ASYMPTOTIC CHARACTERISTICS OF NONLINEAR DIFFERENTIAL SYSTEMS¹

Yu. S. BOGDANOV

At the basis of the following arguments lie very simple considerations. In order not to obscure them with purely technical details, we shall restrict ourselves to the consideration of an isolated equilibrium point of an autonomous finite-dimensional differential system

\[ dx/d\tau=f(x) \tag{1} \]

with continuous right-hand side \(f(x)\) and with bilaterally extendable nonintersecting solutions whose trajectories fill the whole phase space—the \(\nu\)-dimensional Euclidean space \(E=\{\xi\}\). The equilibrium point is located at the origin \(O\),

\[ f(0)=0. \]

The solution \(x=x(\tau)\) with initial value \(x(0)=\xi\in E\) will be denoted by \(x(\tau,\xi)\). As is known, the solutions of system (1) depend continuously on the initial values.

The point \(O\) is called stable in the sense of Lyapunov in the positive direction (for brevity, the words “in the sense of... direction” will be omitted everywhere below) if the quantity

\[ l(\delta)=\sup_{\|\xi\|=\delta,\ \tau>0}\|x(\tau,\xi)\| \]

is infinitesimal together with \(\delta\). If, in addition, \(x(\tau,\xi)\to 0\) as \(\tau\to+\infty\) for all \(\xi\) from some neighborhood of \(O\), then \(O\) is asymptotically stable.

The principal methods for establishing the stability of \(O\) remain, to this day, the first and second methods of Lyapunov [1]. The starting point of the first method is the study as a whole (for all positive values of the argument) of each individual solution of system (1). The starting point of the second method is the study in the small (on each infinitesimal interval of variation of \(\tau\)) of all the solutions of (1) at once with sufficiently small initial values.

The scheme of the first method:

\[ \sup_{\|\xi\|=\delta}\ \varlimsup_{\tau\to+\infty}\Lambda_{1}, \]

the scheme of the second method:

\[ \varlimsup_{\tau\to+\infty}\ \sup_{\|\xi\|=\delta}\Lambda_{2}. \]

¹ A report delivered at the First Belarusian Mathematical Conference, January 25–28, 1964.

In the simplest cases \(\Lambda_1\) and \(\Lambda_2\) mean \(\|x(\tau,\xi)\|\), and in more complicated ones, functions of definite properties of \(\xi\) and \(\tau\).

As we see, the two Lyapunov methods are externally, logically connected with one another. However, there is also an essential difference between them. It consists in the fact that from

\[ \varlimsup_{\tau\to+\infty}\ \sup_{\|\xi\|=\delta}\ \|x(\tau,\xi)\|\to 0 \quad \text{as } \delta\to 0 \]

there always follows \(l(\delta)\to 0\), and hence the stability of \(O\). But from

\[ \sup_{\|\xi\|=\delta}\ \varlimsup_{\tau\to+\infty}\ \|x(\tau,\xi)\|\to 0 \quad \text{as } \delta\to 0, \]

as the well-known example of N. N. Krasovskii [2] shows, the stability of \(O\), generally speaking, does not follow.

The fundamental difference noted above was the reason why the fates of the two Lyapunov methods developed differently. A detailed study of the second method showed its universality in the sense that, with the help of the second method, one can detect the asymptotic stability of the rest point of any particular system (1). A suitable Lyapunov function always exists. At the same time, in recent years publications have appeared that cast doubt on the applicability of the first method to the study of asymptotic stability of rest points of systems of a sufficiently general form.

Below an attempt is set forth to extend the domain of application of Lyapunov’s first method, an attempt to prove that this method also turns out to be universal in a sense analogous to that indicated above.

More precisely, we shall speak of studying the applicability of that specialization of the first method which is called the method of characteristic numbers. Other aspects of Lyapunov’s first method are developed in the theory of the method of reduction.

It is precisely the method of characteristic numbers (or, as it is otherwise called, the method of characteristic exponents) that has been regarded as of limited applicability. The immediate reason for this is the following. It is known that if (1) is a linear system and has negative characteristic exponents, then \(O\) is an asymptotically stable rest point. In the example of N. N. Krasovskii mentioned above, a two-dimensional system is constructed with an unstable rest point attracting all trajectories. By suitably changing the time scale on each of the trajectories of the system, R. E. Vinograd [3] succeeded in ensuring that, in a system analogous to Krasovskii’s system, all motions eventually begin to approach the stationary motion under study very rapidly. As a result, the characteristic exponents of all solutions turn out to be essentially negative, while the rest point remains unstable.

Analysis of the revealed facts leads to the idea that, in passing from the study of weakly nonlinear systems to the study of essentially nonlinear systems, the definition of a characteristic number must be changed in accordance with the very meaning of this concept.

