UDC 517.916

ON THE STRUCTURE OF THE COEFFICIENTS OF THE SUCCESSION FUNCTION

N. F. OTROKOV

In the present article we consider the question of the birth of limit cycles in a neighborhood of the periodic solution

\[ x=\varphi(t),\quad y=\psi(t) \tag{1} \]

of the system of differential equations

\[ \frac{dx}{dt}=P(x,y),\quad \frac{dy}{dt}=Q(x,y) \tag{2} \]

under transition to nearby equations of the form

\[ \frac{dx}{dt}=P(x,y)+p(x,y),\quad \frac{dy}{dt}=Q(x,y)+q(x,y). \tag{3} \]

Under the assumptions that the right-hand sides of system (2), and also the small changes \(p\) and \(q\) in equations (3), are functions of \(x,y\), holomorphic and single-valued in some neighborhood of the curve (1), the succession function of system (3) is constructed in the form of a contour integral equation, and it is shown that, in the first approximation, the coefficients of this function are, in their structure, linear functionals with respect to \(p,q\) and their partial derivatives. The formulas obtained make it possible to solve the problem of splitting a compound cycle into simple ones by means of the simplest fractional rational functions*). In § 3 the question of the birth of limit cycles from a rough cycle is considered under certain additional assumptions, and it is proved that polynomials may be taken as splitting additions, the degree of which is specified depending on the geometric properties of the periodic solution under investigation.

§ 1. INTEGRAL REPRESENTATION OF THE SUCCESSION FUNCTION

In the present section the basic concepts and definitions needed below are formulated [5].

1. Consider the differential equation corresponding to system (2)

\[ P\frac{\partial f}{\partial x}+Q\frac{\partial f}{\partial y}=0, \tag{4} \]

and let by parametric equations

*) The problem of splitting a cycle by continuously differentiable, in particular analytic, additions was solved in [4]. Here a more general problem is posed concerning the birth of limit cycles under the condition that the admissible changes belong to a prescribed class of functions, and this problem is solved in the case when the space of small perturbations contains rational functions of a special form.

\[ x=x_0+ac,\quad y=y_0+bc,\quad |c|<\varepsilon \tag{5} \]

defines a contact-free segment intersecting the cycle at the point \(A_0(x_0=\varphi(0),\; y_0=\psi(0))\). We choose the constants \(a\) and \(b\) so that positive values of the parameter \(c\) correspond to points located inside the cycle. Let \(f_0(x,y)\) be an integral of equation (4), satisfying the initial condition

\[ f_0(x_0+ac,\; y_0+bc)=c \tag{6} \]

and regular at the point \(A_0\). Let \(f_1(x,y)\) be the regular branch into which \(f_0\) passes after one complete circuit along the curve (1) in the counterclockwise direction. One may assume that this direction, which we shall call positive, corresponds to motion along \(l\) toward decreasing values of \(t\). Similarly, the branches \(f_n(x,y)\) and \(f_{-n}(x,y)\) \((n=1,2,\ldots)\) may be obtained after an \(n\)-fold circuit of \(l\) in the positive and, respectively, in the negative directions. The collection of branches obtained in this way will be denoted by \(f(x,y)\).

1. In the general case \(f(x,y)\) turns out to be a multivalued function and \(f_1\not\equiv f_0\), but between these functions there exists a dependence

\[ f_1(x,y)=\omega[f_0(x,y)], \tag{7} \]

where the function \(\omega\), which determines the character of the branching of the first integrals of the system, is a succession function. Indeed, consider the arc of the trajectory \(l(c)\), issuing from the point \(B(x=x_0+ac,\; y=y_0+bc)\) at \(t=0\) and intersecting the same straight line at the point \(A(x=x_0+ac_1,\; y=y_0+bc_1)\) at \(t=T(c)\) (see the figure). Then from equations (6) and (7)

\[ (f_1)_{l(c)}=f_1(B)=\omega[f_0(B)]=\omega(c), \]

and this constant value, equal to \(\omega(c)\), the function \(f_1(x,y)\) will preserve on the arc \(l(c)\) up to its intersection with (5) at the point \(A\). At the same time the value which \(f_1\) assumes after the circuit in the indicated direction will coincide with the value of the initial branch \(f_0(A)=c_1\). Hence \(c_1=\omega(c)\). This equality establishes a correspondence between two successive points of intersection of a positive semitrajectory with the contact-free segment.

