepsilon)$ that leaves the domain $G_m$ at $t=t_0$ enters the domain $G_k$ at $t=t_0+2k\pi$ $(0<k\leqslant m)$, and, for sufficiently small $\varepsilon\ne0$, such families are possible with an arbitrarily large number $m$.

In conclusion, we note that the series obtained for the boundary trajectories of system $(6_\mu)$ for $\omega=1/\mu$ have a rather complicated structure. In these series there will necessarily be terms of the form $e^{-\alpha/\mu}$ $(\alpha>0,\ \mu>0)$. This, in particular, means that the widely known method of averaging$^{(2)}$ cannot be adapted to the determination of the boundary trajectories of system $(6_\mu)$ and, consequently, to the analysis of system $(1_\varepsilon)$.

Received
26 VIII 1961

REFERENCES

1. V. Melnikov, DAN, 139, No. 1, 31 (1961).
2. N. N. Bogolyubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1958.