UDC 517.919 : 531.36

APPLICATION OF GENERALIZED LYAPUNOV FUNCTIONS TO THE INVESTIGATION OF NON-HOMOGENEOUS SYSTEMS

Yu. V. Malyshev

Let a system be given

\[ x' = P(x,y), \quad y' = Q(x,y) \quad ({}' = d/dt), \tag{1} \]

where \(P(x,y), Q(x,y) \in C^{(s)}(U(O))\) \((s \geqslant 1)\); \(O(0,0)\) is an isolated singular point of system (1); \(U(O)\) is a finite neighborhood of the point \(O\); \(V(x,y) \in C^{(s+1)}(U(O))\) is a generalized Lyapunov function (generality is understood in the sense that the derivative of the function \(V\) along the system may change sign in the domain \(U(O)\)).

Denote

\[ V_i=\frac{\partial V_{i-1}}{\partial x}P+\frac{\partial V_{i-1}}{\partial y}Q,\quad V_i^*=\frac{\partial V_{i-1}^*}{\partial x}Q-\frac{\partial V_{i-1}^*}{\partial y}P,\quad V_0=V_0^*=V\;(i=0,1,2,3), \]

where the sign \(*\) denotes differentiation along the adjoint system

\[ x'=Q(x,y),\quad y'=-P(x,y). \tag{2} \]

It is required to investigate the behavior of the integral curves of system (1) in a finite neighborhood \(U(O)\). The main assumptions are: 1) the neighborhood \(U(O)\) is divided by branches of the neutral set

\[ \Gamma_s=\{x,y\mid V_1=0\},\quad \Gamma_s^*=\{x,y\mid V_1^*=0\}, \]

passing through the point \(O\), into a finite number of domains; 2) in the domain \(U(O)\)

\[ (\Gamma_s\cap \Gamma_h)\setminus O=\Lambda,\quad (\Gamma_s^*\cap \Gamma_h^*)\setminus O=\Lambda, \]

here \(\Lambda\) is the empty set, \(\Gamma_h=\{x,y\mid V_2=0\}\), \(\Gamma_h^*=\{x,y\mid V_2^*=0\}\).

Such a formulation of the problem is natural after the works of V. V. Nemytskii, P. N. Papush, M. B. Kudaev, and others, in which systems close to homogeneous, elliptic, hyperbolic, and elliptic-hyperbolic systems were considered and necessary and sufficient conditions were found for the existence of the corresponding Lyapunov functions [1—4].

In contrast to the classical works of I. Bendixson, M. Frommer, and H. Forster (see [5]), here a finite neighborhood \(U(O)\) of the singular point is considered (whose size is determined only by the points of intersection of the sets \(\Gamma_h, \Gamma_h^*\)), and differentiation along the adjoint system (2) is used to obtain additional information about the behavior of the integral curves of system (1).

Together with a positive definite generalized Lyapunov function \(V(x,y)\in C^{(3)}\) \((V(0,0)=0\), the closed curve \(V(x,y)=c_1\) is contained insi-

enclosed curve \(V(x,y)=c_2\), if \(c_2>c_1\)) let us consider a rotating function \(\Phi(x,y)\), whose level lines are orthogonal to the level lines of the function \(V(x,y)\). Denote by \(\Phi_i\) and \(\Phi_i^*\) the \(i\)-th derivatives of the function \(\Phi\) by virtue of systems (1) and (2), respectively.

Lemma 1. The isolated branches \(\Gamma_s\setminus O\) and \(\Gamma_s^*\setminus O\) are curves without contact with respect to the family \(V=\mathrm{const}\) (or \(\Phi=\mathrm{const}\)), if along them, respectively, \(V_2\ne 0\) and \(V_2^*\ne 0\) \((\Phi_2^*\ne 0\) and \(\Phi_2\ne 0)\).

