Yu. S. Bogdanov

APPLICATION OF GENERALIZED CHARACTERISTIC NUMBERS TO THE STUDY OF THE STABILITY OF AN EQUILIBRIUM POINT

(Presented by Academician V. I. Smirnov on 6 IV 1964)

§ 1. Definition of generalized characteristic numbers.

In an \(n\)-dimensional real Euclidean space \(E_0\) consider a domain \(\Xi\) containing the origin \(O_0\). Put
\[ \dot{\Xi}\overset{\mathrm{def}}{=}\Xi\setminus O_0. \]
Denote the boundary of \(\Xi\) by \(\Xi_{\mathrm{gr}}\) (\(\Xi_{\mathrm{gr}}\) may also turn out to be the empty set). Assign a continuous function
\[ x:(-\infty,+\infty)\to \dot{\Xi} \]
to the class \(\mathfrak{R}\) if the \(\alpha\)-limit set of \(x\) coincides with \(O_0\), while the \(\omega\)-limit set is contained in \(\Xi_{\mathrm{gr}}\). Thus,
\[ \lim_{\tau\to-\infty} x(\tau)=O_0,\qquad \operatorname{Lim}_{\tau\to+\infty} x(\tau)\subset \Xi_{\mathrm{gr}}. \]

On the domain \(\dot{\Xi}\) define a scalar function \(v\), filling out by its values \((0,+\infty)\), i.e.
\[ v:\dot{\Xi}\xrightarrow{\text{onto}}(0,+\infty). \]
Denote by \(\Gamma_\gamma\) the level set of the function \(v\) corresponding to the number \(\gamma\), i.e.
\[ \Gamma_\gamma \overset{\mathrm{def}}{=} \{\xi\mid \xi\in\dot{\Xi},\ v(\xi)=\gamma\}. \]
For \(x\in\mathfrak{R}\) and \(\gamma>0\) put
\[ \underline{\tau}(x,\gamma)\overset{\mathrm{def}}{=}\inf\{\tau\mid x(\tau)\in\Gamma_\gamma\},\qquad \overline{\tau}(x,\gamma)\overset{\mathrm{def}}{=}\sup\{\tau\mid x(\tau)\in\Gamma_\gamma\}. \]

Suppose that the function \(v\) has the following properties: \(v_1)\) each \(\Gamma_\gamma\), \(\gamma>0\), is compact; \(v_2)\) for any \(x\), \(x\in\mathfrak{R}\), and all \(\gamma,\tilde{\gamma}\) \((0<\gamma<\tilde{\gamma})\) one has
\[ \underline{\tau}(x,\gamma)<\underline{\tau}(x,\tilde{\gamma}),\qquad \overline{\tau}(x,\gamma)<\overline{\tau}(x,\tilde{\gamma}). \]
We note that all sets \(\Gamma_\gamma\), \(\gamma>0\), are nonempty and all quantities \(\underline{\tau}(x,\gamma)\), \(\overline{\tau}(x,\gamma)\) are finite.

Next consider a function \(d(\gamma_1,\gamma_2)\), defined for all positive values of the arguments and assuming all possible real values:
\[ d:(0,+\infty)\times(0,+\infty)\xrightarrow{\text{onto}}(-\infty,+\infty). \]
Suppose that for all \(\gamma_1,\gamma_2,\gamma_3,\gamma\) \((0<\gamma_1<\gamma_2<\gamma_3,\ 0<\gamma)\) the following hold:
\[ \mathrm{d}_1)\ d(\gamma,\gamma)=0;\qquad \mathrm{d}_2)\ 0<d(\gamma_2,\gamma_1)=-d(\gamma_1,\gamma_2); \]
\[ \mathrm{d}_3)\ d(\gamma_2,\gamma)>d(\gamma_1,\gamma);\qquad \mathrm{d}_4)\ d(\gamma_3,\gamma_2)+d(\gamma_2,\gamma_1)\ge d(\gamma_3,\gamma_1). \]

Take any function
\[ x:(-\infty,+\infty)\to\dot{\Xi}. \]
Put
\[ D(x,\tau,\tau_0)\overset{\mathrm{def}}{=}d\{v[x(\tau_0+\tau)],\,v[x(\tau_0)]\}, \]
where \(\tau_0\in(-\infty,+\infty)\), \(\tau\in(0,+\infty)\).

