MATHEMATICS

M. B. KUDAEV

INVESTIGATION OF THE BEHAVIOR OF TRAJECTORIES OF SYSTEMS OF DIFFERENTIAL EQUATIONS BY MEANS OF LYAPUNOV FUNCTIONS

(Presented by Academician I. G. Petrovskii, 11 VII 1962)

1. We consider a system of differential equations

\[ \frac{dx_i}{dt}=X_i(x_1,\ldots,x_n),\qquad X_i(0,\ldots,0)=0,\qquad i=1,\ldots,n. \tag{1} \]

Here \(X_i(x)\in C^2(U(O))\); \(U(O)\) is a neighborhood of the point \(O\), lying in Euclidean space \(E^n\); \(x=(x_1,\ldots,x_n)\); \(O=(0,\ldots,0)\) is the only singular point in \(U(O)\).

Refining certain definitions of P. N. Papusha \((^1)\), we shall assume that the whole neighborhood \(U(O)\) can be partitioned by topological cones \(H_l,\ l=1,\ldots,k\), with vertex at the point \(O\), into \(m\) domains \(\omega_s,\ s=1,\ldots,m\). In addition, it will sometimes be necessary to use topological cones \(h_r\), also with vertex \(O\), which are situated inside the domain \(\omega_s\) (apart from the point \(O\)). We shall call the latter cones additional cones.

Two formulations of the problem are possible \((^2)\): either, as is done in \((^1)\), one prescribes such analytic properties of the right-hand sides of equations (1) from which one or another arrangement of the trajectories of system (1) will follow; or, conversely, one prescribes in the domains \(\omega_s\) and on the surfaces \(H_l\) and \(h_r\) the topological arrangement of the trajectories and seeks those analytic conditions which follow from this arrangement.

Definition 1. We shall call a sign-constant function \(V(x)\) a generalized Lyapunov function if, being nonzero in the domains \(\omega_s\), it retains one and the same sign in each of them, and if its derivative \(dV/dt\) by virtue of system (1) is a sign-variable function, also retaining its sign in the domains \(\omega_s\), changing it each time upon passing through a surface \(H_l,\ l=1,\ldots,k\), while on the surface

\[ \bigcup_{l=1}^{k} H_l \]

\(V(x)\geq 0,\ dV/dt=0\). Those cones \(H_l\) that enter into the boundary of the domain \(\omega_s\) will be called adjacent.

In addition to the first derivative \(F_1\equiv dV/dt\), we shall also consider the second derivative \(F_2\equiv d^2V/dt^2\) of the function \(V(x)\).

Definition 2. Following V. V. Nemytskii, we shall call a system of integral curves hyperbolic if all curves of this system are either \(O^+\)- or \(O^-\)-curves having no \(\alpha\)- and \(\omega\)-limit points except the origin (parabolic curves), or curves going off in both directions (hyperbolic curves). The system is called elliptic if all integral curves are either parabolic or elliptic, i.e. have as their \(\alpha\)- and \(\omega\)-limit set the singular point \(O\). Finally, the system is called elliptic-hyperbolic if, in it, apart from \(O\), there are integral curves of all the three classes mentioned \((^{3,4})\).

II. Let us formulate the following theorems.

Theorem 1. If \(V(x)\) is a generalized Lyapunov function, and the function \(F_2>0\) (\(F_2<0\)) on the cones \(H_l,\ l=1,\ldots,k\), then system (1) will be hyperbolic (elliptic).

This theorem is a generalization of Theorem 7 from [1], since in the latter constancy of sign of the function \(V(x)\) was not allowed.

Let us note that, for a generalized Lyapunov function \(V(x)\), simultaneous fulfillment of the relations
\[ V(p)=0,\quad F_2(p)<0,\quad \text{where } p\in \bigcup_{l=1}^{k}H_l,\quad p\ne 0, \]
is impossible.

Theorem 2. Suppose:

a) the function \(V(x)\) is positive definite throughout the neighborhood under consideration;

b) the function \(F_1\) is equal to zero on the surface \(\displaystyle \bigcup_{l=1}^{k}H_l\), preserves its sign in each domain \(\omega_s\), and changes it when passing through the surface \(H_l,\ l=1,\ldots,k\);

c) the function \(F_2\) preserves its sign on each cone \(H_l\), but there is at least one pair of adjacent cones forming part of the boundary surfaces of some domain \(\omega_r\), on which \(F_2\) has opposite signs;

d) there is an additional cone \(h_r,\ h_r\setminus O\subset \omega_s\), separating this pair of cones and on which the conditions \(F_2=0,\ F_1F_3>0\) are satisfied, or there exists a pair of additional cones \(h_{r_i},\ (h_{r_i}\setminus O)\subset \omega_r,\ i=1,2\), on which the conditions \(F_2=0,\ F_1F_3<0\) are satisfied, and the mentioned cones \(h_{r_i}\) are not separated by other additional cones.

Then system (1) is elliptic-hyperbolic. In this case the domain \(\omega_r\) has an open subset in \(E^n\), including respectively the surface \(h_r\setminus O\) or the surfaces \(h_{r_i}\setminus O\), consisting entirely of points of parabolic trajectories.

Remark. When speaking of the sign of a function on any of the cones, we never include the singular point in this cone. All the functions considered by us vanish at \(x=0\).

