UDC 517.916.925

ON THE THEORY OF CANONICAL SYSTEMS

N. P. ERUGIN

§ 1. Consider a system of two Hamilton equations:

\[ \dot{x}=\frac{dx}{dt}=-\frac{\partial H_0(x,y)}{\partial y},\qquad \dot{y}=\frac{dy}{dt}=\frac{\partial H_0(x,y)}{\partial x}, \tag{1.1} \]

where

\[ H_0(x,y)=\frac{\lambda}{2}(x^2+y^2)+P(x,y), \tag{1.2} \]

\[ P(x,y)=\sum_{m=3}^{\infty} P_m(x,y), \tag{1.3} \]

\(P_m(x,y)\) is a homogeneous polynomial of degree \(m\), and this series converges in a neighborhood of the origin. System (1.1) has the integral

\[ H_0(x,y)=c^2, \tag{1.4} \]

from which we see that the point \((0,0)\) will be a center, i.e., near this point we have only closed integral curves—periodic solutions—which we obtain from

\[ H_0(r\cos\vartheta,\ r\sin\vartheta)=c^2 \tag{1.5} \]

in the form

\[ r=\varphi(\sin\vartheta,\ \cos\vartheta,\ c) \tag{1.6} \]

or

\[ r=\sum_{k=1}^{\infty} r_k(\sin\vartheta,\ \cos\vartheta)c^k, \tag{1.7} \]

where this series converges for \(|c|<\Delta\). This was shown by Poincaré and Lyapunov. It is known that if we have a nonlinear system of differential equations whose first approximation (linear system) has negative characteristic numbers (c.n.) in the sense of Lyapunov, then the zero solution of this system will be stable, no matter how one chooses the coefficients of the nonlinear terms of the series representing the right-hand sides. But if there is even one zero c.n., then the question of the stability of the zero solution is decided by the choice of the coefficients of the nonlinear terms. Lyapunov called these cases critical. The question of critical cases is different if we consider the Hamiltonian system (1.1). Indeed, no matter how we change the coefficients of the series \(P(x,y)\) in system (1.1), the point \((0,0)\) will always be a center, as is seen from (1.4), although the linear system corresponding to system (1.1),

\[ \dot{x}=-\lambda y,\qquad \dot{y}=\lambda x, \tag{1.8} \]

has purely imaginary characteristic roots: \(\lambda i,\ -\lambda i\), i.e., zero characteristic numbers in the sense of Lyapunov.

Thus the qualitative picture—a center about the origin for system (1.1)—is rough from the point of view of perturbations of the coefficients of the nonlinear terms. And the point \((0,0)\) is not asymptotically stable. The equilibrium point \((0,0)\) cannot be asymptotically stable for system (1.1), no matter how the quadratic terms of the function \(H_0\) are chosen. Indeed, if the quadratic terms of the function \(H_0\) are such that the first approximation of system (1.1) has a negative characteristic number, then the second characteristic number, as Poincaré and Lyapunov showed, will be positive\(^1\). Then the zero solution will be stable neither as \(t \to \infty\) nor as \(t \to -\infty\). But if both characteristic numbers are zero, i.e., the first approximation has the form (1.6), then we have a center.

We now pose the question of the stability of the zero solution of system (1.1) differently. If the characteristic numbers of the first approximation are negative, then, as is known, the zero solution will be stable whatever the nonlinear parts of the right-hand sides may be, provided these nonlinear parts are infinitesimals of order higher than \(r=\sqrt{x^2+y^2}\). These nonlinear parts may also be nonholomorphic in a neighborhood of the point \((0,0)\). If, however, we have the doubtful case, then, as we have already said above, even with holomorphic right-hand sides the question of the stability of the equilibrium point \(x=y=0\) depends on the choice of the nonlinear terms. If we have a Hamiltonian system, where
\[ H=\frac{\lambda}{2}(x^2+y^2)+P(x,y), \]
then the stability of the equilibrium point \(x=y=0\) is not destroyed for any choice of a holomorphic \(P(x,y)\) containing no terms below the third degree. We now pose the question of the stability of the equilibrium point \(x=y=0\) in the case where in system (1.1) nonholomorphic \(P(x,y)\) are allowed, having order of smallness in a neighborhood of the point \(x=y=0\) higher than \(r=\sqrt{x^2+y^2}\).

First we consider an example of N. I. Gavrilov [6]
\[ \dot{x}=-\frac{\partial H}{\partial y},\qquad \dot{y}=\frac{\partial H}{\partial x}, \tag{1.9} \]
\[ H=\sin\left(\varphi+\ln\frac{r}{1-r}\right),\qquad x=r\cos\varphi,\quad y=r\sin\varphi . \tag{1.10} \]

We have the integral \(H=c\), or
\[ \varphi+\ln\frac{r}{1-r}=\sigma=\mathrm{const},\qquad r=\frac{e^{\sigma-\varphi}}{1+e^{\sigma-\varphi}} . \tag{1.11} \]

These are spirals: \(r\to 0\) as \(\varphi\to\infty\), and \(r\to 1\) as \(\varphi\to-\infty\). System (1.9), in expanded form, is:
\[ \dot{x}=-\cos\left(\varphi+\ln\frac{r}{1-r}\right) \left[ \frac{x(1-\sqrt{x^2+y^2})+y}{1-\sqrt{x^2+y^2}} \right]\frac{1}{x^2+y^2}, \]
\[ \dot{y}=\cos\left(\varphi+\ln\frac{r}{1-r}\right) \left[ \frac{x-y(1-\sqrt{x^2+y^2})}{1-\sqrt{x^2+y^2}} \right]\frac{1}{x^2+y^2}. \tag{1.12} \]

Here the right-hand sides are undefined at the point \(x=y=0\) and have no limit as \(x^2+y^2\to0\). In a neighborhood of each point \((x_0,y_0)\), different from the ori—

\(^1\) For system (1.1) this is easy to see if \(H=ax^2+2bxy+cy^2+P(x,y)\).