Associate with the function \(x\) the generalized characteristic numbers: the lower \(vd\)-number
\[ \overline{\Omega}\,vd\,x \overset{\mathrm{def}}{=} \max\left\{ \varlimsup_{\tau\to+\infty}\frac{1}{\tau}D(x,\tau,\tau_0),\, -\varliminf_{\tau\to+\infty}\frac{1}{\tau}D(x,-\tau,\tau_0) \right\} \]
and the \(vd\)-number
\[ \Omega^{*}vd\,x \overset{\mathrm{def}}{=} \varlimsup_{\tau\to+\infty}\frac{1}{\tau} \sup_{-\infty<\tau_0<+\infty}D(x,\tau,\tau_0). \]

We note that from \(\mathrm{d}_{1-4})\) it follows that
\[ \sup_{\gamma>0}\left|d(\gamma_2,\gamma)-d(\gamma_1,\gamma)\right|\le 2|d(\gamma_2,\gamma_1)|, \]
and therefore \(\overline{\Omega}\,vd\,x\) does not depend on \(\tau_0\). The lower \(vd\)-number is either finite ...

number, or by an improper number \(-\infty\), or by an improper number \(+\infty\), i.e. \(\overline{\Omega}\,vd\,x\in[-\infty,+\infty]\). Similarly \(\overset{*}{\Omega}\,vd\,x\in[-\infty,+\infty]\). As is not hard to see,

\[ \overline{\Omega}\,vd\,x \leq \overset{*}{\Omega}\,vd\,x . \]

§ 2. Structure of a neighborhood of an unstable equilibrium point.

Consider the differential system

\[ dx/dt=f(x), \qquad x\in \Xi, \tag{1} \]

with right-hand side \(f(x)\) satisfying a local Lipschitz condition on \(\Xi\) and vanishing at \(O_0\). The origin \(O_0\) is therefore an equilibrium point. Suppose that all solutions of (1) are continuable in both directions, i.e. are defined for all values of the argument \(\tau\). The solution \(x(\tau)\) with initial value \(x(0)=\xi\) will be denoted by \(x(\tau,\xi)\). The set of all \(\alpha\)-limit points of solutions of (1) distinct from the trivial one will be denoted by \(A\).

Assume that \(O_0\) is unstable (here and below instability is understood in the sense of Lyapunov). If \(\nu=2\) and \(O_0\) is an isolated equilibrium point, then there exists a solution \(x(\tau,\xi_0)\), \(\xi_0\in\Xi\), such that \(x(\tau,\xi_0)\to O_0\) as \(\tau\to-\infty\), i.e. \(O_0\in A\). Examples show that for some systems of type (1), when \(\nu>2\), an isolated unstable equilibrium point may fail to be an \(\alpha\)-limit point for any solution \(x(\tau,\xi)\), \(\xi\in\Xi\), and, consequently, \(O_0\notin A\). However, the following theorem is valid:

Theorem. Let \(O_0\) be unstable. Then \(O_0\in\overline{A}\).

Outline of the proof. Suppose, to the contrary, that \(O_0\notin\overline{A}\). Denote by \(\varepsilon_0\) a number such that the entire closed sphere \(S(\varepsilon_0)\) of radius \(\varepsilon_0\) with center at \(O_0\) is contained in \(\Xi\) and has no common points with \(A\). Take an arbitrary \(\varepsilon\) from the interval \((0,\varepsilon_0)\). To each point \(\sigma\) of the surface \(S_{\mathrm{rp}}(\varepsilon)\) of the sphere \(S(\varepsilon)\) assign the number \(\tau(\sigma)\)—the greatest of the nonpositive moments of strict entry of \(x(\tau,\sigma)\) into \(S(\varepsilon)\), i.e. \(x(\tau,\sigma)\in S(\varepsilon)\) for all \(\tau\in[\tau(\sigma),0]\) and \(x(\tau_n,\sigma)\notin S(\varepsilon)\), where \(\{\tau_n\}\) is some increasing sequence converging to \(\tau(\sigma)\). By virtue of the continuous dependence of solutions of (1) on initial values, the function \(\tau(\sigma)\) is lower semicontinuous, i.e.

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

Consequently, \(\tau(\sigma)\) is bounded below on \(S_{\mathrm{rp}}(\varepsilon)\) by some number \(\tau^*\). The vector function \(f(x)\) is continuous on \(S(\varepsilon)\) and vanishes at \(O_0\); therefore there is a positive \(\delta=\delta(\varepsilon)\) such that every solution of (1) requires a time, greater in absolute value than \(|\tau^*|\), to pass from \(S_{\mathrm{rp}}(\delta)\) to \(S_{\mathrm{rp}}(\varepsilon)\). Consequently, none of the solutions \(x(\tau,\xi)\), \(\xi\in S(\delta)\), intersects the surface \(S_{\mathrm{rp}}(\varepsilon)\) for \(\tau>0\), i.e. the point \(O_0\) turns out to be stable, contrary to the hypothesis of the theorem.