III. In this section we shall show that the conditions of Theorems 1 and 2 are, in a certain sense, also necessary.

Definition 3. We shall call some domain \(\omega_s\) an attracting (repelling) hyperbolic domain if it contains \(O^+\)-(\(O^-\))-curves of system (1), as well as arcs \(f(p,t<0)\) (\(f(p,t>0)\)) of negative (positive) hyperbolic semitrajectories, where \(p\) is an arbitrary (nonzero) point from the union of adjacent cones \(H_l\) corresponding to the domain \(\omega_s\). The domain \(\omega_s\) is called an attracting (repelling) elliptic domain if every trajectory that enters it approaches the singular point \(O\) as \(t\to +\infty\) (\(t\to -\infty\)) and leaves \(\omega_s\) for some finite or infinite \(t<0\) (\(t>0\)), respectively.

Theorem 3. Let system (1) be hyperbolic (elliptic), and let each surface \(H_l\setminus O\in C^2(U(O))\), \(l=1,\ldots,k\). Suppose that each hyperbolic (elliptic) curve of system (1) has with the surface
\[ \bigcup_{l=1}^{k}H_l \]
one and only one common point, and that, of any two adjacent domains \(\omega_s'\) and \(\omega_s''\) (whose boundaries have a nonempty intersection distinct from the point \(O\)), one is attracting if the other is repelling, and conversely.

Then there exists such a sign-definite (positive definite) function \(V(x)\in C^2(U_1(O))\), where \(U_1(O)\subset U(O)\) is some neighborhood of the point \(O\), that on the surface \(\displaystyle \bigcup_{l=1}^{k}H_l\)
\[ V(x)=F_1(x)=0\quad (V(x)>0,\ F_1(x)=0\ \text{respectively}); \]
the product of the functions \(VF_1<0\) in each attracting

repelling domain and \(VF_1>0\) in each repelling domain \(\omega_s\); on the surface \(\bigcup_{l=1}^{k} H_l\) the function \(F_2>0\) (\(F_2<0\), respectively).

We note that for a hyperbolic system (1) satisfying the conditions of Theorem 3, there also exists a positive definite function having all the properties of the function \(V\) of Theorem 3, except, of course, for vanishing at nonsingular points of the surface \(\bigcup_{l=1}^{k} H_l\).

In what follows we shall assume that the right-hand sides of equations (1) are functions of class \(C^3(U(O))\), and also that all \((n-1)\)-dimensional cones \(h_r\setminus O\) and \(H_l\setminus O\) are surfaces of class \(C^3(U(O))\).

Definition 4. A domain \(\omega_s\) is called attracting elliptic-hyperbolic if every point \(f(p,t)\), where \(p\) is an arbitrary (different from \(O\)) point of the union of the adjacent cones \(H_l\) corresponding to the domain \(\omega_s\), belongs to the domain \(\omega_s\) either only for positive values of \(t\), or only for negative values of \(t\); in the first case, as \(t\to +\infty\), it tends to the singular point \(O\) (points of arcs of elliptic semitrajectories), while in the second it leaves \(U(O)\), no longer intersecting the surface \(\bigcup_{l=1}^{k} H_l\), for some finite or infinite \(t<0\) (points of arcs of hyperbolic semitrajectories); the remaining points of the domain \(\omega_s\) lie on \(O^+\)-trajectories. A repelling elliptic-hyperbolic domain is defined analogously.

Theorem 4. Suppose that system (1) is elliptic-hyperbolic and that, in the neighborhood under consideration, there is an attracting (repelling) elliptic-hyperbolic domain \(\omega_s\), which is divided by an additional cone \(h_r\) into two subdomains, one of which contains arcs of all elliptic, and the other arcs of all hyperbolic, semitrajectories; moreover the surface \(h_r\setminus O\) consists of points only of \(O^+\)- (\(O^-\)-) trajectories.

Then in some neighborhood \(U_1(O)\) of the point \(O\), \(U_1(O)\subset U(O)\), there exists a positive definite function \(V(x)\in C^3(U_1(O))\) having the following properties:

1) \(F_1<0\) (\(F_1>0\)) at points of the domain \(\omega_s'=\omega_s\cap U_1(O)\);

2) \(F_2>0\) at those points of the adjacent cones \(H_l\cap U_1(O)\) corresponding to the domain \(\omega_s'\) through which the hyperbolic semitrajectories pass; \(F_2<0\) at those points of these cones through which the elliptic semitrajectories of the domain \(\omega_s\) pass; and at points of the surface \((h_r\setminus O)\cap \omega_s'\) the conditions
\[ F_2=0,\qquad F_1F_3>0, \]
hold, where \(F_3\) is the third derivative of the function \(V(x)\) along system (1).

With the aid of Theorem 4 it is easy to obtain the converse of Theorem 2 (the case in which, on \(h_r\setminus O\), \(F_1F_3>0\)).

As for the converse of Theorem 2 in the case in which it is assumed there that on two additional cones the conditions \(F_2=0\) and \(F_1F_3<0\) hold, this is done with the aid of a theorem similar to Theorem 4. In proving these converse theorems, some ideas and results from the works \((5\text{--}8)\) are used.

In conclusion, I take this opportunity to express my deep gratitude to V. V. Nemytskii for posing the problems and for his guidance.

Moscow State University
named after M. V. Lomonosov

Received
10 VII 1962

CITED LITERATURE

1. P. N. Papush, Matem. sborn., 38, issue 3 (1956).
2. V. V. Nemytskii, UMN, 9, issue 3 (1954).
3. V. V. Nemytskii, Vestn. Mosk. univ., No. 6 (1960).
4. V. Niemiytzki, Ann. di Mat. pura ed appl., 49, 11 (1960).
5. A. M. Lyapunov, The General Problem of the Stability of Motion, 1950.
6. N. N. Krasovskii, Prikl. matem. i mekh., 18, issue 5 (1954).
7. E. A. Barbashin, Matem. sborn., 29, issue 2 (1951).
8. I. M. Gelfand, G. E. Shilov, Generalized Functions and Operations on Them, Moscow, 1959, p. 182.