CLASSIFICATION OF MULTIDIMENSIONAL SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS BY THE METHOD OF LYAPUNOV FUNCTIONS

M. B. KUDAEV

1. The method of Lyapunov functions has recently begun to be applied to the topological classification of systems of ordinary differential equations (see [1—3]) (classification of singular points), that is, to the solution of one of the principal problems of the qualitative theory of differential equations. The power and, consequently, the reliability of this method are confirmed by proving the existence, and even by constructing, a Lyapunov function that establishes a particular direct theorem on the belonging of an isolated singular point to a definite class. If, however, we have succeeded in determining the topological class of a simpler system (and the method of Lyapunov functions also makes it possible to classify singular points of higher order [4]), then by the comparison method we establish the class of a system “close” to the given one in a certain sense, but already considerably more complicated. All this gives the method of Lyapunov functions an important place among methods for studying nonlinear systems ([5], [6]).

Below we consider the system of differential equations

\[ \frac{dx}{dt}=X(x), \tag{1} \]

where the vector

\[ x= \begin{bmatrix} x_1\\ \vdots\\ x_n \end{bmatrix} \]

is a point of the \(n\)-dimensional Euclidean space \(E^n\), \(n>2\), and the vector

\[ X(x)= \begin{bmatrix} X_1(x)\\ \vdots\\ X_n(x) \end{bmatrix} \]

has components \(X_i(x)\) of class \(C^2\) in the considered neighborhood \(U=U(0)\) of the unique point \(x=0\) in \(U\) with the property \(X_i(0)=0\), \(i=1,\ldots,n\) (\(x=0\) is an isolated singular point of system (1)).

Let \(V(x)\) be some function of class \(C^2(U)\). Then its first two derivatives with respect to \(t\) by virtue of system (1) will be continuous functions in \(U\). More precisely,

\[ F_1(x)\equiv \frac{dV}{dt} =\sum_{i=1}^{n}\frac{\partial V}{\partial x_i}X_i\in C^1(U), \qquad F_2(x)\equiv \frac{d^2V}{dt^2} =\sum_{i=1}^{n}\frac{\partial F_1}{\partial x_i}X_i\in C(U). \]

Definition 1. A generalized Lyapunov function (g. L. f.) for system (1) is a sign-constant function \(V(x)\in C^2(U)\) possessing the properties:

a) the manifold \(F_1^0=\{F_1(x)=0\}\) is a pencil of a finite number of topological \((n-1)\)-dimensional cones \(H_i\) with a unique common point—the vertex \(O\), and \(V(x)\) can vanish only at points \(x\) of \(F_1^0\);

b) the function \(F_1(x)\) changes sign at each passage from one part of the neighborhood \(U\) to another with respect to any \(H_i\);

c) the function \(F_2(x)\) is sign-definite on each of the \(H_i\).

A L.f. may be nonnegative or nonpositive, as well as positive or negative definite. For definiteness, in what follows we shall assume that \(V(x)\) is nonnegative.

Definition 2. The regions \(\omega_l\) (\(l\) assumes a finite number of values), into which \(U\) is divided by the cones \(H_i\), are called normal regions (n.r.), and the union of the boundary cones n.r. \(\omega_l\) from \(H_i\) is called the lateral surface of n.r. \(\omega_l\) and is denoted by \(F(\omega_l)\). Depending on whether the function \(F_2(x)\) on \(F(\omega_l)\setminus O\) is positive, negative, or sign-changing, the n.r. is respectively called hyperbolic, elliptic, or mixed. Obviously, in each n.r. the function \(F_1(x)\) preserves its sign, and in adjacent normal regions it has opposite signs (Definition 1).

Definition 3. An n.r. \(\omega_l\) is called attracting (respectively repelling) if in it \(F_1(x)<0\) (\(F_1(x)>0\)).