1. Along with \(f(x,y)\), other multivalued functions \(F(x,y)\) will be used; their definition must be clarified by means of the following construction. Make a cut of the domain \(S\) along the analytic curve \(AB\). In the resulting simply connected domain \(\sigma\), we shall regard the boundary \(AB\) as consisting of two segments: the positive \(A_1B_1\) and the negative \(A_2B_2\).

Let in the closed domain \(\bar{\sigma}\) there be defined an infinite sequence of single-valued analytic functions \(F_n(x,y)\) \((n=\pm1,\pm2,\ldots)\), regular at every point of \(\bar{\sigma}\) and satisfying the boundary conditions

\[ (F_n)^{-}_{A_2B_2}=(F_{n+1})^{+}_{A_1B_1}\quad (n=0,\pm1,\ldots). \]

We take one of these functions, for example \(F_0\), as the initial branch, and for constructing its subsequent branches we apply the principle of uniqueness of an analytic function: as long as the variable point \(E\) does not leave the domain \(\sigma\), the values of \(F\) coincide with \(F_n\) for some completely determined \(n\), and only when the segment \(AB\) is crossed does \(F\) pass, depending on the direction of motion, either into \(F_{n+1}\) or into \(F_{n-1}\). A sequence of single-valued functions ordered in this way will be called a complete

by a function \(F(x,y)\) of class \(\Omega\). Obviously, \(\Omega\) contains all single-valued functions, the integral \(f(x,y)\) of the differential equation (4), normalized according to condition (6), all other solutions of this equation having regular branches in the domain \(S(\varepsilon,l)\), as well as their partial derivatives with respect to \(x\) and \(y\). The integrating factor of system (2), determined by one of the equalities

\[ \frac{\partial f}{\partial x}=\mu(x,y)Q,\qquad \frac{\partial f}{\partial y}=-\mu(x,y)P. \tag{8} \]

will belong to \(\Omega\).

1. We shall call a point \(E\subset S(\varepsilon,l)\) a point of single-valuedness of a function \(F\subset\Omega\) if all branches of this function have at \(E\) one and the same value, equal to \(F_0(E)\):

\[ F_k(E)=F_0(E)\qquad (k=\pm1,\ \pm2,\ldots). \tag{9} \]

We shall call \(E\) a point of single-valuedness of order \(n\) if at this point conditions (9) are satisfied for the function \(F\) and for all its partial derivatives up to order \(n-1\) inclusive, but for at least one of the derivatives of order \(n\), \(E\) is not a point of single-valuedness.

In the general case, for an arbitrary function \(F\), the set of its points of single-valuedness consists either of a finite number of points and curves, or coincides with \(S(\varepsilon,l)\), which is possible only in the case when \(F\) is a single-valued function. But if \(F\) is an integral of equation (4), then, as follows from (6) and (7), the set of points of single-valuedness consists exclusively of closed trajectories of the system determined by the equation

\[ \omega(f_0)-f_0=0. \]

Moreover, there is a direct dependence between the order of single-valuedness and the multiplicity of the cycle, as is shown by the following

Theorem 1. A closed trajectory \(l\) will be an \(n\)-fold limit cycle of system (2) if and only if one of the following conditions is satisfied:

\[ \text{1) }\omega'(0)=1,\quad \omega''(0)=0,\quad \omega^{(n-1)}(0)=0,\quad \omega^{(n)}(0)\ne0. \tag{10} \]

2) There exists a first integral of the system, single-valued on \(l\), of order \(n\).

3) There exists an integrating factor of the system, single-valued on \(l\), of order \(n-1\).

1. The condition that a closed integral curve passes through a given point \(E\subset S(\varepsilon,l)\) can be obtained in the form of the integral equation

\[ g(E)\equiv-\int_{L(E)}\mu(x,y)(P\,dy-Q\,dx)=0, \tag{11} \]

where \(E\) is the initial and terminal point of a piecewise smooth closed contour \(L(E)\) having the same winding index as \(l\). Let us note that the value of the integral (11) does not depend on the form of the path, but, owing to the multivaluedness of the integrand, depends on the choice of the initial point. \(g(E)\) is a correspondence function, for the construction of which only the existence of a returning domain is required.