... of the origin and of the points \(x^2+y^2=1\), the right-hand sides are holomorphic. But in a neighborhood of the origin the right-hand sides are not representable in the form of a series in positive powers of \(x\) and \(y\), and the point \(x=y=0\) is not an equilibrium point. However, the discrete set of spirals (1.11) consists of equilibrium points:

\[ \varphi+\ln\frac{r}{1-r}=(2k+1)\frac{\pi}{2},\quad k=0,\ \pm 1,\ \pm 2,\ldots \tag{1.13} \]

Introduce into (1.12) a new independent variable:

\[ d\tau=\cos\left(\varphi+\ln\frac{r}{1-r}\right)\frac{1}{(1-r)r^2}\,dt. \]

Then the system (1.12) passes into the system

\[ \frac{dx}{d\tau}=-x-y+x\sqrt{x^2+y^2},\quad \frac{dy}{d\tau}=x-y+y\sqrt{x^2+y^2}. \tag{1.14} \]

Here the right-hand sides are defined and continuous in a neighborhood of the point \((0,0)\), and the uniqueness theorem is satisfied. But this system is no longer a Hamiltonian system. It is easy to see\(^1\) that for the system (1.12), between any two neighboring spirals (1.13), alternately one has \(\varphi\to-\infty,\ r\to 1\) as \(t\to t_0\), and \(\varphi\to+\infty,\ r\to 0\) as \(t\to t_1\). This qualitative picture is determined by the system (1.9). For the system (1.14), however, the point \((0,0)\) will be an asymptotically stable equilibrium point, since the characteristic roots of the coefficient matrix corresponding to the linear system are
\(\lambda_1=-1+i,\ \lambda_2=-1-i\).

If we take

\[ H=\varphi+\ln\frac{r}{1-r}, \tag{1.15} \]

then the Hamiltonian system will be\(^2\)

\[ \dot{x}=-\frac{x+y-x\sqrt{x^2+y^2}}{1-\sqrt{x^2+y^2}}\,\frac{1}{x^2+y^2}, \]

\[ \dot{y}=\frac{x-y+y\sqrt{x^2+y^2}}{1-\sqrt{x^2+y^2}}\,\frac{1}{x^2+y^2}. \tag{1.16} \]

If we put here

\[ d\tau=\frac{dt}{(1-r)r^2}, \tag{1.17} \]

then we again obtain the system (1.14).

In all these examples either the point \((0,0)\) is not an equilibrium point, and in a neighborhood of this point the existence and uniqueness theorem is not satisfied—

\(^1\) In polar coordinates the system (1.12) is written as

\[ \dot{\varphi}=\frac{\cos\left(\varphi+\ln\frac{r}{1-r}\right)}{(1-r)r^2},\quad \dot{r}=-\cos\left(\varphi+\ln\frac{r}{1-r}\right). \]

\(^2\) And in polar coordinates
\(\dot{r}=-1,\quad \dot{\varphi}=\dfrac{1}{r^2(1-r)}\), whence we see that \(r\to0\) as \(t\to t_0\), where \(r_0\) is the initial value of \(r\) at \(t=0\).

...of stability, or this system is not a Hamiltonian system (system (1.14)).

For the system

\[ \dot{x}=-\frac{\partial H}{\partial y}, \qquad \dot{y}=\frac{\partial H}{\partial x}; \qquad H=\frac{\lambda}{2}(x^2+y^2)+a(x^2+y^2)^2+ \]

\[ +P(x,y,t) \tag{1.18} \]

with constant \(\lambda\) and \(a\), an integral will be\(^1\)

\[ a(x^2+y^2)^2+P(x,y,t)=c, \tag{1.19} \]

if

\[ P=\Phi(x^2+y^2,\ x\sin\lambda t-y\cos\lambda t,\ x\cos\lambda t+y\sin\lambda t). \]

It follows from this that if \(\Phi(u,v,w)\) is holomorphic in a neighborhood of the point \(u=v=w=0\), \(a\ne0\), and \(P\cdot(x^2+y^2)^{-2}\to0\) as \(x^2+y^2\to0\), then the equilibrium point \(x=y=0\) of system (1.18) is non-asymptotically stable. But if \(a=0\), it is easy to indicate such holomorphic functions \(\Phi\) for which the point \(x=y=0\) for system (1.18) will be unstable. For example, this will be so for

\[ \Phi(u,v,w)=v^3+w^3. \]

§ 2. We shall consider the following problem. Suppose a Hamiltonian system (1.1) is given. Let us introduce the perturbed system

\[ H=H_0(x,y)+P(x,y,t), \qquad P(0,0,t)=0, \tag{2.1} \]

\[ \dot{x}=-\frac{\partial H_0}{\partial y}-\frac{\partial P}{\partial y}, \qquad \dot{y}=\frac{\partial H_0}{\partial x}+\frac{\partial P}{\partial x}. \tag{2.2} \]

We shall show that there exist perturbations \(P(x,y,t)\) for which the equilibrium point \((0,0)\) of system (2.2) will be stable, asymptotically stable, or unstable.

Let us first note that if the perturbed system is not Hamiltonian, then all these three cases are possible, when the perturbation is holomorphic with respect to \(x,y\) in a neighborhood of the origin of coordinates. This follows from the general theory developed by Lyapunov. This is also seen from the example

\[ \dot{x}=-\frac{\partial H_0}{\partial y}+\mu H_0^n x\varphi(t), \qquad \dot{y}=\frac{\partial H_0}{\partial x}+\mu H_0^n y\varphi(t), \tag{2.3} \]

since, by virtue of these equations, we have

\[ \frac{dH_0}{dt} = \frac{\partial H_0}{\partial x}\dot{x} + \frac{\partial H_0}{\partial y}\dot{y} = \mu\varphi(t)H_0^n \left[ x\frac{\partial H_0}{\partial x} + y\frac{\partial H_0}{\partial y} \right]. \tag{2.4} \]

Taking into account the value (1.2) of the function \(H_0\), we obtain

\[ x\frac{\partial H_0}{\partial x} + y\frac{\partial H_0}{\partial y} = \lambda r^2+ \sum_{m=3}^{\infty} mr^m P_m(\cos\vartheta,\sin\vartheta), \tag{2.5} \]

\(x=r\cos\vartheta,\ y=r\sin\vartheta\). Here \(\mu>0\) is a parameter. Now from (2.4) we see that for \(\varphi(t)>0\) the equilibrium point \(x=y=0\) will be unstable, while for \(\varphi(t)<0\) it will be asymptotically stable. But system (2.3) is not canonical.

\[ \rule{8em}{0.4pt} \]

\(^1\) It is easy, of course, to indicate also the most general form of such a function \(P\).