Definition 4. System (1) is called parabolic if all its trajectories in \(U\) are \(O^+\)- or \(O^-\)-curves with \(\omega\)- or \(\alpha\)-limit point \(O\) (respectively). System (1) in \(U\) is called hyperbolic if in \(U\), in any neighborhood of the point \(O\), there pass hyperbolic trajectories—trajectories leaving \(U\) at both ends. System (1) in \(U\) is called elliptic if in \(U\), in any neighborhood of the point \(O\), there are elliptic trajectories—trajectories entering \(O\) with both ends as \(t\to \pm\infty\) (respectively). System (1) is called elliptic-hyperbolic in \(U\) if, in any neighborhood of \(O\) in \(U\), there pass both elliptic and hyperbolic trajectories.

It can be proved that hyperbolic, elliptic, and elliptic-hyperbolic systems in \(U\) have both \(O^+\)- and \(O^-\)-curves (see, for example, [1], [7], [8]).

The following theorem has been proved: if \(F_2(x)\) is positive (negative) definite on \(F_1^0\), then system (1) is hyperbolic (elliptic) ([3], [8]).

The case in which \(F_2(x)\) is sign-changing on \(F_1^0\) was studied in [3], [8].

Definition 5. System (1) is called mixed (of mixed type) if the second derivative with respect to \(t\) of the L.f. \(F_2(x)\) is sign-changing on the manifold \(F_1^0\).

If system (1) is mixed with respect to the L.f. \(V(x)\), then the set of mixed normal regions (m.n.r.) in \(U\) is nonempty. The disposition of integral curves in hyperbolic and elliptic n.r. is completely determined. As for an m.n.r., this cannot be said of it. For example, it may have whole semitrajectories (\(O\)-curves), or it may not. In [3] and [8] the first of these cases is considered. Below the second case will be considered.

2. SOME THEOREMS ON MIXED SYSTEMS OF GENERAL FORM

Theorem 1. If system (1) is mixed, then in the neighborhood \(U(O)\) under consideration it has at least one \(O^+\)-curve or one \(O^-\)-curve.

Proof. By the definition of a mixed system, the neighborhood \(U\) is divided by \(k\) cones \(H_i,\ i=1,\ldots,k\), into \(m\) normal domains \(\omega_l,\ l=1,\ldots,m\). Among the latter there is necessarily at least one stable normal domain.

Moreover, if system (1) is mixed with respect to the Lyapunov function \(V(x)\), then among the cones \(H_i\) of the manifold \(F_1^0\) there are \(s\) cones, \(2\le s\le k\),

\[ H_{l_1},\ H_{l_2},\ \ldots,\ H_{l_s}, \tag{2} \]

each of which divides the neighborhood \(U(O)\) into two parts \(I(H_{l_i})\) and \(E(H_{l_i})\) (the interior and the exterior of the cone \(H_{l_i},\ i=1,\ldots,s\)), of which one contains all the points of the manifold \(F_1^0\), except for the points of the cone \(H_{l_i}\) itself, while the other contains not a single point of \(F_1^0\). For definiteness, let us assume that the inner domain \(I(H_{l_1})\) of the cone \(H_{l_1}\) and the outer domains \(E(H_{l_j}),\ j=2,\ldots,s\), of the cones \(H_{l_2},\ldots,H_{l_s}\) contain no points of the manifold \(F_1^0\).

Obviously, the domains \(\omega_{l_1}=I(H_{l_1})\), \(\omega_j=E(H_{l_j})\), \(j=2,\ldots,s\), are normal domains whose lateral surfaces are \(F(\omega_{l_i})=H_{l_i},\ i=1,\ldots,s\).

Two cases may occur:

1) in all normal domains \(\omega_{l_i},\ i=1,\ldots,s\), the function \(F_1(x)>0\) (or \(F_1(x)<0\));

2) in at least two of these normal domains the signs of the function \(F_1(x)\) are different.