Proof. Denote \(X=\{P,Q\}\), \(Y=\{Q,-P\}\), \(\operatorname{grad} V=\{V_x,V_y\}\), \(\operatorname{grad}\Phi=\{\Phi_x,\Phi_y\}\). Consider the isolated branch \(\Gamma_s\setminus O\). Suppose that on it there is a point of contact; then at this point
\(V_1=(\operatorname{grad}V\cdot X)=0\), \([\operatorname{grad}V_1\cdot \operatorname{grad}V]=0\). Hence \(V_2=(\operatorname{grad}V_1\cdot X)=0\), which contradicts the condition of the lemma. Suppose now that the point of contact lies on \(\Gamma_s^*\setminus O\); then at this point

\[ V_1^*=(\operatorname{grad}V\cdot Y)=0,\quad [\operatorname{grad}V_1^*\cdot \operatorname{grad}V]=0. \]

Hence \(V_2^*=(\operatorname{grad}V_1^*\cdot Y)=0\), which contradicts the condition of the lemma. Thus, \(\Gamma_s\setminus O\) and \(\Gamma_s^*\setminus O\) are curves without contact with respect to the family \(V=\mathrm{const}\). For the family \(\Phi=\mathrm{const}\) the lemma is proved analogously.

Lemma 2. If in a neighborhood \(U(\Gamma_s)\) (or \(U(\Gamma_s^*)\)) \(\Phi_2^*\) (or \(\Phi_2\)) preserves its sign, is different from zero, and 1) \(V_2<0\) \((V_2^*<0)\), then \(\Phi_2^*>0\) \((\Phi_2>0)\) when \(\Phi\) decreases along \(\Gamma_s\) \((\Gamma_s^*)\), starting from the point \(O\), and \(\Phi_2^*<0\) \((\Phi_2<0)\) when \(\Phi\) increases along \(\Gamma_s\) \((\Gamma_s^*)\); 2) \(V_2>0\) \((V_2^*>0)\), then \(\Phi_2^*<0\) \((\Phi_2<0)\) when \(\Phi\) decreases along \(\Gamma_s\) \((\Gamma_s^*)\), and \(\Phi_2^*>0\) \((\Phi_2>0)\) when \(\Phi\) increases along \(\Gamma_s\) \((\Gamma_s^*)\).

Proof. Let along \(\Gamma_s^*\setminus O\), \(V_2^*<0\) and \(\Phi\) decreases; then

\[ V_2^*=(\operatorname{grad}V_1^*\cdot Y)<0 \tag{3} \]

and

\[ (\operatorname{grad}\Phi\cdot \tau)<0 \tag{4} \]

(\(\tau\) is the tangent vector to \(\Gamma_s^*\)). Moreover, let

\[ (\operatorname{grad}V\cdot \tau)>0. \tag{5} \]

By the definition of the functions \(V\) and \(\Phi\) we have

\[ (\operatorname{grad}V\cdot \operatorname{grad}\Phi)=0. \]

It follows from (4) and (5) that the direction of \(\operatorname{grad}\Phi\) is obtained by rotating \(\operatorname{grad}V\) by \(90^\circ\) counterclockwise. This means that

\[ -\frac{\Phi_x}{V_y}=\frac{\Phi_y}{V_x}>0 \]

(if the direction of \(\operatorname{grad}V\) does not coincide with the direction of the coordinate axes).

But

\[ \Phi_1=-\frac{\Phi_x}{V_y}V_1^*,\quad \Phi_2=-\frac{\Phi_x}{V_y}(\operatorname{grad}V_1^*\cdot X) \]

along \(\Gamma_s^*\setminus O\). From (3) it follows that
\((\operatorname{grad}V_1^*\cdot X)>0\), i.e. \(\Phi_2>0\), as was required to prove. The remaining cases are considered analogously.

Definition 1. A branch of the neutral curve \(\Gamma_s\) will be called elliptic if along it the function \(V\) assumes a maximal value (internal tangency of the integral curves with the level lines of the function \(V\)), and hyperbolic of the 2nd kind if along it the function \(V\) assumes a minimal-

value (external tangency of the integral curves with the level lines of the function \(V\)).

Definition 2. A branch of the neutral curve \(\Gamma_s^*\) of system (1) will be called parabolic (hyperbolic of the first kind) if, for the adjoint system (2), it is elliptic (hyperbolic of the second kind).