Now consider the canonical system (2.2), where

\[ P(x,y,t)=\mu H_0^n p(x,y)\varphi(t). \tag{2.6} \]

It is easy to see that for such a system we obtain

\[ \frac{dH_0}{dt} = \frac{\partial H_0}{\partial x}\dot{x} + \frac{\partial H_0}{\partial y}\dot{y} = \mu H_0^n \left[ \frac{\partial H_0}{\partial y}\frac{\partial p}{\partial x} - \frac{\partial H_0}{\partial x}\frac{\partial p}{\partial y} \right]\varphi(t). \tag{2.7} \]

Let us find a function \(p(x,y)\) such that

\[ \frac{\partial H_0}{\partial y}\frac{\partial p}{\partial x} - \frac{\partial H_0}{\partial x}\frac{\partial p}{\partial y} = q(x,y)\geq 0. \tag{2.8} \]

The corresponding system of ordinary equations has the form

\[ \frac{dx}{\dfrac{\partial H_0}{\partial y}} = \frac{dy}{-\dfrac{\partial H_0}{\partial x}} = \frac{dp}{q(x,y)}. \tag{2.9} \]

We cannot find a holomorphic function \(p(x,y)\) satisfying equation (2.8), since then, according to (2.7), for constant \(\varphi(t)<0\) the equilibrium point of the system (2.2) would be asymptotically stable, which is impossible for a holomorphic \(H\) given by equality (2.1). If we seek such a function \(p\) that

\[ \frac{\partial p}{\partial y}=-M(x,y)x,\qquad \frac{\partial p}{\partial x}=M(x,y)y, \tag{2.10} \]

then we find

\[ p(x,y)=\Phi\left(\frac{y}{x}\right),\qquad M=-\frac{1}{x^2}\Phi'\left(\frac{y}{x}\right), \tag{2.11} \]

\[ \frac{\partial H_0}{\partial y}\frac{\partial p}{\partial x} - \frac{\partial H_0}{\partial x}\frac{\partial p}{\partial y} = -\frac{1}{x^2}\Phi'\left(\frac{y}{x}\right) \left[ \frac{\partial H_0}{\partial x}x + \frac{\partial H_0}{\partial y}y \right]. \tag{2.11\(_1\)} \]

To satisfy condition (2.8), one must choose such a \(\Phi\left(\dfrac{y}{x}\right)\) that

\[ -\frac{1}{x^2}\Phi'\left(\frac{y}{x}\right)>0. \]

Take the function

\[ \Phi\left(\frac{y}{x}\right)=\left(1+e^{\frac{y}{x}}\right)^{-1}, \]

which satisfies this condition. Then (2.6) takes the form

\[ P(x,y,t)=\mu H_0^n\left(1+e^{\frac{y}{x}}\right)^{-1}\varphi(t),\qquad P(0,0,t)=0 \tag{2.12} \]

and (2.7)

\[ \frac{dH_0}{dt} = \frac{\mu}{x^2} \left(1+e^{\frac{y}{x}}\right)^{-2} e^{\frac{y}{x}} H_0^n \left[ x\frac{\partial H_0}{\partial x} + y\frac{\partial H_0}{\partial y} \right]\varphi(t). \tag{2.13} \]

Let us now write equations (2.2):

\[ \dot{x} = -\frac{\partial H_0}{\partial y} - \mu n H_0^{\,n-1}\frac{\partial H_0}{\partial y} \left(1+e^{\frac{y}{x}}\right)^{-1}\varphi(t) + \]

\[ +\frac{\mu}{x} H_0^n \left(1+e^{\frac{y}{x}}\right)^{-2} e^{\frac{y}{x}} \varphi(t), \]

\[ \dot{y}=\frac{\partial H_0}{\partial x} +\mu n H_0^{n-1}\frac{\partial H_0}{\partial x} \left(1+e^{\frac{y}{x}}\right)^{-1}\varphi(t)- \]

\[ -\frac{\mu y}{x^2}H_0^n\left(1+e^{\frac{y}{x}}\right)^{-2} e^{\frac{y}{x}}\varphi(t). \tag{2.14} \]

We see that both the perturbation (2.12) and the perturbations in equations (2.14) are bounded in any disk with center at the origin and are arbitrarily small for sufficiently small values of the parameter \(\mu\). They are infinitely small as \(x^2+y^2\to 0\), of order \(H_0^n\). But both the perturbation (2.12) and the second terms in equations (2.14) are discontinuous at points of the \(y\)-axis. The right-hand sides of equations (2.14) have \(n-2\) bounded (but discontinuous at points of the \(y\)-axis) partial derivatives. This ensures uniqueness of solutions at every point. At points of the \(y\)-axis we take as the values of the right-hand sides of equations (2.14) their limits from the right for \(y>0\) and from the left for \(y<0\). Thus,

\[ \left.\dot{x}\right|_{x=0} =-\frac{\partial H_0(0,y)}{\partial y} =-\lambda y+\ldots \begin{cases} <0 & \text{for } y>0 \text{ and small},\\ >0 & \text{for } y<0, \end{cases} \]

\[ \left.\dot{y}\right|_{x=0} =\frac{\partial H_0(0,y)}{\partial x}. \]

The limits of the right-hand sides of equations (2.14) as \(x^2+y^2\to 0\) are equal to zero; therefore, for \(x=y=0\) we set the right-hand sides of equations (2.14) equal to zero. From (2.13) we see [1] that for \(\varphi(t+2\pi)=\varphi(t)<0\) the equilibrium point \((0,0)\) will be asymptotically stable, and for \(\varphi(t)>0\) unstable.

Remark. If we take as the perturbation

\[ P(x,y,t)=\mu e^{-\frac{1}{H_0(x,y)}}\left(1+e^{\frac{y}{x}}\right)^{-1}\varphi(t), \]

then we obtain system (2.2), whose perturbations together with all partial derivatives will be small of order higher than any positive power of \(x^2+y^2\) as \(x^2+y^2\to 0\). And instead of (2.13) we obtain

\[ \frac{dH_0}{dt} =\frac{\mu}{x^2}\left(1+e^{\frac{y}{x}}\right)^{-2} e^{\frac{y}{x}}e^{-\frac{1}{H_0(x,y)}} \left[ x\frac{\partial H_0}{\partial x} +y\frac{\partial H_0}{\partial y} \right]\varphi(t), \]

whence we obtain the same results as for system (2.14).