In the first case, system (1) has a certain nonempty set of \(O^-\)- (or \(O^+\)-) curves in each of the normal domains \(\omega_{l_i},\ i=1,\ldots,s\). In the second case, in those normal domains where \(F_1(x)<0\), there are nonempty sets of \(O^+\)-, and in those where \(F_1(x)>0\), nonempty sets of \(O^-\)-curves. The last two assertions were proved in [8] for a Lyapunov function, and for a positive-definite Lyapunov function in [1].

Corollary. If system (1) is of mixed type with respect to the Lyapunov function \(V(x)\), then in the neighborhood \(U(O)\) there is at least one \(O\)-curve that does not intersect the hypersurface \(F_1^0\) (for finite \(|t|\)).

Indeed, in \(U\) there is at least one hyperbolic or one elliptic normal domain. But the set of rays (parabolas), i.e. \(O\)-curves, in such domains is nonempty [8].

Theorem 2. Suppose the conditions of Theorem 1 are fulfilled. The subset of ray points of the level surface \(V(x)=\varepsilon,\ \varepsilon>0\), lying in any of the normal domains \(\omega_i,\ i=1,\ldots,s\), is closed in \(\{V(x)=\varepsilon\}\). If, moreover, on the lateral surface \(F(\omega_{l_i})=H_{l_i}\) of the normal domain \(\omega_{l_i}\) the function \(F_2(x)\) is positive definite, then this set is also connected.

The theorem is proved in the same way as the corresponding theorems for elliptic and hyperbolic nonmixed systems in [1] and [8].

Theorem 3. A mixed system (1) cannot have in \(U(O)\) integral curves different from parabolic, hyperbolic, and elliptic ones (not counting the singular point \(O\)); moreover, a hyperbolic (elliptic) integral curve intersects transversely the hypersurface \(F_1^0\) at least once.

Proof. In an arbitrary normal domain \(\omega_1\subset U\), consider a point \(p\). The point \(f(p,t)^*\) intersects level surfaces of the function \(V(x)\) or

\[ \text{* } f(p,\tau)\text{ is the point at which the point }p\in U(O)\text{ finds itself at }t=\tau, \text{ moving along the trajectory }L_p\text{ passing through }p\text{ at }t=0,\ f(p,0)\equiv p. \]

in the direction toward \(O\), or conversely, depending on whether in \(\omega_l\) the inequality \(F_1(x)<0\) (or \(F_1(x)>0\)) holds. Moreover,

\[ \lim_{t\to+\infty} f(p;t)=0 \quad \left(\text{or } \lim_{t\to-\infty} f(p;t)=0\right), \]

and for some \(t<0\) (or \(t>0\)) \(f(p,t)\) intersects the lateral surface \(F(\omega_l)\) at a point different from \(O\). After this \(f(p,t)\) will pass into an adjacent n. r. \(\omega_l'\). In the region \(\omega_l'\) it will either tend to \(O\) as \(t\to-\infty\) (\(t\to+\infty\), respectively), or will pass, for some \(t<0\) (\(t>0\)), into a third n. r. adjacent to \(\omega_l'\), etc.

In general, if in one of the n. r. of the point \(f(p,t)\) remains for all \(t>0\) (\(t<0\)), then it remains for us to establish the behavior of the point \(f(p,t)\) for all those \(t<0\) (\(t>0\)) for which \(f(p,t)\in U\).

The trajectory of the point \(f(p,t)\) will be elliptic if its negative (positive) semitrajectory in the n. r. \(\omega_l\), or in some other n. r., tends to \(O\) as \(t\to-\infty\) (\(t\to+\infty\)). If, however, \(f(p,t)\) leaves the n. r. \(\omega_l\), or another n. r., through the rear wall, then the trajectory of the point \(f(p,t)\) will be an \(O^+\)-(\(O^-\))-trajectory.