1. We shall assume that system (2) satisfies the multiplicity conditions (10) for some finite \(n\ge2\). We write system (3) in the form

\[ \frac{dx}{dt}=P+\lambda p,\qquad \frac{dy}{dt}=Q+\lambda q, \tag{12} \]

where \(\lambda\) is a small parameter, and \(p,q\) are arbitrary functions from the given

linear metric space of functions \(D(p,q)\). By \(F(x,y,\lambda)\) we shall mean a solution of the differential equation

\[ (P+\lambda p)\frac{\partial F}{\partial x}+(Q+\lambda q)\frac{\partial F}{\partial y}=0 \tag{13} \]

in the sense of a complete analytic function of two real variables \(x\) and \(y\) and of the small parameter \(\lambda\). The function \(F\) consists of an infinite set of distinct branches passing into one another in accordance with the branching of \(f(x,y)\). The initial branch satisfies the same condition (6) as \(f(x,y)\), and for \(\lambda=0\) everywhere in the domain \(S(e,l)\) we have \(F(x,y;0)\equiv f(x,y)\). Following (11), we write the succession function for system (12) in the form

\[ G(E,\lambda)=-\int_{L(E)} M(x,y,\lambda)[(P+\lambda p)\,dy-(Q+\lambda q)\,dx], \tag{14} \]

where \(M\) is the integrating factor of system (12), connected with the integral \(F\) by equations of the form (8). Here \(G(E,\lambda)=0\) if and only if a closed trajectory of system (12) passes through the point \(E\).

As the path of integration in (14), choose the closed curve \(L(B)=(B,A,l(c),B)\), consisting of the segment of the straight line \(s\) between the points \(B(x=x_0+ac,\ y=y_0+bc)\) and \(A(x=x_0+a\omega(c),\ y=y_0+b\omega(c))\), and of another trajectory \(l(c)\):

\[ x=\varphi(t,c),\qquad y=\psi(t,c), \tag{15} \]

which leaves the point \(B\) at \(t=0\) and intersects \(s\) at the point \(A\) when \(t=T(c)\) (see the figure). The succession function \(\omega(c)\), regular at the point \(c=0\), under the assumptions made has the expansion

\[ \omega(c)=c+\sum_{k=n}^{\infty}\gamma_k c^k,\qquad \gamma_n\ne0,\quad n\ge2. \tag{16} \]

The point \(B\) is taken as the initial point, and at the beginning of the path of integration the initial branch of the function \(M(x,y,\lambda)\) is taken. Substituting all the indicated values, we obtain

\[ G(c,\lambda)=-\int_{c}^{\omega(c)} M[(P+\lambda p)b-(Q+\lambda q)a]\,ds -\lambda\int_{T(c)}^{0} M(Qp-Pq)\,dt, \tag{17} \]

where, in the functions under the integral signs, the variables \(x\) and \(y\) should be replaced, in the first integral, by the values \(x=x_0+as,\ y=y_0+bs\), and in the second by those from equations (15). From the very manner of constructing the integral \(F\) it follows that, whatever \(p,q\in D\) may be, there exist positive numbers \(\varepsilon_0\) and \(\delta_0\) such that \(G(c,\lambda)\) will be regular with respect to \(c\) and \(\lambda\) for \(|c|<\varepsilon_0\) and \(|\lambda|<\delta_0\). The integral over the segment \(BA\), in view of (16), will begin with the term \(\gamma_n c^n\), and for \(\lambda=0\) takes the value equal to \(-\omega(c)+c\). Hence

\[ G(c,\lambda)=\sum_{k=0}^{\infty}\frac{c^k}{k!}\,G_k(\lambda). \tag{18} \]

The coefficients \(G_k\) are functions of the parameter \(\lambda\), of which \(G_n(0)\ne0\), while the expansions of the first \(n\) in powers of \(\lambda\) begin with terms of the first degree. The coefficients of these linear terms are expressed by integrals

\[ g_k=-\left(\frac{d^k}{dc^k}\int_{T(c)}^0 \mu(Qp-Pq)\,dt\right)_{c=0} \quad (k=0,1,\ldots,n-1). \tag{19} \]

1. To distinct roots \(c_1\) and \(c_2\) of the equation

\[ G(c,\lambda)=0 \tag{20} \]

with fixed \(p,q,\lambda\) there correspond cycles of the system (12) that intersect the contactless segment \(s\) at points close to \(A_0\), and conversely. Therefore the number of cycles that are born is equal to the number of real branches of the function \(c(\lambda)\), defined implicitly by equation (20), that are continuous and simultaneously vanish at \(\lambda=0\). The number of real roots, which in general cannot exceed \(n\), is determined by the first coefficients of the expansion (18). In a number of cases the question is decided by the linear terms of these coefficients. The following lemma, obvious in itself, shows precisely which properties of these integrals, and at the same time of the space \(D\), are decisive.