§ 3. Criteria for stability and instability of an equilibrium point.

Necessary criterion for instability. If \(O_0\) is unstable, then in every neighborhood \(u\) of this point there are initial values of nontrivial solutions (1) with nonnegative small \(vd\)-numbers.

Proof. On the basis of the theorem of § 2 there exists a solution \(x(\tau,\xi_0)\), \(\xi_0\in\Xi\), and points \(a\in u\) such that \(x(\tau_n,\xi_0)\to a\) for some sequence \(\{\tau_n\}\to-\infty\). Without loss of generality, \(\xi_0\in u\). The sequence \(\{x(\tau_n,\xi_0)\}\) is contained inside some surface \(\Gamma_{\gamma_0}\); therefore there exists a number \(\Delta\) such that for all \(n\)

\[ d\{v[x(0,\xi_0)],\,v[x(\tau_n,\xi_0)]\}>\Delta, \]

\[ \frac{1}{\tau_n}\,d\{v[x(\tau_n,\xi_0)],\,v[x(0,\xi_0)]\}>\frac{\Delta}{-\tau_n}, \]

\[ \tau_n^* \overset{\mathrm{def}}{=} -\tau_n,\qquad -\lim_{n\to\infty}\frac{1}{\tau_n^*}\, d\{v[x(-\tau_n,\xi_0)],\,v[x(0,\xi_0)]\}\geq 0, \]

i.e.

\[ \overline{\Omega}\,vd\,[x(\tau,\xi_0)]\geq 0. \]

Sufficient criterion for instability. If in any neighborhood of the point \(O_0\) there are initial values of solutions of (1) with positive \(\mathrm{vd}\)-numbers, then \(O_0\) is unstable.

Proof follows from the fact that \(\stackrel{*}{\Omega}\mathrm{vd}[x(\tau,\xi)]>0\) entails the existence of a sequence \(\{\tau_n\}\to+\infty\) such that either \(\{x(-\tau_n,\xi)\}\to O_0\), or \(\{x(\tau_n,\xi)\}\to \Xi_{\mathrm{gr}}\).

Sufficient criterion for asymptotic stability. If the small \(\mathrm{vd}\)-numbers of all nontrivial solutions (1) with initial values from a sufficiently small neighborhood of the point \(O_0\) are negative, then \(O_0\) is asymptotically stable.

Proof. The stability of \(O_0\) follows from the necessary criterion for instability. Moreover, from \(\overline{\Omega}\mathrm{vd}[x(\tau,\xi)]<0\) it follows that \(x(\tau,\xi)\to O_0\) as \(\tau\to+\infty\), and hence \(O_0\) is a point of attraction for every solution \(x(\tau,\xi)\), and therefore \(O_0\) is asymptotically stable.

Necessary criterion for asymptotic stability. If \(O_0\) is asymptotically stable, then there exist functions \(v_0\) and \(d_0\), satisfying the conditions of §1 and such that the \(v_0d_0\)-numbers of all nontrivial solutions (1) with sufficiently small initial values are negative.

Proof. Without loss of generality, we take \(\Xi\) to be the domain of attraction of \(O_0\). From the asymptotic stability of \(O_0\) it follows that there exists a Lyapunov function \(v_0(x)\), satisfying conditions \(\mathrm{v}_{1-2}\) and strictly decreasing along every nontrivial solution of (1) (see, for example, \((^1)\), pp. 214–218). Take arbitrary \(\gamma_1\) and \(\gamma_2\), \(0<\gamma_1<\gamma_2\). For any point
\[ \xi\in \Gamma_{\gamma_1\gamma_2}\overset{\mathrm{def}}{=}\bigcup_{\gamma_1<\gamma<\gamma_2}\Gamma_\gamma \]
define the number \(T(\xi,\gamma_2,\gamma_1)\)—the time during which \(x(\tau,\xi)\) remains strictly inside \(\Gamma_{\gamma_1\gamma_2}\). For given \(\gamma_1\) and \(\gamma_2\) the set \(\{T(\xi,\gamma_2,\gamma_1)\}\) is bounded above. Put
\[ d_0(\gamma_2,\gamma_1)\overset{\mathrm{def}}{=}\sup\{T(\xi,\gamma_2,\gamma_1)\},\qquad d(\gamma_1,\gamma_2)\overset{\mathrm{def}}{=}-d(\gamma_2,\gamma_1),\qquad d(\gamma_1,\gamma_1)\overset{\mathrm{def}}{=}0, \]
where the supremum is taken over all \(\xi\in\Gamma_{\gamma_1\gamma_2}\). The function \(d_0(\gamma_2,\gamma_1)\) satisfies conditions \(\mathrm{d}_{1-4}\). From the definition of \(d_0(\gamma_2,\gamma_1)\) it follows that
\[ \sup_{-\infty<\tau_0<+\infty} d_0\{v_0[x(\tau_0+\tau)],\,v_0[x(\tau_0)]\}\le -\tau \]
and
\[ \stackrel{*}{\Omega}v_0d_0[x(\tau,\xi)]\le -1 \]
for any \(\xi\in\Xi\).