If the point \(f(p,t)\) does not remain in any one of the n. r. as \(|t|\to+\infty\), then its trajectory \(L_p\) will be hyperbolic, since the number of normal regions in \(U\) is finite, and the hypersurfaces \(H_i<0,\ i=1,\ldots,k,\) possess one-sided passability. Moreover, \(f(p,t)\) leaves \(U\) through the rear walls of two different n. r., one of which is attracting and the other repelling.

Corollary. The trajectories of the mixed system (1) can intersect the hypersurface \(F_1^0<0\) only a finite number of times. Each hyperbolic or elliptic trajectory intersects the hypersurface \(F_1^0<0\) at least once; a parabolic trajectory may intersect, or may fail to intersect, \(F_1^0<0\).

In this connection it is appropriate to recall that parabolic trajectories of an elliptic or hyperbolic system (with respect to the o. f. L. \(V(x)\)), which is not mixed with respect to the same function \(V(x)\), do not intersect \(F_1^0<0\), while every elliptic or hyperbolic trajectory intersects \(F_1^0<0\) once and only once. From this, however, it does not follow that there are no mixed systems each elliptic or hyperbolic trajectory of which has with \(F_1^0<0\) one and only one common point, while each parabolic trajectory has no points on \(F_1^0<0\).

Fig. 1. \((n=2)\)

For example, the elliptic-hyperbolic system shown in Fig. 1, for which, say, Theorem 2 [3] is valid, is such a system.

Remark. With respect to mixed systems the plane \(E^2\) is exceptional (Fig. 2).

We shall give another formulation of the theorem just proved—the main theorem of this section.

Theorem 4. If system (1) is mixed with respect to the generalized Lyapunov function, then it is either parabolic, or elliptic, or hyperbolic, or elliptic-hyperbolic.

Fig. 2. \((n=2)\)

Thus, when there exists a generalized Lyapunov function \(V(x)\) with respect to which (1) is of mixed type, then in the neighborhood \(U\) under consideration the singular point \(O\) is the unique limit point for the trajectories of (1), and (1) can be neither a center nor a focus (in \(U\)).

3. SOME SUFFICIENT CRITERIA FOR PARABOLICITY, ELLIPTICITY, HYPERBOLICITY, AND ELLIPTIC-HYPERBOLICITY OF A MIXED SYSTEM

We now wish to give several sufficient criteria for a mixed system to belong to one of the possible types listed above (Theorem 4). For this we shall need the following definition.

Definition 6. If the normal region \(\omega\) contains no entire semitrajectories of system (1), i.e. no trajectory remains in \(\omega\) as \(t \to +\infty\) or \(t \to -\infty\), then it is called an unstable normal region.

The particular type of system (1), when it is mixed, depends a) on the distribution of the signs of the function
\[ F_2(x)=\frac{d^2 V}{dt^2} \]
on the cones (2) \(H_i,\ i=1,\ldots,s,\ 2\le s\le k\), which we considered in the proof of Theorem 1, and b) on the number of normal regions lying between all possible pairs of these cones, if it is assumed that all these normal regions are of mixed type and unstable.

Thus, in what follows up to Theorem 8, we shall have in mind only the case of system (1) when, in passing at least from one of the cones
\[ H_{l_1},\ H_{l_2},\ \ldots,\ H_{l_s} \tag{2} \]
(we shall assume, for definiteness, from \(H_{l_k}\)) to any other cone from (2), the signs of the functions
\[ F_2(x)=\frac{d^2 V}{dt^2} \]
on the cones from \(F_1^0 \setminus O\), lying between \(H_{l_k}\) and \(H_{l_j}\), \(j\ne k,\ 1\le j\le s\), alternate. In this case we shall call the cones \(H_{l_k}\) orienting cones. It can be proved that, for an arbitrary mixed system (1), the number of orienting cones is no more than two.

In what follows we shall assume that \(H_{l_1}\) is one of the orienting cones, which, as is easy to see, in no way violates the generality of the arguments.