Let us take as the function \(V\) the function \(V=-\dfrac{1}{2}(x^2+y^2)\); then the neutral sets \(\Gamma_s\) and \(\Gamma_s^*\) coincide respectively with the lines of normal directions \(\Gamma_N\) and with the lines of convergent directions \(\Gamma_C\), the angular coefficients of which at the singular point (directions \(N, C\)) determine the exceptional directions of systems (2) and (1) [7—9]

\[ \Gamma_N=\{x,y\mid V_1\equiv xP+yQ=0\},\qquad \Gamma_C=\{x,y\mid V_1^*\equiv xQ-yP=0\}. \]

The functions \(V_2\) and \(V_2^*\) along \(\Gamma_N\) and \(\Gamma_C\) take the form

\[ V_2=-\frac{(P^2+Q^2)P^2}{Q^2} \left[ \frac{1}{k^2}-\frac{d}{dt}\left(\frac{Q}{P}\right)\cdot\frac{dt}{dk} \right], \]

\[ V_2^*=(P^2+Q^2) \left[ 1-\frac{d^*}{dt}\left(\frac{Q}{P}\right)\cdot\frac{d^*t}{dk} \right] \qquad (k=y/x). \]

Hence analytic criteria are obtained for the ellipticity, hyperbolicity, and parabolicity of the branches \(\Gamma_N,\Gamma_C\). Namely, \(\Gamma_C\) \((\Gamma_N)\) is parabolic (elliptic) if along it

\[ \frac{d^*}{dt}\left(\frac{Q}{P}\right)\cdot\frac{d^*t}{dk}>1 \qquad \left( \frac{d}{dt}\left(\frac{Q}{P}\right)\cdot\frac{dt}{dk}>\frac{1}{k^2} \right), \tag{6} \]

and \(\Gamma_C\) \((\Gamma_N)\) is hyperbolic of the 1st (2nd) kind if along it

\[ \frac{d^*}{dt}\left(\frac{Q}{P}\right)\cdot\frac{d^*t}{dk}<1 \qquad \left( \frac{d}{dt}\left(\frac{Q}{P}\right)\cdot\frac{dt}{dk}<\frac{1}{k^2} \right). \tag{7} \]

If \(Q/P\) is a homogeneous function, then conditions (6) and (7) coincide with the well-known conditions of G. E. Shilov, which determine the type of integral rays [6].

Lemma 3. If an isolated branch \(\Gamma_C\) of system (1) 1) passes through the point \(O\), 2) does not coincide with an integral curve, 3) has no common tangents at the point \(O\) with the set \(\Gamma_h^*\), and \(V_2^*<0\) \((V_2^*>0)\) along \(\Gamma_C\setminus O\), then there exists a neighborhood \(U(\Gamma_C)\subset U(O)\) such that every trajectory \(q=f(p,t)\), where \(p\in\Gamma_C\), enters the singular point as \(t\to+\infty\) or as \(t\to-\infty\), touching \(\Gamma_C\) (leaves the neighborhood \(U(\Gamma_C)\) in a finite interval of variation of the parameter \(t\)).

Proof. Consider a region \(\sigma\subset U(O)\) between the branch \(\Gamma_C\) and the radius-vector \(r_p\) of the point \(p\in\Gamma_C\), inside which \(V_2^*<0\) (according to the assumptions made, such a neighborhood will always be found). By Lemma 2, the motion \(q=f(p,t)\) as \(t\) varies will enter this region and, upon continuing, will remain there. The absence in the region \(\sigma\) of other singular points, except the point \(O\), ensures the entrance of the trajectory \(q=f(p,t)\) into the point \(O\). Since the exceptional directions coincide with the directions \(C\) [7], the trajectories \(q=f(p,t)\) will enter the singular point while touching \(\Gamma_C\). Conversely, if \(V_2^*>0\) along \(\Gamma_C\), the trajectory \(q=f(p,t)\) cannot enter the region \(\sigma\) either for \(t>0\) or for \(t<0\). Thus, there exists a neigh-

neighborhood \(U(\Gamma_C) \subset U(O)\), which all trajectories \(q=f(p,t)\), \(p \in \Gamma_C\), leave over a finite interval of variation of \(t\).

The lemma is proved.