§ 4. Criterion for conditional stability. Denote by \(m\) a certain closed subset of \(\Xi\). Theorem §2 can be strengthened in the direction that, if \(O_0\) is unstable with respect to perturbations from \(m\), then \(O_0\in\bar A\cap m\), and hence in \(m\) there exist nontrivial solutions (1) with arbitrarily small (in norm) initial values and nonnegative small \(\mathrm{vd}\)-numbers (necessary criterion for instability). Hence there follows a sufficient criterion for conditional asymptotic stability.

If the small \(\mathrm{vd}\)-numbers of all nontrivial solutions (1) with initial values from \(m\) are negative, then \(O_0\) is asymptotically stable with respect to perturbations from \(m\).

§ 5. Generalized exponents. The \(\mathrm{vd}\)-number and the small \(\mathrm{vd}\)-number are a natural generalization of the Lyapunov characteristic exponent (with sign changed)
\[ \bar{\omega}x\overset{\mathrm{def}}{=}\lim_{\tau\to+\infty}\frac{1}{\tau}\ln\{\|x(\tau_0+\tau)\|/\|x(\tau_0)\|\}. \]
The results of §3 show that the method of generalized characteristic numbers is universal in detecting the asymptotic stability of an equilibrium point in the same sense in which Lyapunov’s second method is universal. A direct generalization—the generalized characteristic exponent of Lyapunov—obобщ...

the generalized exponent \(\overline{\Omega}x\) and the small generalized exponent \(\overset{*}{\Omega}x\) can be obtained if one sets \(v(x)=\|x\|\), \(d(\gamma_1,\gamma_2)\equiv \ln \frac{\gamma_1}{\gamma_2}\), i.e.

\[ \overline{\Omega}x \overset{\mathrm{def}}{=} \max \left\{ \overline{\lim_{\tau\to+\infty}} \frac{1}{\tau} \ln \frac{\|x(\tau_0+\tau)\|}{\|x(\tau_0)\|}, \; -\underline{\lim_{\tau\to+\infty}} \frac{1}{\tau} \ln \frac{\|x(\tau_0+\tau)\|}{\|x(\tau_0)\|} \right\}, \]

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

If \(x\) is a solution of the linear system (1), then \(\overline{\Omega}x=\overset{*}{\Omega}x=\overline{\omega}x\). In contrast to \(\overline{\omega}x\) \((^{2,3})\), the generalized characteristic number \(\Omega x\), and still more \(\overset{*}{\Omega}x\), has the property that the negativity of the generalized characteristic numbers of all nontrivial solutions of (1) with sufficiently small initial values in norm ensures the asymptotic stability of the rest point \(O_0\) (see § 3). It is not difficult to show that the relation

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

where \(\underline{\alpha}x\) is the lower exponent of the function \(x(-\tau)\), i.e.

\[ \underline{\alpha}x \overset{\mathrm{def}}{=} \underline{\lim_{\tau\to+\infty}} \frac{1}{\tau} \ln \frac{\|x(\tau_0-\tau)\|}{\|x(\tau_0)\|}, \]

or, in other words, the minus-exponent of the solution \(x\) \((^4)\), always holds. For weakly nonlinear systems the criteria of § 3 for the specific values of \(v\) and \(d\) indicated in the present paragraph are consequences of known results \((^4)\).

Belorussian State University
named after V. I. Lenin

Received
30 III 1964

REFERENCES

\(^{1}\) Proceedings of the International Symposium on Nonlinear Oscillations, Kiev, 12–18 IX 1961, 1. Analytical Methods in the Theory of Nonlinear Oscillations, Kiev, 1963.
\(^{2}\) R. E. Vinograd, DAN, 114, No. 2, 239 (1957).
\(^{3}\) R. E. Vinograd, Matem. sborn., 41, No. 1, 431 (1957).
\(^{4}\) D. M. Grobman, Matem. sborn., 46, No. 3, 343 (1958).