Now take

\[ P(x,y,t)=\delta \mu H_0^n(x,y)\arcsin \frac{x}{\sqrt{x^2+y^2}}\varphi(t), \tag{2.15} \]

where

\[ P(0,0,t)=0,\qquad \arcsin z \text{ is the principal value}, \]

\[ \delta= \begin{cases} 1, & y>0,\\ -1, & y\le 0, \end{cases} \quad x>0; \qquad \delta= \begin{cases} 1, & y\ge 0,\\ -1, & y<0, \end{cases} \quad x<0. \tag{2.16} \]

The system (2.2) will have the form\(^1\)

\[ \begin{aligned} \dot{x}={}&-\frac{\partial H_0}{\partial y} -\delta n\mu H_0^{\,n-1}(x,y)\frac{\partial H_0}{\partial y} \arcsin \frac{x}{\sqrt{x^2+y^2}}\,\varphi(t) \\ &+\mu H_0^{\,n}(x,y)\frac{x}{x^2+y^2}\,\varphi(t), \end{aligned} \tag{2.17} \]

\[ \begin{aligned} \dot{y}={}&\frac{\partial H_0}{\partial x} +\delta n\mu H_0^{\,n-1}\frac{\partial H_0}{\partial x} \arcsin \frac{x}{\sqrt{x^2+y^2}}\,\varphi(t) \\ &+\mu H_0^{\,n}\frac{y}{x^2+y^2}\,\varphi(t). \end{aligned} \]

By virtue of these equations we obtain

\[ \frac{dH_0}{dt} =\mu H_0^{\,n}\left[\lambda+\sum_{k=3}^{\infty} k r^{k-2}P_k(\cos\vartheta,\sin\vartheta)\right]\varphi(t). \tag{2.18} \]

As is seen from (2.15), the perturbation \(P(x,y,t)\) has a finite jump at the points of the \(x\)-axis (owing to the presence of the factor \(\delta\)), equal to \(\pi\mu H_0^{\,n}(x,0)\). The second terms in the right-hand sides of equations (2.17) also have a finite jump tending to zero together with \(x\). But the perturbations of the right-hand sides, together with \(n-1\) derivatives, are bounded and tend to zero as \(x^2+y^2\to 0\). At points of the \(x\)-axis we take as the values of the right-hand sides their limits as \(y\to -0\), when \(x>0\), and as \(y\to +0\), when \(x<0\). Thus,

\[ \begin{aligned} \left.\dot{x}\right|_{y=0,\ x>0} ={}&-\left.\frac{\partial H_0}{\partial y}\right|_{y=0} +n\mu H_0^{\,n-1}(x,y)\times \\ &\times\left. \frac{\partial H_0(x,y)}{\partial y}\,\frac{\pi}{2}\,\varphi(t) \right|_{y=0} +\mu H_0^{\,n}(x,0)\varphi(t), \end{aligned} \]

\[ \begin{aligned} \left.\dot{y}\right|_{y=0,\ x>0} ={}&\left.\frac{\partial H_0(x,y)}{\partial x}\right|_{y=0} -n\mu H_0^{\,n-1}(x,y)\times \\ &\times\left. \frac{\partial H_0(x,y)}{\partial x}\,\frac{\pi}{2}\,\varphi(t) \right|_{y=0}, \end{aligned} \]

\[ \begin{aligned} \left.\dot{x}\right|_{y=0,\ x<0} ={}&-\left.\frac{\partial H_0(x,y)}{\partial y}\right|_{y=0} +n\mu H_0^{\,n-1}(x,y)\times \\ &\times\left. \frac{\partial H_0(x,y)}{\partial y}\,\frac{\pi}{2}\,\varphi(t) \right|_{y=0} +\mu H_0^{\,n}(x,0)\varphi(t), \end{aligned} \]

\[ \left.\dot{y}\right|_{y=0,\ x<0} = \frac{\partial H_0(x,0)}{\partial x} -\left. n\mu H_0^{\,n-1}(x,y)\frac{\partial H_0(x,y)}{\partial x}\, \frac{\pi}{2}\,\varphi(t) \right|_{y=0}. \]

From (2.18), as before, we conclude that for \(\varphi(t+2\pi)=\varphi(t)<0\) the equilibrium point \(x=y=0\) is asymptotically stable, while for \(\varphi(t)>0\) it is unstable.

Remark. If instead of (2.15) we take

\(^1\) When differentiating \(\arcsin \dfrac{x}{\sqrt{x^2+y^2}}\), \(\delta\) appears; therefore in the last terms there will be the factor \(\delta^2=1\).

\[ P(x,y,t)=\delta\mu e^{-\frac{1}{H_0(x,y)}}\arcsin\frac{x}{\sqrt{x^2+y^2}}\,\varphi(t), \]

then the perturbations of the right-hand sides of equations (2.2), as \(x^2+y^2\to 0\), will be small of higher order than any positive power of the quantity \(x^2+y^2\), and we shall arrive at the same conclusions concerning the stability of the equilibrium point \(x=y=0\) as in the case (2.15).

If, instead of (1.2), we take

\[ H_0(x,y)=\frac{\lambda}{2}(x^2+y^2)+P_m(x^2+y^2), \]

where \(P_m(z)\) is a polynomial of degree \(m\), and instead of (2.15)

\[ P(x,y,t)=\delta\mu(x^2+y^2)^n\arcsin\frac{x}{\sqrt{x^2+y^2}}\,\frac{\varphi(t)}{2}, \tag{2.19} \]

where the integer \(n>m\), then instead of equality (2.18) we obtain

\[ \frac{d(x^2+y^2)}{dt}=\mu(x^2+y^2)^n\varphi(t). \]

Hence we find

\[ (x^2+y^2)^{n-1} = \frac{1}{(x_0^2+y_0^2)^{1-n}-(n-1)\mu\displaystyle\int_{t_0}^{t}\varphi(t)\,dt}, \]

\[ x(t_0)=x_0,\qquad y(t_0)=y_0. \]

In addition to the preceding conclusions concerning the stability of the zero solution \(x=y=0\), we see that here, for all sufficiently small \(x_0^2+y_0^2\) and when

\[ \int_0^{2\pi}\varphi(t)\,dt=0, \]

the solutions will be periodic with period \(2\pi\). This follows from the fact that in this case the function

\[ \int_{t_0}^{t}\varphi(t)\,dt \]

is periodic with period \(2\pi\).