From Lyapunov’s definition it follows that the characteristic number (with the sign changed, i.e., the characteristic exponent) \(\bar\omega x\) of the vector function \(x(\tau)\) is equal to

\[ \bar\omega x=\varlimsup_{\tau\to+\infty}\frac{1}{\tau}\,h(\tau,\tau_0);\qquad h(\tau,\tau_0)=\ln[\|x(\tau+\tau_0)\|/\|x(\tau_0)\|]. \]

If (1) is a linear system, then for the given solution \(x(\tau)\) the quantity \(h(\tau,\tau_0)\), for sufficiently large \(\tau\), depends mainly on \(\tau\). For nonlinear systems the situation is different; therefore, instead of \(h(\tau,\tau_0)\) it is natural to introduce

\[ \sup_{-\infty<\tau_0<+\infty} h(\tau,\tau_0) \]

and, instead of the characteristic exponent \(\overline{\omega}x\), to consider the generalized exponent

\[ \overset{*}{\Omega}x=\overline{\lim_{\tau\to+\infty}}\ \sup_{-\infty<\tau_0<+\infty}\frac{1}{\tau}h(\tau,\tau_0). \]

The generalized exponent has the following important property:

I. If \(O\) is unstable, then through every neighborhood of it there passes at least one trajectory of a nontrivial solution whose generalized exponent is positive or zero.

For two-dimensional systems the stated property of the generalized exponent is obvious. Indeed, from the instability of \(O\) there follows the existence of an \(O^{-}\)-solution \(\tilde{x}(\tau)\), i.e., a solution whose trajectory approaches \(O\) as \(\tau\to-\infty\):

\[ \lim_{\tau\to-\infty}\|\tilde{x}(\tau)\|=0. \]

For such a solution, for every sufficiently large \(\tau_0\) one has \(\|\tilde{x}(-\tau_0)\|<\|\tilde{x}(0)\|\), and therefore

\[ \tilde{h}(\tau_0,-\tau_0)>1, \]

\[ \sup_{-\infty<\tau_0<+\infty}\frac{1}{\tau}\tilde{h}(\tau,\tau_0)>0 \quad \text{for } \tau>\tau^{*}, \]

and hence

\[ \overset{*}{\Omega}\tilde{x}\geqslant 0. \]

The following example shows that already for \(n=3\) there exist unstable equilibrium points to which no trajectory approaches as \(\tau\to-\infty\).

Example. Consider the system

\[ \left\{ \begin{aligned} \frac{dx_1}{d\tau} &= \frac{-x_2+x_1(x_1^2+x_2^2-x_3^2)K(x)} {1+x_1^2+x_2^2+x_3^2}, \\[6pt] \frac{dx_2}{d\tau} &= \frac{x_1+x_2(x_1^2+x_2^2-x_3^2)K(x)} {1+x_1^2+x_2^2+x_3^2}, \\[6pt] \frac{dx_3}{d\tau} &= \frac{(x_1^2+x_2^2-x_3^2)L(x)} {1+x_1^2+x_2^2+x_3^2}, \end{aligned} \right. \tag{2} \]

where

\[ K(\xi)= \begin{cases} 1, & \xi_1^2+\xi_2^2\leqslant \xi_3^2,\\ \xi_3/(\xi_1^2+\xi_2^2), & \xi_1^2+\xi_2^2>\xi_3^2, \end{cases} \]

\[ L(\xi)= \begin{cases} \xi_3, & \xi_1^2+\xi_2^2\leqslant \xi_3^2,\\ 1, & \xi_1^2+\xi_2^2>\xi_3^2. \end{cases} \]

System (2) belongs to systems of type (1), since the right-hand sides of (2) are continuous and satisfy a local Lipschitz condition with respect to their arguments in the entire three-dimensional phase space, and at infinity have order no higher than the first (relative to the norm of the vector argument). The unique equilibrium point of (2) is the origin \(O=(0,0,0)\).

The solution \(\tilde{x}(\tau)=[\tilde{x}_1(\tau),\tilde{x}_2(\tau),\tilde{x}_3(\tau)]\) with initial values

\[ \tilde{x}(0)=[0,0,\tilde{\xi}_3],\qquad \tilde{\xi}_3\ne 0 \]

has as its trajectory the half-axis \(\xi_3\) (positive for \(\tilde{\xi}_3>0\) and negative otherwise), motion along which, as \(\tau\) increases, is directed toward \(O\) and goes to infinity as \(\tau\to-\infty\).

Every solution \(\overset{*}{x}(\tau)\) with initial values

\[ \overset{*}{x}(0)=[\overset{*}{\xi}_1,\overset{*}{\xi}_2,\overset{*}{\xi}_3],\qquad \overset{*}{\xi}_1^{\,2}+\overset{*}{\xi}_2^{\,2}=\overset{*}{\xi}_3^{\,2}>0 \]

is periodic; the trajectory of such a motion is a horizontal circle lying on the cone (\(\xi_1,\xi_2,\xi_3\) are always the current coordinates)

\[ \xi_1^2+\xi_2^2=\xi_3^2 . \tag{3} \]