Theorem 5. Let (1) be a mixed system with respect to the ordinary Lyapunov function \(V(x)\), and let all n.o. \(\omega_i\) lying between one of the orienting cones, say \(H_{l_1}\), and each of the remaining cones (2) be unstable s.n.o., and let their number be odd (for any \(H_{l_j}\), \(j=2,\ldots,s\)).

Then (1) is a parabolic system in \(U\); moreover, if on \(H_{l_1}\setminus O\) the inequality \(F_2(x)>0\) \((F_2(x)<0)\) holds, then all trajectories (except \(O\)) will be \(O^+\)-curves (\(O^-\)-curves), respectively, i.e. \(\lim_{t\to+\infty} f(p,t)=0\) (respectively \(\lim_{t\to-\infty} f(p,t)=0\)) for every \(p\in U,\ p\ne0\).

Proof. We shall use the notation \(\omega_{l_1}=I(H_{l_1})\), \(\omega_{l_j}=E(H_{l_j})\), \(j=2,\ldots,s\), introduced in the proof of Theorem 1. We restrict ourselves to considering the case when \(F_2(x)>0\) on \(H_{l_1}\setminus O\) and, consequently, \(F_2(x)<0\) on \(H_{l_j}\setminus O\), \(j=2,\ldots,s\) (the other case is treated analogously).

For definiteness suppose that in the n.o. \(\omega_{l_j}\) \(F_1(x)<0\). Then from the conditions of the theorem it follows that in the n.o. \(\omega_{l_j}\) \(F_1(x)<0\) and on the cones \(H_{l_j}\setminus O\) \(F_2(x)<0\), \((j=2,\ldots,s)\). Since the n.o. \(\omega_{l_i}\), \(i=1,\ldots,s\), have one-cone lateral surfaces \(H_{l_i}\), respectively, the information obtained is sufficient to determine (up to a topological mapping) the arrangement of the trajectories in each of these regions (not speaking of the cardinality of the set of rays of each n.o.) [1, 8]. The regions \(\omega_{l_2},\ldots,\omega_{l_s}\) will be attracting elliptic n.o. (Fig. 3, \(a\)), while \(\omega_{l_1}\) is an attracting hyperbolic n.o. (Fig. 3, \(b\)). Since the part of the neighborhood \(U\) between the cone

Fig. 3. \((n=3,\ j=2,\ldots,s)\)

\(H_{l_1}\) and any of the remaining cones (2) is filled with unstable n.o., the trajectories \(L_p\) starting at points \(p\) of the union of the n.o. \(\displaystyle \bigcup_{j=2}^{s}\omega_{l_j}\), as \(t\to+\infty\), within the limits of the n.o. to which \(p\) belongs, enter the point \(O\). With increase in the negative direction, the point \(f(p,t)\) leaves \(U(O)\) at some finite \(t=t_1<0\) through the back wall of one of the n.o. \(\omega_i,\ i=1,\ldots,m\), after first crossing a finite number \((\ge0)\) of hypersurfaces \(F_1^{0}\setminus O\) (since the number

regions is odd). If, in addition, the point \(f(p,t)\) crosses \(H_{l_1}\setminus O\) at \(t=t'<0\), then for all \(t<t'\) it will be in \(\omega_{l_1}\), until it leaves \(U\) through the rear wall of the n. r. \(\omega_{l_1}\). In \(\omega_{l_1}\) the set of parabolic trajectories (more precisely, \(O^{-}\)-curves) is nonempty. The remaining arcs in \(\omega_{l_1}\), as well as all arcs lying in the n. r. neighborhood \(U\), will have finite time length. We assume (see the remark made below) that one end of each of these arcs belongs to the union of the cones \(H_{l_j}\setminus O\), \(j=2,\ldots,s\), while the other belongs either to the cone \(H_{l_1}\setminus O\), or to the boundary \(\Gamma\) of the neighborhood \(U(O)\). All trajectories in \(U\) are \(O^{+}\)-curves entering \(O\) within the n. r. \(\omega_{l_i}\), \(i=1,\ldots,s\) (Fig. 4). The theorem is proved.