We have proved the theorem.

For every Hamiltonian system (1.1) there exists a perturbation \(P(x,y,t)\) such that the equilibrium point \(x=y=0\) of the Hamiltonian system (2.2) will be asymptotically stable or unstable. Here \(P(x,y,t)\) may or may not depend on \(t\). But \(P(x,y,t)\), or its partial derivatives, will have a finite discontinuity along some curve, although, in general, both \(P(x,y,t)\) and its partial derivatives may be small of order higher than any positive power of the quantity \(x^2+y^2\) as \(x^2+y^2\to 0\).

Let us make a few remarks.

1. In our construction the indicated discontinuities occurred along straight lines, but it is not difficult to prove that these discontinuities may also occur along curves of a fairly broad class.

2. One may ask: is such a perturbation \(P(x,y,t)\) possible which, together with its first derivatives, would be continuous in a neighborhood of the origin of coordinates? In our construction of the perturbations \(P(x,y,t)\) this is impossible, since \(\Phi(z)\) in (2.11) must be discontinuous, satisfy the equality

\[ \lim_{z\to+\infty}\Phi(z)=\lim_{z\to-\infty}\Phi(z), \]

and have a continuous sign-constant derivative, which is impossible.

1. There also exists a perturbation \(P(x,y,t)\) under which all solutions in a neighborhood of the origin will be periodic with period \(2\pi\). This is shown for the example of a system in which \(P(x,y,t)\) is taken in the form (2.19). Here, as in the general case, the perturbation and the right-hand sides of the perturbed Hamiltonian system have a finite discontinuity along the \(x\)-axis. In the next section we shall obtain this last case of periodic solutions of the perturbed system for an arbitrary Hamiltonian system (1.1), considering perturbations of another nature and holomorphic ones.

§ 3. Consider the Hamiltonian system (1.1) and the perturbed system of the form1

\[ \dot{x}=-\frac{\partial H_0}{\partial y}\{1+\Phi(H_0,t)\},\qquad \dot{y}=\frac{\partial H_0}{\partial x}\{1+\Phi(H_0,t)\}, \tag{3.1} \]

where \(H_0\) is the function (1.2); \(\Phi(z,t)\) is a continuously differentiable function in \(z\) and \(t\), periodic with period \(\omega\) in \(t\), and

\[ \frac{|\Phi(z,t)|}{|z|}\to 0 \]

as \(|z|\to 0\).

The system (3.1) is canonical with

\[ H(z,t)=z+\int_0^z \Phi(z,t)\,dz,\qquad z=H_0 \tag{3.2} \]

and has the integral

\[ H_0(x,y)=c^2, \tag{3.3} \]

therefore the equilibrium point \(x=y=0\) of the system (3.1) will be stable (non-asymptotically), and the integral curves in a neighborhood of the origin will be closed (but the solutions need not be periodic).

We now consider a special case of the system (3.1), putting in (1.2)

\[ P(x,y)=a^2x^4+b^2y^4,\quad a\text{ and }b\text{ are constants}, \tag{3.4} \]

and

\[ \Phi(z,t)=\sum_{k\geq 1}\Phi_k(z)\varphi_k(t),\qquad \varphi_k(t+2\pi)=\varphi_k(t), \tag{3.5} \]

where \(k\) assumes a finite or infinite set of integer values,2 and the \(\Phi_k(z)\) are polynomials or series having no constant terms. The integral (3.3) now has the form

\[ z=\frac{\lambda}{2}(x^2+y^2)+a^2x^4+b^2y^4=c^2. \tag{3.6} \]

Hence we see that all integral curves of the system (3.1) will be closed.

Let us consider, for this system, the question of the existence of periodic solutions.

Putting in (3.6) \(x=r\cos\vartheta,\ y=r\sin\vartheta\), we obtain

\[ \frac{\lambda}{2}r^2+r^4(a^2\cos^4\vartheta+b^2\sin^4\vartheta)=c^2. \tag{3.7} \]

For \(\vartheta\) we obtain the differential equation

\[ \dot{\vartheta}=\left[\lambda+4r^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)\right] \left[1+\sum_{k\geqslant 1}\Phi_k(c^2)\varphi_k(t)\right]. \]

The positive value of \(r^2\) from (3.7) we find in the form

\[ 4\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)r^2 = \sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)}-\lambda . \]

Substituting this into the preceding equality, we obtain

\[ \dot{\vartheta} = \sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)} \left[1+\sum_{k\geqslant 1}\Phi_k(c^2)\varphi_k(t)\right], \]

whence

\[ \int_0^\vartheta \frac{d\vartheta} {\sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)}} = t+\sum_{k\geqslant 1}\Phi_k(c^2)\int_0^t\varphi_k(t)\,dt . \tag{3.8} \]

From this we see that if \(\Phi_k\equiv 0\) \((k=1,2,\ldots)\), i.e. if instead of the system (3.1) we have the unperturbed system, then (3.8) takes the form

\[ \int_0^\vartheta \frac{d\vartheta} {\sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)}} = t . \tag{3.9} \]

Here the function \(\vartheta=\vartheta(t)\) has the property

\[ \vartheta(t+T)=\vartheta(t)+2\pi, \tag{3.10} \]

where

\[ T=\int_0^{2\pi} \frac{d\vartheta} {\sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)}} . \tag{3.11} \]

Therefore the functions \(\sin\vartheta(t)\), \(\cos\vartheta(t)\) are periodic with period \(t=T\). And then, according to (3.7), \(r=r(t)\) is periodic with period \(T\). It follows that all closed integral curves (3.7) correspond to periodic solutions with period \(T\), depending on \(c^2\). What can be said about the function \(\vartheta=\vartheta(t)\) defined by equality (3.8)? Can it have property (3.10)?