The trajectory of the solution \(\check{x}(\tau)\) with initial values

\[ \check{x}(0)=[\check{\xi}_1,\check{\xi}_2,\check{\xi}_3],\qquad 0<\check{\xi}_1^{\,2}+\check{\xi}_2^{\,2}<\check{\xi}_3^{\,2} \]

is a conical spiral lying on the circular cone

\[ (\xi_1^2+\xi_2^2)\check{\xi}_3^{\,2} = \xi_3^2(\check{\xi}_1^{\,2}+\check{\xi}_2^{\,2}), \tag{4} \]

approaching \(O\) as \(\tau\to+\infty\) and going to infinity as \(\tau\to-\infty\). The trajectory of the solution \(\hat{x}(\tau)\) with initial value

\[ \hat{x}(0)=[\hat{\xi}_1,\hat{\xi}_2,\hat{\xi}_3],\qquad \hat{\xi}_1^{\,2}+\hat{\xi}_2^{\,2}>\hat{\xi}_3^{\,2} \]

is a spiral running along the hyperboloid of revolution

\[ \xi_1^2+\xi_2^2+\hat{\xi}_3^{\,2} = \hat{\xi}_1^{\,2}+\hat{\xi}_2^{\,2}+\xi_3^2 \tag{5} \]

from infinity to infinity (from bottom to top). If one moves along the cones (4) or along the hyperboloids (5) toward the principal cone (3), then the “pitch” of the corresponding spiral \(\check{x}(\tau)\) or \(\hat{x}(\tau)\) decreases without bound.

The equilibrium point \(O\) of system (2) is unstable, since through every neighborhood of \(O\) there pass cones (5) carrying trajectories that go upward to infinity as \(\tau\to+\infty\). At the same time, \(O\) is not an \(\alpha\)-limit point of any trajectory corresponding to a nontrivial solution of (2). However, \(O\) has property I (in any neighborhood of \(O\) there are initial values of periodic solutions \(\overset{*}{x}(\tau)\), \(\Omega\,\overset{*}{x}=0\)). The decisive role here is played by the circumstance that, although \(O\) is not an \(\alpha\)-limit point for any nontrivial solution (2), in every neighborhood of \(O\) there are the indicated \(\alpha\)-limit points. The situation is analogous in the general case.

Let us denote by \(\Delta\) the set of those points \(E\) which are \(\alpha\)-limit points for at least one trajectory (1), distinct from \(O\).

II. If \(O\) is unstable, then \(O \in \overline{A}\). In other words, in any neighborhood of \(O\) there are points of \(A\).

Proof of II.
Suppose, to the contrary, that

\[ O \in \overline{\overline{A}}. \tag{6} \]

Denote by \(\varepsilon_0\) a number such that the entire closed sphere \(S(\varepsilon_0)\) with center at \(O\) and radius \(\varepsilon_0\) has no points in common with \(A\),

\[ A \cap S(\varepsilon)=O. \tag{7} \]

Take an arbitrary \(\varepsilon,\ 0<\varepsilon<\varepsilon_0\). The negative semitrajectory of the solution \(x(\tau,\sigma)\), where \(\sigma\) is an arbitrary point of the surface \(S_{\mathrm{gr}}(\varepsilon)\) of the sphere \(S(\varepsilon)\), cannot entirely belong to \(S(\varepsilon)\), since from \(x(\tau,\sigma)\in S(\varepsilon)\) for all \(\tau<0\) (or even for some \(\{\tau_n\}\to-\infty\)) there follows the existence of an \(\alpha\)-limit point of \(x(\tau,\sigma)\) in \(S(\varepsilon)\), which contradicts (7). To the point \(\sigma\) we assign the number \(t(\sigma)\)—the greatest of the nonpositive instants of strict entry of \(x(\tau,\sigma)\) into \(S(\varepsilon)\), i.e. \(x(\tau,\sigma)\in S(\varepsilon)\) for all \(\tau\in[t(\sigma),0]\) and \(x(\tau_n,\sigma)\in \overline{S(\varepsilon)}\), where \(\{\tau_n\}\) is some increasing sequence converging to \(t(\sigma)\). In particular, \(t(\sigma)\) may turn out to be zero.

By virtue of the continuous dependence of the solutions (1) on the initial values and the closedness of \(S(\varepsilon)\), it follows that \(x(\tau_n,\sigma)\in \overline{S(\varepsilon)}\) for all \(\sigma\) sufficiently close to \(\sigma_0\). Therefore the function \(t(\sigma)\) is lower semicontinuous, i.e.

\[ \lim_{\sigma\to\sigma_0} t(\sigma) \geq t(\sigma_0). \]

The lower semicontinuity of \(t(\sigma)\) on \(S_{\mathrm{gr}}(\varepsilon)\) ensures that this function is bounded below by some number \(\tau^*\):

\[ t(\sigma)\geq \tau^*, \qquad \sigma\in S_{\mathrm{gr}}(\varepsilon). \]

The vector-function \(f(x)\) is continuous on \(S_{\mathrm{gr}}(\varepsilon)\) and vanishes at \(O\); therefore there exists a positive \(\delta=\delta(\varepsilon)\), \(\delta(\varepsilon)\in(0,\varepsilon)\), such that every trajectory (1) requires, in going from \(S_{\mathrm{gr}}(\delta)\) to \(S_{\mathrm{gr}}(\varepsilon)\), a time greater in absolute value than \(|\tau^*|\), since otherwise there would be found a trajectory of a nontrivial solution (1) reaching \(O\) in finite time. By the choice of \(\delta\), none of the negative semitrajectories of the solutions \(x(\tau,\sigma)\), \(\sigma\in S_{\mathrm{gr}}(\varepsilon)\), has common points with \(S(\varepsilon)\). Consequently, no positive semitrajectory (1) with initial values from \(S(\delta)\) reaches the boundary \(S_{\mathrm{gr}}(\varepsilon)\) and remains inside \(S(\varepsilon)\), i.e.

\[ \|\xi\|\leq \delta \]

implies

\[ \|x(\tau,\xi)\|<\varepsilon \tag{8} \]

for all \(\tau>0\). The number \(\varepsilon\) is arbitrary; therefore from (8) the stability of \(O\) follows, which contradicts condition II.