Lemma 1. Suppose that in the equation

\[ \sum_{k=0}^{\infty}\frac{c^k}{k!}\,G_k(\lambda_1,\ldots,\lambda_m)=0, \tag{21} \]

which contains in the left-hand side a function of the variable \(c\) and the parameters \(\lambda=(\lambda_1,\ldots,\lambda_m)\), regular for \(|c|<c_0\) and \(|\lambda_i|<\lambda_0\), the first \(n+1\) coefficients of the expansion satisfy the conditions:

1) \(G_k(0)=0\) \((k=0,1,\ldots,n-1)\), \(G_n(0)\ne0\).

2) The rank of the matrix of the linear terms of the expansions of \(G_k(\lambda)\) \((k=0,1,\ldots,r-1)\) in powers of \(\lambda\) is equal to \(r\), where \(0\le r\le n\).

Then there exist positive numbers \(\delta_0\) and \(\varepsilon_0\) such that, for all \(|\lambda_i|<\delta_0\) \((i=1,\ldots,m)\), equation (21) cannot have more than \(n\) roots \(|c_i|<\varepsilon_0\); but whatever \(\varepsilon<\varepsilon_0\) and \(\delta<\delta_0\) may be, there are always \(\lambda_i=\lambda_i^*\), \(|\lambda_i^*|<\delta\), such that the corresponding equation (21) will have no fewer than \(r\) distinct real roots in the interval \(|c|<\varepsilon\) if \(n-r\) is even, and no fewer than \(r+1\) if \(n-r\) is odd.

§ 2. FORM OF THE COEFFICIENTS

1. The linear terms expressed by formulas (19) are computed if the curves (15) close to the cycle are known. Below it is shown that these quantities can be computed in the form of integrals along a prescribed trajectory.

Lemma 2. Let \(l\) be a limiting cycle of multiplicity \(n\ge2\), and let \(F(x,y)\) be an analytic function of \(x\) and \(y\) in the domain \(S(\varepsilon,l)\), \(F\subset\Omega\), and have order of single-valuedness on \(l\) equal to \(k\ge1\). Under these conditions there exist functions \(\alpha(x,y)\) and \(\beta(x,y)\) of order of single-valuedness not lower than \(k\), satisfying the equation

\[ \alpha Q-\beta P=F(x,y), \tag{22} \]

and such that

\[ \frac{d}{dc}\int_{T(c)}^0 F\,dt = \int_{T(c)}^0\left(\alpha_x' + \beta_y'\right)\frac{dt}{\mu} +c^k R_1(c)+c^{n-1}R_2(c), \tag{23} \]

where \(R_1\) and \(R_2\) are functions of \(c\), regular at the point \(c=0\). Under the integral sign

... there are certain single-valued branches of the functions \(F,\mu,\alpha,\beta\), in which \(x\) and \(y\) are replaced by the values from equations (15).

Proof. Let \(F_0(x,y)\) denote the initial branch. Fix a sufficiently small value of the parameter \(c\) and consider the arc of the trajectory \(l(c+\Delta c)\) corresponding to the increment \(\Delta c\). Then, by definition, we have

\[ \frac{d}{dc}\int_{T(c)}^{0} F_0\,dt = \lim_{\Delta c\to 0}\frac{1}{\Delta c} \left[ \int_{A_1m_1B_1} F_0\,dt - \int_{AmB} F_0\,dt \right]. \tag{24} \]

For definiteness we shall assume that the straight line \(s\) serves as the cut on which the branches of \(F\) pass one into the other. In view of this, the second of the integrals on the right-hand side of (24) changes only sign if the limits of integration are interchanged and, at the same time, the branch \(F_1\) is substituted in place of the branch \(F_0\). The expression under the limit sign in (24) will then be rewritten in the form

\[ \Gamma(c)= \int_{A_1m_1B_1} F_0\,dt + \int_{BmA} F_1\,dt . \tag{25} \]

We now define two functions \(\alpha(x,y)\) and \(\beta(x,y)\) by means of equation (22). Since \(P\) and \(Q\) are single-valued, there exists an infinite set of pairs of such functions, analytic and of the same order of single-valuedness as \(F\). For example, one may set

\[ \alpha = F\,\frac{Q}{P^2+Q^2}, \qquad \beta = -F\,\frac{P}{P^2+Q^2}. \]