Let us first turn to the system (3.1), where \(\Phi(z,t)\) is given by equality (3.5). Can this system have a periodic solution with period \(b\)? Suppose such a solution exists, i.e.

\[ x(t+b)=x(t),\qquad \dot{x}(t+b)=\dot{x}(t), \]

\[ y(t+b)=y(t),\qquad \dot{y}(t+b)=\dot{y}(t). \]

Then from (3.1) we have\(^1\)

\[ \left(\lambda y+\frac{\partial P}{\partial y}\right) \sum \Phi_k(c^2)\,[\varphi_k(t+b)-\varphi_k(t)]=0, \]

\[ \left(\lambda x+\frac{\partial P}{\partial x}\right) \sum \Phi_k(c^2)\,[\varphi_k(t+b)-\varphi_k(t)]=0. \tag{3.12} \]

\(^1\) We repeat the argument carried out in [2].

These equalities can hold under the following circumstances:

\[ \text{I.}\quad \lambda y+\frac{\partial P}{\partial y}=0,\qquad \lambda x+\frac{\partial P}{\partial x}=0, \]

\[ \text{II.}\quad \sum_{k\geqslant 1}\Phi_k(c^2)\,[\varphi_k(t+b)-\varphi_k(t)]=0. \]

Taking into account the value of \(P\) given by equality (3.4), from I we obtain the trivial solution \(x=0,\ y=0\). When, then, can equality II hold? In this equality we have replaced the argument \(z\) by \(c^2\), since on the integral curves we have (3.7). Equality II can hold under two circumstances:

\[ \varphi_k(t+b)-\varphi_k(t)=0, \tag{3.13} \]

\[ \Phi_k(c^2)=0\quad (k=1,\,2,\ldots). \tag{3.14} \]

If

\[ b=\omega m,\quad m\text{ an integer}, \tag{3.15} \]

then the equalities (3.13) are satisfied, since the \(\varphi_k(t)\) have period \(\omega\). If, however, (3.13) does not hold, then (3.14) must hold. And this means that \(p=c^2\) is a common root of all the functions \(\Phi_k(z)\), or of \(\Phi_1(z)\), if
\[ \sum \Phi_k(z)\varphi_k(t)=\Phi_1(c^2)\varphi_1(t), \]
i.e., if there is only one term in the sum.

Now we turn again to equality (3.8) and see that if \(c_k^2\) \((k=1,2,\ldots)\) are the roots of the equations (under our assumptions, a discrete or countable set), then the period \(T\) of the periodic solution will be determined by equality (3.11), where one must substitute \(c^2=c_k^2\).

Let us consider possibility (3.15). In this case, putting (which is always possible)

\[ \varphi_k(t)=a_k+\Psi_k(t),\qquad \int_0^\omega \Psi_k(t)\,dt=0, \tag{3.16} \]

from equality (3.8), for \(\vartheta=2\pi l,\ t=m\omega\), we obtain

\[ \int_0^{2\pi l} \frac{d\vartheta} {\sqrt{\lambda^2+16c^2\left(a^2\cos^4\vartheta+b^2\sin^4\vartheta\right)}} = m\omega\left(1+\sum \Phi_k(c^2)a_k\right). \tag{3.17} \]

Hence we obtain a countable sequence of values \(c^2=c_{m,l}^2\) \((m=1,2,\ldots;\ l=1,2,\ldots)\), for which the solutions of system (3.1) will be periodic with periods \(T_{m,l}=m\omega\). Here \(c_{m,l}^2\) must be found from equations (3.17) for different \(m\) and \(l\) (integers).

Let us note that as \(c^2\to 0\) we have \(\sum \Phi_k(c^2)a_k\to 0\), since the functions \(\Phi_k(z)\) have no constant terms. At the same time the left-hand side of equality (3.17) approaches the value \(\dfrac{2\pi l}{\lambda}\). In other words, for small \(c\) we have

\[ \frac{2\pi l}{\lambda}\,(1+\delta(c))=m\omega,\qquad \delta(c)\to 0\quad \text{as } c\to 0, \]

or

\[ \frac{2l(1+\delta(c))}{m}=\frac{\omega\lambda}{\pi}. \tag{3.18} \]

It follows from this that, for some sequence \(c_{l,m}\to 0\) as \(l,m\to\infty\), we have the functions

\[ \vartheta=\vartheta_{l,m}(t+T_{l,m})=\vartheta_{l,m}(t)+2\pi l . \]

To these \(c_{l,m}\) there correspond periodic solutions of system (3.1), contracting to the origin as \(c_{l,m}\to 0\). Thus, we have obtained two countable sequences (and the second sequence certainly exists) of periodic solutions of system (3.1), contracting to the origin. The remaining solutions of system (3.1) (corresponding to the integral curves (3.6)) will not be periodic, but they will all be closed according to (3.6). By a theorem of N. A. Lukashevich [3], there exists such a change of the independent variable after which they become periodic. However, for the given system this is clear from (3.1). Indeed, if we introduce a new independent variable

\[ d\tau=\left[1+\Sigma\Phi_k(H_0)\varphi_k(t)\right]dt, \]

then we obtain the system

\[ \frac{dx}{d\tau}=-\lambda y-\frac{\partial P}{\partial y}, \qquad \frac{dy}{d\tau}=\lambda x+\frac{\partial P}{\partial x}, \]

for which all closed integral curves (3.6) will be periodic with period (3.11).

In conclusion, let us note that here the fact of the rationality or irrationality of the number \(\dfrac{\lambda\omega}{\pi}\) (which, under other methods of solution, is of significance) plays no role in solving the question of stability of the equilibrium point—the zero solution of system (3.1) is always stable (not asymptotically).

Let us also note that we can likewise consider the general system (3.1). In this case, instead of (3.8) we obtain

\[ \int_{0}^{\vartheta} \frac{d\vartheta} {\lambda+\sum_{k=3}^{\infty}kr^{k-2}P_k(\cos\vartheta,\sin\vartheta)} = t+\int_{0}^{t}\Phi(c^2,t)\,dt, \]

where \(r\) is given by equality (1.6) or (1.7).

Thus, we see that perturbations of canonical systems (1.1), without violations of the first approximation, which has purely imaginary characteristic roots (or zero characteristic exponents in the sense of Lyapunov), in the class of holomorphic Hamiltonian functions do not disturb the stability (non-asymptotic) of the equilibrium point \(x=y=0\). But perturbations in the class of nonanalytic functions, having a finite discontinuity along a line and arbitrarily small together with all partial derivatives as \(x^2+y^2\to 0\), can lead to a system whose equilibrium point will be stable, asymptotically stable, or unstable. Hence there follows the important problem of finding criteria for distinguishing perturbations\({}^{1)}\) under which one of these cases will occur.