Thus, (6) leads to a contradiction, and, consequently, it is true that

\[ O \in \overline{A}. \]

Remark. Theorem II remains valid also in other spaces with a compact sphere about \(O\) and may fail if \(O\) has no compact neighborhoods.

Proof I.
Suppose that \(O\) is unstable. Take any neighborhood \(u\) of the point \(O\). On the basis of II there exist a trajectory \(x(\tau,\xi_0)\), \(\xi_0 \in u\setminus O\), and a point \(\xi_\alpha \in u\) such that \(\{x(-\tau_n,\xi_0)\}\to \xi_\alpha\) for some sequence \(\{\tau_n\}\to +\infty\). The vector \(x(0,\xi_0)=\xi_0\) is different from \(O\). The sequence \(\{\|x(-\tau_n,\xi_0)\|\}\) is bounded above. Consequently, there exists such a \(\Delta\) that

\[ \|x(0,\xi_0)\|/\|x(-\tau_n,\xi_0)\|\geq \Delta, \]

and therefore

\[ h(\tau_n,-\tau_n)\geq \ln \Delta, \]

\[ \sup_{-\infty<\tau_0<+\infty}\frac{1}{\tau_n}h(\tau_n,\tau_0)\geq 0, \]

\[ \overset{*}{\Omega}x(\tau,\xi_0) = \overline{\lim_{\tau\to+\infty}} \sup_{-\infty<\tau_0<+\infty} \frac{1}{\tau}h(\tau,\tau_0)\geq 0. \]

From I there follows a sufficient criterion for the asymptotic stability of \(O\).

III. If the generalized exponents of all nontrivial solutions of (1) with initial values from a sufficiently small neighborhood of \(O\) are negative, then \(O\) is asymptotically stable.

Indeed, the stability of \(O\) follows from I. Moreover, from the negativity of the generalized exponent of the solution \(x(\tau,\xi)\) it follows that

\[ \sup_{-\infty<\tau_0<+\infty}\frac{1}{\tau}h(\tau,\tau_0)<a=\mathrm{const}<0 \quad \text{for } \tau>\tau^*; \]

\[ \frac{1}{\tau}h(\tau,0) = \frac{1}{\tau}\ln[\|x(\tau,\xi)\|/\|\xi\|] <a,\quad \tau>\tau^*; \]

\[ \ln\|x(\tau,\xi)\|\to -\infty \quad \text{as } \tau\to+\infty, \]

\[ \|x(\tau,\xi)\|\to 0 \quad \text{as } \tau\to+\infty. \]

Thus the equilibrium point \(O\) is not only stable, but also serves as an attracting point for all neighboring trajectories. Consequently, \(O\) is asymptotically stable.

Remark. In connection with proposition III the following problem arises:
if the generalized exponents of all nontrivial solutions of some system (1) with sufficiently small initial conditions are negative, must they necessarily be essentially negative? In other words, if for the given system (1)

\[ \overset{*}{\Omega}x(\tau,\xi)<0 \]

for all \(0<\|\xi\|<\delta\), must it necessarily follow that

\[ \sup\{\overset{*}{\Omega}x(\tau,\xi)\}<0? \]

If the generalized exponents of the solution \(x(\tau,\xi)\) are positive, then there exist sequences \(\{\tau_n\}\) and \(\{\tau_{0n}\}\) such that

\[ \{\tau_n\}\to +\infty,\quad -\infty<\{\tau_{0n}\}<+\infty, \]

\[ \frac{1}{\tau_n}h(\tau_n,\tau_{0n})>\beta=\mathrm{const}>0, \]

\[ \|x(\tau_n+\tau_{0n},\xi)\|>\|x(\tau_{0n},\xi)\|e^{\beta\tau_n}. \]

Two cases are possible:

1) \(\|x(\tau_{0n}, \xi)\| \to 0\) as \(n \to \infty\);

2) \(\displaystyle \lim_{n\to\infty}\|x(\tau_{0n}, \xi)\| > 0\).

In the first case \(O\) is an \(\alpha\)-limit point for \(x(\tau,\xi)\); in the second, \(\|x(\tau,\xi)\|\) increases without bound. If there exists \(\{\xi_n\}\to 0\), \(\overset{*}{\Omega}x(\tau,\xi_n)>0\), then it follows from what has been set forth that \(O\) cannot be stable, i.e.

IV. If in every neighborhood of \(O\) there are initial values of solutions of (1) with positive generalized exponents, then \(O\) is unstable.