Fig. 4. \((n=3,\ s=3)\)

Remark. In an arbitrary neighborhood \(U(O)\) of a singular point there may, of course, also exist “additional arcs” [8]; they may also lie in a separately taken n. r. \(\omega_i\subset U(O)\). These will be arcs of trajectories \(L_p,\ p\in U(O)\) (or \(p\in\omega_i\)) of finite time length, with both ends lying on the boundary \(\Gamma\) of the neighborhood \(U(O)\). However, by changing \(U(O)\), one can ensure that in \(U(O)\) there lies only an entire semi-trajectory of the point \(f(p,t)\). This possibility follows from the fact that in \(U(O)\) the point \(O\) is the unique limit point for the points \(f(p,t)\) (Theorem 3 or 4). Therefore, whenever we speak of elliptic, elliptic-hyperbolic, hyperbolic, and parabolic systems, we shall have in mind that the neighborhood \(U(O)\) contains no “additional arcs” [8].

Theorem 6. If 1) system (1) satisfies the conditions of Theorem 5, and \(F_1(x)<0\) (\(F_1(x)>0\)) in the n. r. \(\omega_{l_1}\) lying inside the orienting cone \(H_{l_1}\); 2) system (1) is positively (negatively) stable in the sense of Lyapunov, then system (1) is asymptotically stable in the sense of Lyapunov in the positive (negative) direction of variation of \(t\).

Theorem 6 is an obvious consequence of Theorem 5. It remains only, perhaps, to repeat once more the known fact that from the fact that all trajectories are \(O^{+}\)-(\(O^{-}\))-curves it by no means follows that the singular trajectory is stable, and therefore neither does asymptotic stability of the singular trajectory follow, respectively, as \(t\to+\infty\) (\(t\to-\infty\)). This explains the second condition of the theorem.

The following theorem gives a sufficient criterion for ellipticity of (1).

Theorem 7. Let (1) be such a mixed system with respect to the o. f. L. \(V(x)\) that

1) on one of the orienting cones, say on \(H_{l_1}\) of the set (2), the function \(F_2(x)\) is negative definite;

Remark. In Figs. 4, 6–8 the plus or minus signs standing inside normal regions indicate that in the corresponding normal region \(F_1(x)>0\) or \(F_1(x)<0\).

The inscriptions placed next to the cones \(F_2>0\) or \(F_2<0\) indicate that on the corresponding cone \(F_2\) is positive or negative definite.

2) all normal regions \(\omega_1\) lying between the cone \(H_{l_1}\) and each of the remaining cones \(H_{l_j}\) from (2) are unstable s.n.r., and their number, for at least one \(j=2,\ldots,s\), is even.

Then (1) is an elliptic system.

Proof. Let us assume, for definiteness, that in the normal region \(\omega_{l_1}\) with lateral surface \(H_{l_1}\) the function \(F_1(x)<0\). Then in the normal regions \(\omega_{l_j}\), \(j=2,\ldots,s\), \(F_1(x)>0\), since the number of normal regions lying between \(H_{l_1}\) and \(H_{l_j}\), \(j=2,\ldots,s\), is even by condition 2), while \(F_1(x)\) changes sign to the opposite sign at each passage from one normal region to an adjacent one (by the definition of the generalized Lyapunov function). From conditions 1) and 2) it follows that on the surface \(H_{l_j}\), for at least one \(j=2,\ldots,s\), the function
\[ F_2(x)=\frac{d^2V}{dt^2} \]
is negative definite. Hence some part or all of the \(s-1\) simple (i.e., nonmixed) normal regions \(\omega_{l_j}\), \(j=2,\ldots,s\), are elliptic, as is \(\omega_{l_1}\), with \(\omega_{l_1}\) attracting and the former repelling. The arrangement of the arcs of trajectories in them for \(E^3\) is given in Fig. 5, \(a\) and \(b\). Those of the normal regions \(\omega_{l_i}\), \(i=1,\ldots,s\), which turn out not to be elliptic will be repelling hyperbolic, as is easy to see (Fig. 5, \(c\)), and will be filled with arcs of negative ray semitrajectories. In the remaining normal regions of the neighborhood \(U\), as follows from the condition of their instability, we shall have only arcs of trajectories of finite time length with endpoints at points of the lateral surfaces and at the back walls of these regions, distinct from \(O\). Their positive semitrajectories lie in \(\omega_{l_1}\). Thus, system (1) is elliptic (Fig. 6). The theorem is proved.