Theorem 1. Let \(S_C \subset U(O)\) be the sector between two neighboring branches \(\Gamma_C\) of system (1) passing through the point \(O\), having at the point \(O\) no common tangents with the set \(\Gamma_h^*\), and let \(V_2^*<0\) \((V_2^*>0)\) along \(\Gamma_C \setminus O\). Then there exists a neighborhood \(U(S_C) \subset U(O)\) within which there is contained a connected set of elliptic (hyperbolic) integral curves of system (1).

Proof. Let \(V_2^*<0\) along \(\Gamma_C \setminus O\). According to Lemma 3, for the branches \(\Gamma_C\) there exist neighborhoods \(U_1(\Gamma_C)\) and \(U_2(\Gamma_C)\) within which normal Frommer sectors of the first type [5] are located. Let \(\rho_1\) and \(\rho_2\) be the radii of the circles bounding these sectors, \(\rho=\inf\{\rho_1,\rho_2\}\), and let \(g\) be the region between the branches \(\Gamma_C\), bounded above by an arc of the circle of radius \(\rho\) (the back wall), with center at the singular point. Represent the set \(g\) as the sum of two nonintersecting sets \(g \cap U(\Gamma_C)\) and
\[ G=g \setminus g \cap U(\Gamma_C). \]
In the region \(g \cap U(\Gamma_C)\) the behavior of the integral curves is determined by Lemma 3.

Let us study the behavior of the integral curves in the region \(G\). For the adjoint system (2) in this region (by Papush’s theorem [3]) there are no branches \(\Gamma_N\) and directions \(N\), i.e., for the given system (1) in this region there are no branches \(\Gamma_C\) and directions \(C\), and hence all trajectories leave the region \(G\) over a finite interval of variation of \(t\).

Consider a point \(P\) on the lateral boundary of the region \(G\). For the trajectory \(q=f(P,t)\) inside \(G\) there are two possibilities: 1) either it intersects the second lateral wall of the region \(G\), and then every trajectory \(q=f(P,t)\) as \(P \to O\) will behave analogously and the theorem is proved; or 2) it intersects the level line \(V=c\) at some point \(Q\). In this case, as \(P \to O\), the point \(Q\) moves along the level line \(V=c\) in the direction toward the second wall of the region \(G\). If, for some position of the point \(P\), the point \(Q\) lies on the other lateral wall of the region \(G\), then, as \(P \to O\) further, all trajectories will intersect both lateral walls of the region \(G\), and the theorem is proved. If, however, as \(P \to O\), \(Q \to Q_0\), where \(Q_0\) does not belong to the lateral boundary of the region \(G\), then a \(O\)-curve passes through \(Q_0\), which contradicts the absence of \(\Gamma_C\) and of exceptional directions \(C\) in the region \(G\). Thus, in the region \(g \cup U(\Gamma_C)\) there indeed exists a connected set of elliptic curves without branches.

The case when, along \(\Gamma_C \setminus O\), \(V_2^*>0\) is proved analogously.

Remark. The theorem remains valid also in the case when the branches \(\Gamma_C\) forming the boundaries of the sector \(S_C\) are tangent to each other.

Theorem 2. Let \(S_C \subset U(O)\) be the sector between two neighboring branches \(\Gamma_C\) passing through the point \(O\), on which the function \(V_2^*\) preserves its sign, different on each branch \(\Gamma_C\), and suppose that on the set \(\Gamma_h^*\), having no common tangents with \(\Gamma_C\) at the point \(O\) and situated between the branches \(\Gamma_C\), the inequality \(V_1^*V_3^*>0\) is satisfied. Then there exists a neighborhood \(U(S_C) \subset U(O)\) within which there is contained a simply connected set of parabolic integral curves that does not degenerate into a single integral curve.

Proof. As in Theorem 1, consider the regions \(g \cap U(\Gamma_C)\) and
\[ G=g \setminus g \cap U(\Gamma_C). \]
In the region \(g \cap U(\Gamma_C)\) the behavior of the integral curves is determined by Lemma 3. In the region \(G\) (by Kudaev’s theorem [4]) for the adjoint system (2) there are no branches \(\Gamma_N\) and directions \(N\), i.e., for the given system (1) there are no branches \(\Gamma_C\) and directions \(C\). Just as in Theorem 1, it can be shown that between the points \(P\) and \(Q\) there is established

such a correspondence under which, beginning from some instant, as \(P\to O\) all trajectories \(q=f(P,t)\) intersect both lateral boundaries of the region \(G\). Since these trajectories in the regions \(U_1(\Gamma_C)\) and \(U_2(\Gamma_C)\) have, respectively, elliptic and hyperbolic character, in the whole they will be parabolic.