Let us take some vector function \(x(\tau)\), \(-\infty<\tau<+\infty\). From the definition of the generalized exponent \(\overset{*}{\Omega}x\) it follows that for any scalar function \(\varphi(\tau)\)

\[ \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}h[\tau,\varphi(\tau)]\leq \overset{*}{\Omega}x. \tag{9} \]

Put \(\varphi(\tau)\equiv 0\). Then from (9) it follows that

\[ \overset{*}{\Omega}x\geq \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}\ln[\|x(\tau)\|/\|x(0)\|] =\overline{\omega}x. \]

Put \(\varphi(\tau)\equiv-\tau\). Then

\[ \overset{*}{\Omega}x\geq \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}\ln[\|x(0)\|/\|x(-\tau)\|] = \]

\[ =-\underline{\lim_{\tau\to+\infty}}\frac{1}{\tau}\ln[\|x(-\tau)\|/\|x(0)\|] =-\underline{\alpha}x, \]

where \(\underline{\alpha}x\) is the lower exponent of the vector function \(x(-\tau)\), or, otherwise, the minus-exponent of \(x(\tau)\), introduced by D. M. Grobman [4]. Finally,

\[ \overset{*}{\Omega}x\geq \max\{\overline{\omega}x,\,-\underline{\alpha}x\}. \tag{10} \]

For systems (1) possessing sufficient regularity of solutions, in relation (10) the equality sign holds for all solutions. However, there exist systems (1) for some solutions of which

\[ \overset{*}{\Omega}x>\max\{\overline{\omega}x,\,-\underline{\alpha}x\}. \tag{11} \]

Remark. Systems (1) naturally split into three classes: all solutions (here and below, all solutions with sufficiently small initial values) of systems of the 1st class are such that \(\overset{*}{\Omega}x=\overline{\omega}x=-\underline{\alpha}x\). In systems of the 2nd class there exist solutions \(x_0(\tau)\) such that \(\overline{\omega}x_0\neq-\underline{\alpha}x_0\), but for all solutions

\[ \overset{*}{\Omega}x=\max\{\overline{\omega}x,\,-\underline{\alpha}x\}. \]

Finally, systems of the 3rd class have solutions for which (11) is valid.

The number (possibly improper) \(\max\{\overline{\omega}x,\,-\underline{\alpha}x\}\) will be called the small generalized exponent and denoted by \(\overline{\Omega}x\). On the basis of (10), the small generalized exponent of a vector function never ne-

exceed the generalized exponent of the very same function. It is not difficult to see that Propositions I and III remain valid if, in the formulation of these propositions, one speaks of small generalized exponents rather than of generalized exponents (Proposition IV remains valid in a trivial way).

With the aid of generalized exponents one can also detect conditional stability.

Let \(m\) denote a certain closed subset of the space \(E\), invariant with respect to system (1) and containing the point \(O\). The point \(O\) is called stable with respect to perturbations from \(m\) if the quantity
\[ M(\delta)=\sup \|x(\tau,\xi)\|, \]
where the supremum is taken over all \(\tau\in[0,+\infty)\) and over all \(\xi\in S(\delta)\cap m\), is infinitesimal together with \(\delta\). The point \(O\) is unstable with respect to perturbations from \(m\) if
\[ \lim_{\delta\to+0} M(\delta)>0. \]

Proposition II can be strengthened in the sense that from the instability of \(O\) with respect to perturbations from \(m\) it follows that \(O\in \overline{A}\cap m\), where \(A\) still denotes the set of those points of \(E\) which are \(\alpha\)-limit points for at least one trajectory distinct from \(O\). Thus, from the instability of \(O\) it follows that, in every neighborhood of \(O\), there are points of \(m\) serving as \(\alpha\)-limit points for trajectories of nonzero solutions of (1). From the strengthened Proposition II there immediately follow criteria for conditional stability or instability analogous to Propositions I, III, IV. For example, a sufficient criterion for conditional asymptotic stability (the analogue of III): if the small generalized exponents of all nontrivial solutions with initial values (sufficiently small in norm) from \(m\) are negative, then \(O\) is asymptotically stable with respect to perturbations from \(m\).

The generalized exponents are asymptotic invariants in the following sense. Suppose that system (1) is subjected to the change of variables
\[ \eta=h(\xi), \]
where \(h\) is continuously differentiable, has a continuously differentiable inverse function \(\xi=h^{-1}(\eta)\), and
\[ \alpha\|\xi\|\leq\|\eta\|\leq \beta\|\xi\|,\qquad \tilde\alpha\|\eta\|\leq\|h^{-1}(\eta)\|\leq \tilde\beta\|\eta\|, \]
\[ \alpha,\ \beta,\ \tilde\alpha,\ \tilde\beta=\mathrm{const}>0;\qquad \xi,\eta\in E \]
(a “Lyapunov transformation”). Then the corresponding solutions of (1) and of the transformed system
\[ dy/d\tau=g(y) \tag{12} \]
have the same generalized exponents and small generalized exponents. Indeed, for example,
\[ y(\tau,\eta)=h[x(\tau,\xi)],\qquad \eta=h(\xi), \]
\[ \ln[\|y(\tau+\tau_0)\|/\|y(\tau_0)\|]\leq \ln[\|x(\tau+\tau_0)\|/\|x(\tau_0)\|]+\ln\frac{\beta}{\alpha}, \]
\[ \overset{*}{\Omega}y= \lim_{\tau\to+\infty}\ \sup_{-\infty<\tau_0<+\infty} \frac{1}{\tau}\ln[\|y(\tau+\tau_0)\|/\|y(\tau_0)\|]\leq \overset{*}{\Omega}x, \]
but \(x\) and \(y\) are equivalent; therefore
\[ \overset{*}{\Omega}y=\overset{*}{\Omega}x. \]