Replacing the branches \(F_1\) and \(F_0\) by the corresponding expressions of the form (22), and taking into account that the arcs \(l(c)\) and \(l(c+\Delta c)\) are trajectories of the system, we shall have

\[ \Gamma(c)= \int_{A_1m_1B_1} \alpha_0\,dy-\beta_0\,dx + \int_{BmA} \alpha_1\,dy-\beta_1\,dx . \tag{26} \]

Add to and subtract from the right-hand side of (26) the sum of the integrals

\[ K= \int_{AA_1} \alpha_0\,dy-\beta_0\,dx + \int_{B_1B} \alpha_1\,dy-\beta_1\,dx, \tag{27} \]

taken respectively along the rectilinear segments \(AA_1\) and \(B_1B\) of the straight line \(s\) in the indicated directions. We shall have

\[ \Gamma= \int_{L(A)} \alpha\,dy-\beta\,dx - K, \tag{28} \]

where \(L(A)\) is the closed contour \((AA_1m_1B_1BmA)\), bounding the region \(g\), in which \(\alpha\) and \(\beta\) are single-valued analytic functions. Here the common points of the segments \(AA_1\) and \(B_1B\), which enter into the boundary of \(g\), are to be regarded as distinct. Applying Green’s formula, we obtain

\[ \int_{L(A)} \alpha\,dy-\beta\,dx = \iint_g \left(\alpha'_x+\beta'_y\right)\,dxdy . \tag{29} \]

In the double integral, instead of \(x\) and \(y\) one may introduce the parameters \(c\) and \(t\) as new variables of integration. The functional determinant of the substitution (15) is equal to \(\mu^{-1}\). In this it is not hard to verify, differentiating

the identity \(f(\varphi(t,c), \psi(t,c)) \equiv c\) and taking into account the equalities (8). We may assume that \(\mu>0\). Then, taking into account the other previously made conventions of a normalizing character, for \(\Delta c<0\) we shall have

\[ \int_{L(A)} \alpha\,dy-\beta\,dx = \int_c^{c+\Delta c}\int_{T(c)}^0 (\alpha'_x+\beta'_y)\,\frac{dt\,dc}{\mu}. \tag{30} \]

We transform the integral \(K\), occurring in (28), in the following way, putting, for brevity, \(\Delta F=\alpha\,dy-\beta\,dx=(\alpha b-\beta a)\,ds\):

\[ K=\int_{B_1A_1}\Delta F_0-\int_{BA}\Delta F_0+\int_{BB_1}(\Delta F_0-\Delta F_1). \tag{31} \]

The integral over the segment \(BA\) is equal to the product of \(c^n\) by a function regular at the point \(c=0\). The integral over \(B_1A_1\) is obtained from this by replacing \(c\) by \(c+\Delta c\). Consequently, the difference of the first two terms in (31) is the product of \(\Delta c\,c^{n-1}\) by a function holomorphic for \(c=0,\Delta c=0\). Under the integral sign over the segment \(BB_1\), equal to \(\Delta c\), there stands the difference of two consecutive branches of a single-valued function of order \(k\). Expanding it in powers of \(s\) and integrating, we represent it in the form of the product \(\Delta c\cdot c^k\) by a function regular at \(c=0\). Thus,

\[ K=\Delta c\left(c^{\,n-1}\chi_1(c,\Delta c)+c^k\chi_2(c,\Delta c)\right). \tag{32} \]

Substituting (28), (30), (32) into (24) and passing to the limit, we obtain the final formula in the form (23).

1. Let us apply Lemma 2 to the calculation of the coefficients (19). For this purpose define auxiliary functions \(\alpha_k,\beta_k\) \((k=1,2,\ldots,n-1)\) by means of the recurrence equations

\[ \alpha_1=\mu p,\quad \beta_1=\mu q,\quad \alpha_k Q-\beta_k P= \left(\frac{\partial \alpha_{k-1}}{\partial x} + \frac{\partial \beta_{k-1}}{\partial y}\right)\frac{1}{\mu} \tag{33} \]

\[ (k=2,3,\ldots,n-1). \]

Since \(P\) and \(Q\) are single-valued functions, and the factor \(\mu\) is single-valued of order \(n-1\), the integrand in (19) is a single-valued function of order also equal to \(n-1\). Therefore, to compute the derivative one may apply the differentiation formula (23) for \(k=n-1\), putting \(\alpha=\alpha_1\) and \(\beta=\beta_1\). The subsequent derivatives are found by the same formula in the form

\[ -\frac{d^k}{dc^k}\int_{T(c)}^0 \mu(Qp-Pq)\,dt = \int_{T(c)}^0 \left(\frac{\partial \alpha_k}{\partial x} + \frac{\partial \beta_k}{\partial y}\right)\frac{dt}{\mu} + R_k c^{\,n-k} \tag{34} \]

\[ (k=1,2,\ldots,n-1). \]