In particular, therefore, there arises the problem of determining stability criteria for the zero solution of a canonical system where

\[ H=\frac{\lambda}{2}(x^2+y^2)+a_2(x^2+y^2)^2+\cdots+a_n(x^2+y^2)^n+\mu P(x,y,t), \tag{3.19} \]

\({}^{1)}\) And, possibly, with preservation of the continuity of \(H(x,y,t)\) and of its derivatives.

\[ P(x,y,t+2\pi)=P(x,y,t)\quad \text{and}\quad \frac{P}{r^{2n+1}}\to 0 \quad \text{as } r\to 0. \tag{3.19} \]

Here it is necessary to give criteria for the stability of the zero solution in terms of the properties of the function \(P(x,y,t)\), nonanalytic at the point \(x=y=0\).

As we have seen, no smallness, together with the derivatives, of the function \(P(x,y,t)\) guarantees the stability of the zero solution. Some other properties of the function \(P(x,y,t)\) ensure the stability of the zero solution.

It also follows from the preceding that nonasymptotic stability can be destroyed by nonanalytic perturbations (for example, perturbations \(\varphi(t)\)); such stability is always unstable. But if we have uniform asymptotic stability (under perturbations nonanalytic at the point \(x=y=0\), or on the line of functions \(H\)), then small perturbations of the function \(H\) will not destroy this stability, since such stability is ensured by some inequality. This follows from the existence theorems for a Lyapunov function under uniform asymptotic stability [1].

The following problem is also worth posing. Given an unperturbed canonical system (1.1), all integral curves of which, in a neighborhood of the origin, are closed, and whose solutions are periodic.

Consider the perturbed system

\[ \dot{x}=-\frac{\partial H}{\partial y},\qquad \dot{y}=\frac{\partial H}{\partial x},\qquad H=H_0+\mu P(x,y,t), \tag{3.20} \]

where \(P(x,y,t+2\pi)=P(x,y,t)\).

Find that class of functions \(P\) for which, for small \(\mu\), all solutions of the perturbed system in a neighborhood of the origin will be closed (not necessarily periodic). We have indicated one such class: this is system (3.1). It is, of course, interesting also to indicate those among these solutions that will be periodic.

Consider, in particular, the system corresponding to \(H\) given by formula (3.19), and the unperturbed system

\[ \dot{x}=-\frac{\partial H_0}{\partial y},\qquad \dot{y}=\frac{\partial H_0}{\partial x}, \tag{3.21} \]

where \(H_0\) is obtained from (3.19) for \(\mu=0\). An integral of (3.21) will be

\[ x^2+y^2=c^2, \tag{3.22} \]

and the general solution is

\[ \begin{aligned} x&=-c_2\sin\left[\left(\lambda+4a_2c^2+\ldots+2nc^{2(n-1)}\right)t-c_1\right],\\ y&=c_2\cos\left[\left(\lambda+4a_2c^2+\ldots+2nc^{2(n-1)}\right)t-c_1\right]. \end{aligned} \tag{3.23} \]

This solution will be periodic with period \(T=2l\pi\), if \(c^2\) is determined from the equation

\[ \lambda+4a_2c^2+\ldots+2na_nc^{2(n-1)}=\frac{k}{l};\qquad k,l\text{ are integers.} \tag{3.24} \]

If \(\lambda,a_2,\ldots,a_n\) are constants such that this equation has no positive roots \(c^2\) for any integers \(k\) and \(l\), then the solutions (3.23) cannot give rise to periodic solutions of the canonical system with \(H\) given by equality (3.19) for small \(\mu\). Periodic solutions ...

of this system with period \(\omega \ne 2l\pi\) can be sought as indicated in [2], i.e., from the equalities

\[ \frac{\partial P(x,y,t+\omega)}{\partial y} - \frac{\partial P(x,y,t)}{\partial y} =0, \]

\[ \frac{\partial P(x,y,t+\omega)}{\partial x} - \frac{\partial P(x,y,t)}{\partial x} =0. \]

But there may also be closed integral curves to which nonperiodic solutions correspond. What is the criterion for the closedness of all integral curves of system (3.19), for example, for sufficiently small \(\mu\)? Or does the smallness of \(\mu\) have no significance?

The following question also arises: do there exist such \(\lambda\) and \(P\) in system (3.19), where not all \(a_k\) are equal to zero and \(P(x,y,t)\) is holomorphic with respect to \(x,y\), while the equilibrium point \(x=y=0\) is unstable or asymptotically stable? We have seen that for systems (3.1) and (1.18) the equilibrium point is always nonasymptotically stable. True, for system (1.18) with \(a=0\) and \(\Phi(u,v,w)=v^3+w^3\), the equilibrium point \(x=y=0\) will be unstable. But this system is not of type (3.19), since here all \(a_k=0\). It seems to us that for any holomorphic \(P\) independent of \(\lambda\), the zero solution of system (3.19) will be nonasymptotically stable if one of the \(a_k\) is different from zero.

§ 4. Here we shall indicate a canonical system

\[ \dot{x}=-\frac{\partial H(x,y,t)}{\partial y},\qquad \dot{y}=\frac{\partial H(x,y,t)}{\partial x}, \tag{4.1} \]

having a prescribed integral curve [3]

\[ \omega(x,y)=0. \tag{4.2} \]

The general form of the Hamiltonian function \(H\) in this case will be

\[ H(x,y,t)= \int_{x_0}^{x} \left[ F_2(\omega(x,y),x,y,t) + \frac{\partial \omega(x,y)}{\partial x}M(x,y,t) \right]\,dx + \]

\[ + \int_{y_0}^{y} \left[ M(x_0,y,t)\frac{\partial \omega(x_0,y)}{\partial y} - F_1(\omega(x_0,y),x_0,y,t) \right]\,dy. \tag{4.3} \]

Here \(F_1\) and \(F_2\) possess the property

\[ F_1(0,x,y,t)=F_2(0,x,y,t)=0 \tag{4.4} \]

and satisfy the equality

\[ \frac{ \partial\left[ F_2(\omega(x,y),x,y,t) + \frac{\partial \omega(x,y)}{\partial x}M(x,y,t) \right] }{\partial y} = \]

\[ = - \frac{ \partial\left[ \frac{\partial \omega(x,y)}{\partial y}M(x,y,t) - F_1(\omega(x,y),x,y,t) \right] }{\partial x}. \tag{4.5} \]

Of the three functions \(F_1,F_2\), and \(M\), two may be assigned, and the third is then found from (4.5). In general, if desired, one may choose \(M,F_1\), and \(F_2\) so that the ապահով

uniqueness of the solutions of the system (4.1) is ensured. But this need not be required.