The preceding considerations show that the method of characteristic numbers, with a suitable generalization of the very concept of “characteristic number,” can be applied to establish the stability of the equilibrium point of system (1) in the sense that, from the negativity of the generalized exponents of all nonzero solutions of (1) with sufficiently small initial values, the asymptotic stability of \(O\) follows. Thus uniform-exponential stability can always be detected. In order to make the method universal, one must, of course, give the concept of “characteristic number” greater flexibility. First, it must be taken into account that exponential decrease of the norms of solutions of (1) close to \(O\) is typical only for systems of a comparatively narrow type. Second, the solutions of the given system (1) may be distributed among several classes which differ from one another in essentially different types of asymptotic behavior of the norms of solutions.

We shall restrict ourselves to considering (1) in some domain \(\Xi \subseteq E\) containing the origin of coordinates. Put \(\dot{\Xi}=\Xi\setminus O\). Denote the boundary of \(\Xi\) by \(\Xi_{\mathrm{gr}}\) (\(\Xi_{\mathrm{gr}}\) may also be the empty set). A continuous vector function \(x(\tau)\), defined for all \(\tau\in(-\infty,+\infty)\) and taking values in \(\dot{\Xi}\), will be assigned to the class \(R\) (“the class of radii”) if \(x(\tau)\to O\) as \(\tau\to+\infty\) and if the \(\omega\)-limit set of \(x\) is contained in \(\Xi_{\mathrm{gr}}\),

\[ \operatorname*{Lim}_{\tau\to+\infty} x(\tau)\subset \Xi_{\mathrm{gr}}. \]

On the domain \(\dot{\Xi}\) we define a scalar function \(v\), whose values fill \((0,+\infty)\):

\[ v:\dot{\Xi}\xrightarrow{\text{onto}}(0,+\infty). \]

Let \(\Gamma_\gamma\) denote the level set of the function \(v\) corresponding to the number \(\gamma\), i.e.

\[ \Gamma_\gamma=\{\xi\mid \xi\in\dot{\Xi},\quad v(\xi)=\gamma\}. \]

For \(x\in R\) and \(\gamma>0\) put

\[ \underline{\tau}(x,\gamma)=\inf\{\tau\mid x(\tau)\in\Gamma_\gamma\}, \]

\[ \overline{\tau}(x,\gamma)=\sup\{\tau\mid x(\tau)\in\Gamma_\gamma\}. \]

Suppose that the function \(v\) has the following properties:

\(v_1\). Each \(\Gamma_\gamma\), \(\gamma>0\), is compact;

\(v_2\). For any \(x\in R\) and all \(\gamma,\tilde{\gamma}\) \((0<\gamma<\tilde{\gamma})\), one has

\[ \underline{\tau}(x,\gamma)<\underline{\tau}(x,\tilde{\gamma}),\qquad \overline{\tau}(x,\gamma)<\overline{\tau}(x,\tilde{\gamma}). \]

From properties \(v_{1-2}\) it follows that all sets \(\Gamma_\gamma\), \(\gamma>0\), are nonempty and all quantities \(\underline{\tau}(x,\gamma)\), \(\overline{\tau}(x,\gamma)\) are finite.

As \(v\) one may use, in particular, a nonnegative function which vanishes only at \(O\), and such that each level set is a closed surface, with the level surfaces corresponding to larger values of \(\gamma\) enclosing the level surfaces corresponding to smaller \(\gamma\).

We next consider a function \(d(\gamma_1,\gamma_2)\), defined for all positive values of its arguments and taking all possible real values,

\[ d:(0,+\infty)\times(0,+\infty)\xrightarrow{\text{onto}}(-\infty,+\infty). \]

Suppose that for all positive \(\gamma, \gamma_1, \gamma_2, \gamma_3\) \((\gamma_1<\gamma_2<\gamma_3)\) the following hold:
\[ \begin{aligned} d_1.\quad & d(\gamma,\gamma)=0;\\ d_2.\quad & 0<d(\gamma_2,\gamma_1)=-d(\gamma_1,\gamma_2);\\ d_3.\quad & d(\gamma_2,\gamma)>d(\gamma_1,\gamma);\\ d_4.\quad & d(\gamma_3,\gamma_2)+d(\gamma_2,\gamma_1)\geq d(\gamma_3,\gamma_1). \end{aligned} \]
From the properties \(d_{1-4}\) it follows without difficulty that for arbitrary \(\gamma,\gamma_1,\gamma_2\)
\[ \left|d(\gamma_1,\gamma)-d(\gamma_2,\gamma)\right|\leq 2\left|d(\gamma_2,\gamma_1)\right|. \tag{13} \]

As \(d(\gamma_2,\gamma_1)\) one may take, in particular, the functions \(\gamma_2-\gamma_1\), \(\ln(\gamma_2/\gamma_1)\), or, in general, \(\varphi(\gamma_2)-\varphi(\gamma_1)\), where \(\varphi(\gamma)\) is an increasing function.