The theorem is proved.

Example. \(x'=-y^n,\ y'=Q(x,y)\), where

\[ Q(x,y)=\frac{\prod_{i=1}^{n+1}(y+A_i x^\alpha)-y^{n+1}}{x}\qquad(\alpha>0). \]

Consider the adjoint system

\[ x'=Q(x,y),\qquad y'=y^n. \]

Find \(V_1^*\) and \(V_2^*\) for the function \(V=\frac{1}{2}(x^2+y^2)\):

\[ V_1^*=\prod_{i=1}^{n+1}(y+A_i x^\alpha),\qquad \Gamma_C^m=\{\,x,y\mid y+A_m x^\alpha=0\,\}\quad(m=1,2,\ldots,n+1), \]

\[ V_2^*=\sum_{k=1}^{n+1}(y^n+A_k\alpha x^{\alpha-1}Q) \prod_{\substack{i=1\\(i\ne k)}}^{n+1}(y+A_i x^\alpha), \]

\[ V_2^*(\Gamma_C^m)=(-A_m)^n(1+\alpha A_m^2 x^{2(\alpha-1)}) \prod_{\substack{i=1\\(i\ne m)}}^{n+1}(A_i-A_m)x^{2n\alpha}, \]

\[ \operatorname{sign} V_2^*(\Gamma_C^m)= \operatorname{sign}(-A_m)^n \prod_{\substack{i=1\\(i\ne m)}}^{n+1}(A_i-A_m). \]

The last relation determines the type of \(\Gamma_C^m\). In order to clarify the behavior of the integral curves in the regions between the branches \(\Gamma_C^m\), on which the sign of \(V_2^*\) is different, we make, for system (1), the change of variables according to the formula \(y=ux\):

\[ \frac{du}{dx}= \frac{\prod_{i=1}^{n+1}(u+A_i x^{\alpha-1})}{-u^n x}. \tag{8} \]

Suppose \(0<A_1<A_2<\ldots<A_{n+1}\), \(\alpha=2\); then equation (8) will be homogeneous, and the integral rays are determined from the equation

\[ k^{n+1}+\prod_{i=1}^{n+1}(k+A_i)=0. \tag{9} \]

By virtue of the imposed conditions, equation (9) cannot have real roots lying in the interval \([A_1,A_{n+1}]\). This means that between the isoclines \(0\) of equation (8) there are no \(O\)-curves. To the isoclines \(0\) and the \(O\)-curves of equation (8) there correspond lines of converging directions and \(O\)-curves of system (1). Thus, for system (1), in the regions between the parabolas \(y=-A_m x^2\) \((m=1,2,\ldots,n+1)\), there are no \(O\)-curves.

Thus, the arrangement of the integral curves of system (1) in the present case is completely determined (see the figure). The singular point will be a saddle for even \(n\), and a center for odd \(n\) (taking into account the symmetry of the field of directions with respect to the axis \(Oy\)).

I take this opportunity to thank Prof. V. V. Nemytskii for his supervision of the work.

References

1. Nemytskii V. V. Vestn. Mosk. un-ta, ser. matem., mekh., No. 5, 25–43, 1961.
2. Nemytskii V. V. Vestn. Mosk. un-ta, ser. matem., mekh., No. 6, 26–27, 1962.
3. Papush P. N. Matem. sb., 38 (80): 3, 1956, pp. 337–358.
4. Kudaev M. B. DAN SSSR, 147, No. 6, 1285–1287, 1962.
5. Nemytskii V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Moscow–Leningrad, GITTL, 1949.
6. Shilov G. E. UMN, 5, issue 5 (39), 1950.
7. Nemytskii V. V., Malyshev Yu. V. Izv. vuzov, Matematika, No. 3, 1965.
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9. Malyshev Yu. V. Differentsial’nye uravneniya, 1, No. 5, 692–697, 1965.

Received by the editors
September 2, 1965

Kazan Chemical-Technological Institute