Example 1. If in (4.3) we take \(\omega = y - x,\ M = 1,\ F_1 = y - x,\ F_2 = y - x,\ x_0 = y_0 = 0\), then we obtain

\[ H = xy - \frac{x^2 + y^2}{2} - x + y. \tag{4.6} \]

The canonical system will be

\[ \dot{x} = -\frac{\partial H}{\partial y} = y - x - 1,\qquad \dot{y} = \frac{\partial H}{\partial x} = y - x - 1, \]

for which the integral will be \(y - x = c\), which also contains the integral curve \(\omega = y - x = 0\).

Example 2. Let \(\omega = y^2 - x\). Then from (4.5) we find

\[ F_1(\omega, x, y, t) = 2yF_2(\omega, x, y, t) + y^2 - x + 2y(x - y^2)\varphi(t), \]

where \(F_2\) is arbitrary, having the property \(F_2(0, x, y, t) = 0\).

Let \(F_2 \equiv 0\). From (4.3) we find \(H\) in the form

\[ H = xy - y^3 + \frac{1}{2}(y^4 - x^2)\varphi(t). \]

Hamilton’s system will be

\[ \dot{x} = -\frac{\partial H}{\partial y} = -x + 3y^2 - 2y^3\varphi(t), \]

\[ \dot{y} = \frac{\partial H}{\partial x} = y - x\varphi(t), \tag{4.7} \]

for which, as is easy to see, the curve \(\omega = y^2 - x = 0\) will be integral.

Remark. In [5] it is shown how to find integrals of the system

\[ \dot{x} = P(x, y, t),\qquad \dot{y} = Q(x, y, t), \tag{4.8} \]

\[ R(x, y) = c, \tag{4.9} \]

not containing \(t\).

Using the same method, we can also seek particular integral curves

\[ \omega(x, y) = 0 \tag{4.10} \]

of the system (4.8). We shall, for example, seek\(^1\) an integral curve (4.10) of the system (4.7). By virtue of equations (4.7) we have

\[ \frac{\partial \omega}{\partial x}\bigl[-x + 3y^2 - 2y^3\varphi(t)\bigr] + \frac{\partial \omega}{\partial y}\bigl(y - x\varphi(t)\bigr) = \]

\[ = \frac{\partial \omega}{\partial x}\bigl[-x + 3y^2\bigr] + \frac{\partial \omega}{\partial y}y + \left[ \frac{\partial \omega}{\partial x}(-2y^3) + \frac{\partial \omega}{\partial y}(-x) \right]\varphi(t) = \]

\[ = F(\omega, x, y, t). \]

In order that the curve (4.10) be an integral curve of the equations (4.7), it is necessary and sufficient [4] that \(F(0, x, y, t) = 0\). Or it must be

\(^1\) In this example we also show a general method for finding a particular integral curve.

\[ \frac{\partial \omega}{\partial x}(-x+3y^2)+\frac{\partial \omega}{\partial y}y=F_1(\omega,x,y), \tag{4.11} \]

\[ \frac{\partial \omega}{\partial x}(-2y^3)+\frac{\partial \omega}{\partial y}(-x)=F_2(\omega,x,y), \tag{4.12} \]

where \(F_k(0,x,y)=0\) \((k=1,2)\). Setting \(F_2\equiv 0\) in (4.12), we find an integral of equation (4.12) (for \(F_2\equiv 0\)):

\[ y^4-x^2-c=0. \tag{4.13} \]

Among the curves (4.13) we shall seek one that is an integral for equation (4.11) when \(F_1\equiv 0\). Thus, setting in (4.11) \(\omega=y^4-x^2-c\), we obtain

\[ L(\omega)=-2x(-x+3y^2)+4y^4=4y^4+2x^2-6xy^2. \]

Substitute here the value of \(x\) from (4.13):

\[ L(\omega)=4y^4+2x^2-6xy^2=4y^4+2(y^4-c)-6y^2\sqrt{y^4-c}. \]

For \(c=0\) we obtain \(L(\omega)=0\). Hence we see that for \(c=0\), from (4.13) we have a curve that may contain an integral curve of system (4.7):

\[ y^4-x^2=0. \]

Hence we have

\[ y^2-x=0,\qquad y^2+x=0. \]

The first of these curves is an integral curve for equations (4.7), as is easily verified. Using the integral curve \(y^2-x=0\), we find a one-parameter family of solutions of system (4.7):

\[ x=e^{2t}\left(c+\int_0^t \varphi(t)e^t\,dt\right)^{-2},\qquad y=e^t\left(c+\int_0^t \varphi(t)e^t\,dt\right)^{-1}, \]

whence we see that if \(\varphi(t+\tau)=\varphi(t)\), then for no \(\tau\) will the zero solution of system (4.7) be stable.

References

1. Krasovskii N. N. Some problems in the theory of stability of motion. Fizmatgiz, 1959.
2. Erugin N. P. PMM, vol. XX, issue 1, 1956.
3. Lukashevich N. A. DAN BSSR, 4, No. 3, 1960.
4. Erugin N. P. PMM, vol. XVI, issue 6, 1952.
5. Erugin N. P. DAN BSSR, 2, No. 4, 1958.
6. Gavrilov M. I. XXI Scientific Conference of the Faculties of Mechanics and Mathematics, Physics, and Chemistry. Abstracts of Reports. Odessa, 1966.

Received by the editors
September 1, 1966

Institute of Mathematics, Academy of Sciences of the BSSR

1. The general form of a canonical system that has an integral independent of \(t\), while the right-hand sides of the system contain \(t\) [5]. ↩

2. Let the functions \(\varphi_1(t),\varphi_2(t),\ldots\) be linearly independent. ↩