Take any vector-function \(x:(-\infty,+\infty)\to \Xi\). Put
\[ D(x,\tau,\tau_0)=d\{v[x(\tau_0+\tau)],\,v[x(\tau_0)]\}, \]
where \(\tau_0\in(-\infty,+\infty)\), \(\tau\in(0,+\infty)\). To the function \(x\) we associate the generalized characteristic numbers:

the \(vd\)-number \(\overset{*}{\Omega}\,vd\,x\)
\[ \overset{*}{\Omega}\,vd\,x= \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau} \sup_{-\infty<\tau_0<+\infty} D(x,\tau,\tau_0) \]
and the small \(vd\)-number \(\overline{\Omega}\,vd\,x\)
\[ \overline{\Omega}\,vd\,x= \max\left\{ \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}D(x,\tau,\tau_0),\, -\underline{\lim_{\tau\to+\infty}}\frac{1}{\tau}D(x,\tau,\tau_0) \right\}. \]

From the obvious relations
\[ \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}D(x,\tau,\tau_0) = \overline{\lim_{\tau\to+\infty}}\frac{\tau+\tau_0}{\tau}\, \frac{1}{\tau+\tau_0}D(x,\tau,\tau_0) = \]
\[ = \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}D(x,\tau-\tau_0,\tau_0), \]
\[ \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}D(x,\tau,\tau_0) = \overline{\lim_{\tau\to+\infty}} \left\{ \frac{1}{\tau}D(x,\tau-\tau_*,\tau_*)+ \right. \]
\[ \left. +\frac{1}{\tau}\bigl[D(x,\tau-\tau_0,\tau_0)-D(x,\tau-\tau_*,\tau_*)\bigr] \right\} \]
and (13) it follows that not only \(\overset{*}{\Omega}\,vd\,x\), but also \(\overline{\Omega}\,vd\,x\), does not depend on \(\tau_0\).

Propositions I, III, and IV, as well as their analogues pertaining to cases of conditional stability or instability, extend without any essential change in the proofs also to the case of the \(vd\)-number \(\overset{*}{\Omega}\,vd\,x\) and the small \(vd\)-number \(\overline{\Omega}\,vd\,x\). For example, if the small \(vd\)-numbers of all nontrivial solutions of (1) with initial values from a sufficiently small neighborhood of the point \(O\) are negative, then \(O\) is asymptotically stable.

At the same time, using the new results of the theory of the second Lyapunov method, one can also prove the necessary criterion of asymptotic stability in terms of the theory of the first method.

V. If \(O\) is asymptotically stable, then there exist functions \(v_0\) and \(d_0\), satisfying conditions \(v_{1-2}\) and \(d_{1-4}\), such that the \(v_0d_0\)-numbers of all nontrivial solutions (1) with sufficiently small initial values will be negative.

Proof.
Without loss of generality, we take \(\Xi\) to be the domain of attraction of \(O\). From the asymptotic stability of \(O\) there follows the existence of a Lyapunov function \(v_0(\xi)\), satisfying conditions \(v_{1-2}\) (even stronger conditions: \(\Gamma_\gamma^{0}\) are closed smooth hypersurfaces in \(E\)) and strictly decreasing along each nontrivial solution (1) (see, for example, [5, pp. 214–218]). Take arbitrary \(\gamma_1\) and \(\gamma_2\), \(0<\gamma_1<\gamma_2\). Denote by \(\Gamma_{\gamma_1\gamma_2}^{0}\) the set-theoretic union of all \(\Gamma_\gamma^{0}\), \(\gamma_1\leq \gamma\leq \gamma_2\). From the properties of \(v_0\) it follows that the trajectory of any solution \(x(\tau,\xi)\), \(\xi\in \Gamma_{\gamma_1\gamma_2}^{0}\), remains inside \(\Gamma_{\gamma_1\gamma_2}^{0}\) for a finite time \(T(\xi,\gamma_2,\gamma_1)\). From the continuous dependence of \(x(\tau,\xi)\) on \(\xi\), it follows that the set \(\{T(\xi,\gamma_2,\gamma_1)\}\) for all \(\xi\in\Gamma_{\gamma_1\gamma_2}^{0}\) is bounded above.