Fig. 5.

The following theorem expresses a criterion of hyperbolicity of a mixed system. Its proof is carried out analogously to the proof of Theorem 7. Therefore we give it without proof.

Theorem 8. Let (1) be a mixed system with respect to the generalized nonnegative Lyapunov function \(V(x)\in C^2(U)\). Then all its trajectories in the neighborhood \(U\), except for the singular point \(O\) and rays (parabolas), will be hyperbolic if the following conditions are satisfied:

1) on one of the orienting cones (say, on \(H_{l_1}\)) the function \(F_2(x)\) is positive definite;

2) all normal regions \(\omega_i\) lying between the pair of cones \(H_{l_1}\) and \(H_{l_j}\) are unstable mixed normal regions, and their number is even for at least one \(j=2,\ldots,s\).

In this case the sets of \(O^+\)- and \(O^-\)-curves in \(U\) are nonempty (Fig. 7).

As has already been noted, in [8] several sufficient conditions were given for the elliptic-hyperbolicity of a mixed system. These criteria were formulated using properties of the third derivative with respect to \(t\) of the generalized Lyapunov function \(V(x)\) along system (1).

There are cases when the elliptic-hyperbolicity of the mixed system (1) follows from the properties of the first two derivatives of the function \(V(x)\) with respect to \(t\) along system (1).

Fig. 6. \((n=3,\ s=3)\)

Fig. 7. \((n=3,\ s=3)\)

Let us suppose that system (1) and the function \(V(x)\) are such that:

a) all the conditions of Theorem 6 (respectively, Theorem 7) are satisfied, with those changes which will be described below in b) and c);

b) in addition, suppose that among the cones in (2) there are \(q-1\) cones, \(1<q<s\) (for definiteness we shall assume that these cones are \(H_{l_j}\), \(j=2,\ldots,q<s\)), which are separated from the orienting cone \(H_{l_1}\) by an odd number of unstable c. n. r.;

c) in contrast to the preceding considerations (see Theorems 5–8 and their proofs), assume that at least one of the regions \(E(H_{l_j})\), \(j=2,\ldots,q\), is not a normal region and contains some (finite) number of cones \((i=2,\ldots,q)\) \(H_1,H_2,\ldots,H_{r_i}\) from the variety \(F_1^0\) with common vertex \(O\). Let the regions \(E(H_1),\ldots,E(H_{r_i})\), \(i=2,\ldots,q\),—the “exteriors” of the cones \(H_1,\ldots,H_{r_i}\)—be normal regions, and let on their lateral surfaces \(H_1,\ldots,H_{r_i}\) the function \(F_2(x)\) be positive definite (respectively negative definite) (see a)).

Then the following theorem holds, expressing one of the sufficient criteria for elliptic-hyperbolicity of system (1).

Theorem 9. If, for the mixed system (1), conditions a), b), c) are fulfilled, then it is elliptic-hyperbolic; moreover, through the points of the union of the cones \(H_k\setminus O\), \(k=1,\ldots,R_i,\ i=2,\ldots,q\), pass only hyperbolic (elliptic) trajectories, while through the points of the cones \(H_{l_{q+1}}, H_{l_{q+2}},\ldots,H_{l_s}\) pass only elliptic (hyperbolic) trajectories, respectively.

We shall carry out the proof for the case when, on the orienting cone \(H_{l_1}\setminus O\), the function \(F_2(x)\) is negative (the case when on \(H_{l_1}\) the function \(F_2(x)\) is positive definite is treated analogously). Then it is easy to see that on the cones \(H_{l_2}, \ldots, H_{l_q}\), \(1<q<s\), the function \(F_2(x)\) is positive definite (this follows from a) and b)). And since, by condition c), \(F_2(x)>0\) for \(x\in \bigcup_{j=1}^{r_i} H_j\setminus O\), \(i=1,\ldots,q\), then if the cones \(H_k\setminus O\), \(k=1,\ldots,r_2\), lie, for example, in the domain \(I(H_{l_2})\) (if such cones are also present in other domains \(I(H_{l_i})\), \(i=3,4,\ldots,q\), then everything will proceed analogously), the domain \(\omega_{l_2}\) between \(H_{l_2}\) and \(\bigcup_{k=1}^{r_2} H_k\) will be a hyperbolic normal attracting domain (if we assume that in \(\omega_{l_1}\), \(F_1(x)<0\)). The domains \(I(H_k)\), \(k=1,\ldots,r_2\), will be repelling hyperbolic n. d. Hence, within the union of the domains \(\omega_{l_2}\), \(I(H_k)\), \(k=1,\ldots,r_2\), and their common boundary (without \(O\)), we have hyperbolic trajectories passing through the points of the surface \(\bigcup_{k=1}^{r_2} H_k\setminus O\).

System (1) will be elliptic-hyperbolic, since the trajectories passing through the points of the cones \(H_{l_{q+1}}, H_{l_{q+2}}, \ldots, H_{l_s}\) will be elliptic (which is established in exactly the same way as in the proof of Theorem 6) (Fig. 8,a). Figure 8,b shows the case when on \(H_{l_1}\setminus O\), \(F_2(x)>0\), \(q=0\), \(r_2=1\). It is clear that the number of such criteria can be increased.

Fig. 8. \((n=3,\ q=2,\ r_2=1)\)

All that has been stated above remains valid also for such systems (1) for which the function \(F_1(x)=0\) also in certain sets inside normal domains that do not divide \(U(O)\), and also if the cones \(H_i\) from \(F_1^0\) intersect in manifolds of dimension less than \(n-1\).

Let us also note that if any of the arrangements of trajectories of system (1) in \(U\) considered in this paper is prescribed, then one can prove the existence of a generalized Lyapunov function that yields the corresponding theorem from those given above.

If, for the functions used in the investigation of system (1), one imposes the requirement that their absolute values tend to \(\infty\) as \(|x|\to\infty\), \(x\in\omega_i\) (see [11]), then the results obtained can be extended to an arbitrarily large neighborhood \(U\subset E^n\) (the investigation as a whole).

References

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2. Nemytskii V. V. Vestnik MGU, ser. I, No. 5, 1961.

3. Kudaev M. B. DAN SSSR, 147, No. 6, 1962.

4. Shestakov A. A. On the asymptotic behavior of multidimensional systems of differential equations. All-Union Correspondence Institute of Railway Engineers. Scientific Notes, issue 7. Moscow, 1961.

5. Nemytskii V. V. UMN, 9, issue 3 (61), 1954.

6. Letov A. M. Stability of nonlinear controlled systems. Fizmatgiz, Moscow, 1962.

7. Kudaev M. B. Vestnik MGU, ser. I, No. 1, 1963.

8. Kudaev M. B. Dissertation. Moscow State University, 1962.

9. Nemytskii V. V., Stepanov V. V. Qualitative theory of differential equations. Moscow, 1949.

10. Kudaev M. B. Scientific Notes of Kabardino-Balkarian State University, issue 19, 1963.

11. Barbashin E. A., Krasovskii N. N. DAN SSSR, 86, issue 3, 1952.

Received by the editors October 10, 1964.
Kabardino-Balkarian State University