Put

\[ d_0(\gamma_2,\gamma_1)=\sup_\xi \{T(\xi,\gamma_2,\gamma_1)\}; \]

\[ d_0(\gamma_1,\gamma_2)=-d(\gamma_2,\gamma_1);\qquad d(\gamma_1,\gamma_1)=0. \]

The function \(d_0(\gamma_2,\gamma_1)\) satisfies conditions \(d_{1-4}\). Indeed, \(d_{1-3}\) are obvious, while \(d_4\) follows from the relations \((\xi\in \Gamma_{\gamma_1\gamma_2}^{0},\ \eta,\eta'\in\Gamma_{\gamma_2\gamma_3}^{0},\ \zeta\in\Gamma_{\gamma_3\gamma_1}^{0}\), where \(\eta\) lies on the trajectory \(x(\tau,\xi)\)),

\[ T(\xi,\gamma_2,\gamma_1)+T(\eta,\gamma_3,\gamma_2)=T(\xi,\gamma_3,\gamma_1)=T(\eta,\gamma_3,\gamma_1), \]

\[ \sup_\xi \{T(\xi,\gamma_2,\gamma_1)\}+\sup_{\eta'} \{T(\eta',\gamma_3,\gamma_2)\}\geq \sup_\zeta \{T(\zeta,\gamma_3,\gamma_1)\}. \]

From the definition of \(T\) and \(d_0\) it follows (\(\xi\) lies on the trajectory \(x(\tau)\), \(\tau>0\)) that

\[ T\{\xi, v_0[x(\tau_0)], v_0[x(\tau+\tau_0)]\}=\tau, \]

\[ d_0\{v_0[x(\tau+\tau_0)],\ v_0[x(\tau_0)]\}\leq -\tau, \]

\[ \frac{1}{\tau}\sup_{-\infty<\tau_0<+\infty} d_0\{v_0[x(\tau+\tau_0)],\ v_0[x(\tau_0)]\}\leq -1. \]

Consequently,

\[ \overset{*}{\Omega}\, v_0d_0[x(\tau,\xi)]\leq -1 \]

for any \(\xi\in\Xi\). Thus \(v_0d_0\)-number of any nontrivial solution (1) with initial values from \(\Xi\) turns out to be negative (indeed, substantially negative: less than \(-1\)).

A transformation \(l\), one-to-one and continuously differentiable together with its inverse \(l^{-1}\) of the space \(E\) onto itself, will be called a \(vd\)-transformation if the quantities \(d\{v(\xi),v[l(\xi)]\}\) are bounded in absolute value for all \(\xi\in E\):

\[ \sup_\xi |d\{v(\xi),v[l(\xi)]\}|<+\infty. \]

We shall call \(vd\)-invariants (asymptotic \(vd\)-characteristics) of system (1) the properties (1) that remain unchanged under passage from (1) to any other system (12) by means of a \(vd\)-transformation. The generalized characteristic numbers \(\overset{*}{\Omega}\,vd\,x\) and \(\overline{\Omega}\,vd\,x\) are \(vd\)-invariants. Indeed, let

\[ y(\tau)=l[x(\tau)]\neq 0,\qquad y(0)=\eta=l[x(0)]=l(\xi). \]

Then

\[ D(x,\tau,\tau_0)-D(y,\tau,\tau_0)= \]

\[ = d\{v[x(\tau_0+\tau)],\,v[x(\tau_0)]\} -d\{v[y(\tau_0+\tau)],\,v[y(\tau_0)]\}=A+B, \]

where

\[ A=d\{v[x(\tau+\tau_0)],\,v[x(\tau_0)]\} -d\{v[x(\tau+\tau_0)],\,v[y(\tau_0)]\}, \]

\[ B=d\{v[x(\tau+\tau_0)],\,v[y(\tau_0)]\} -d\{v[y(\tau_0+\tau)],\,v[y(\tau_0)]\}, \]

but, on the basis of (13),

\[ |A+B|\leq 2|d\{v[x(\tau_0)],\,v[y(\tau_0)]\}| +2|d\{v[x(\tau+\tau_0)],\,v[y(\tau+\tau_0)]\}| \]

and by (14)

\[ \sup |D(x,\tau,\tau_0)-D(y,\tau,\tau_0)|<+\infty . \]

Therefore

\[ \overset{*}{\Omega}\operatorname{vd}x = \lim_{\tau\to+\infty}\frac{1}{\tau} \sup_{-\infty<\tau_0<+\infty} \{D(y,\tau,\tau_0)+|D(x,\tau,\tau_0)-D(y,\tau,\tau_0)|\} \leq \]

\[ \leq \overset{*}{\Omega}\operatorname{vd}y \]

and, since \(x\) and \(y\) are on an equal footing,

\[ \overset{*}{\Omega}\operatorname{vd}x = \overset{*}{\Omega}\operatorname{vd}y. \]

Similarly,

\[ \overline{\Omega}\operatorname{vd}x = \overline{\Omega}\operatorname{vd}y. \]

References

1. Lyapunov A. M. Collected Works, 2. Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1956.
2. Krasovskii N. N. PMM, 17, No. 6, 1953, pp. 651–672.
3. Vinograd R. E. DAN SSSR, 114, No. 2, 1957, pp. 239–240.
4. Grobman D. M. Mat. sbornik, 46, No. 3, 1958, pp. 343–358.
5. Proceedings of the International Symposium on Nonlinear Oscillations, Kiev, September 12–18, 1961, 1. Analytical Methods in the Theory of Nonlinear Oscillations. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1963.

Belorussian State University
named after V. I. Lenin