From (34), putting \(c=0\) and taking into account that the integrands on the cycle become periodic functions of \(t\), we obtain

\[ g_0=\int_0^T \mu(Qp-Pq)\,dt,\quad g_k=\int_0^T \left(\frac{\partial \alpha_k}{\partial x} + \frac{\partial \beta_k}{\partial y}\right)\frac{dt}{\mu} \tag{35} \]

\[ (k=1,2,\ldots,n-1). \]

The known arbitrariness in the choice of the functions \(\alpha,\beta\) does not change the values of the integrals. Formulas (35) show that in a neighborhood \(S(\varepsilon,l)\) of a multiple limit cycle the linear terms of the first \(n-1\) coefficients in the expansion of the succession function in powers of the integration constant are linear functionals and, for their computation, do not require integration of the differential equations of the system.

1. Let \(r\) be a fixed natural number, \(1 \leq r \leq n\), and let \((p_i,q_i)\) \((i=1,2,\ldots,r)\) be arbitrary functions taken in the class \(D\). Setting in (35)

\[ p=\sum_{i=1}^{r}\lambda_i p_i(x,y),\qquad q=\sum_{i=1}^{r}\lambda_i q_i(x,y), \tag{36} \]

we obtain linear homogeneous forms with respect to the variables \(\lambda_1,\ldots,\lambda_r\).

Definition 2. The functionals \(g_0,g_1,\ldots,g_{n-2}\) will be called linearly independent in the space \(D(p,q)\) if there exist \((p_i,q_i)\in D\) such that the corresponding forms (35) are linearly independent.

With the help of equation (20), applying Lemma 1 to it, one easily proves

Theorem 2. Let \(l\) be a limit cycle of multiplicity \(n\geq 2\) of system (2), and suppose there exists such an \(r\), \(1\leq r\leq n-1\), that the integrals \(g_i(p,q)\) \((i=0,1,\ldots,r-1)\) are linearly independent in the given space of nearby systems. In that case, for the maximal number \(m\) of limit cycles arising in a neighborhood of \(l\), the estimates \(r\leq m\leq n\) hold if \(n-r\) is even, and \(r+1\leq m\leq n\) if \(n-r\) is an odd number.

Definition 3. A limit cycle of system (2) will be called rough, or structurally stable, with respect to the space of perturbations \(D(p,q)\), if \(m=1\). The cycle \(l\) will be called non-rough, unstable in \(D(p,q)\), if \(m>1\).

If the order of multiplicity of the cycle is \(n=1\), then such a cycle is always rough, i.e., \(m=1\), whatever the set \(D\) [1], provided only that it consists of analytic functions or functions differentiable up to some finite order greater than the multiplicity of the cycle. Under these general smoothness assumptions, non-rough cycles can occur only if \(n\geq 2\), i.e., in the presence of a certain degree of degeneration of the original equations. The actual determination of the number \(m\), in particular the solution of the question of relative roughness or non-roughness, is a problem whose methods of solution depend essentially not only on the degree of degeneracy of the given system, but also on the structure of the space of admissible changes.

New methods for solving the problem of the birth of periodic motions in systems close to conservative ones are given in the monograph of N. N. Bogolyubov and Yu. A. Mitropolsky [2].

Duff [8] studied the behavior of cycles in spaces of nearby systems corresponding to rotation of the field of directions.

In the work of K. S. Sibirsky [7] an estimate was obtained for the number of cycles born in a neighborhood of a complex focus of a second-order algebraic system of special type.

In the simplest case, when \(D\) consists of all analytic functions, the maximal number of cycles born is exactly equal to the order of multiplicity of the cycle under consideration [4].

However, in many applied problems \(D\) consists of a certain finite number of special functions depending on a finite number of slightly varying parameters. Below it is shown that for a complete splitting of the cycle \((m=n)\) there is no need to consider all continuously

differentiable additions; it suffices to use only rational functions of the type of potentials of small intensity with sources located outside the limit cycle.

§ 3. Splitting of the cycle by rational functions

We first prove several auxiliary lemmas.

1. Let \(M_1(x_1=\varphi(t_1),\, y_1=\psi(t_1))\) be some point on the cycle, corresponding to the value \(t=t_1\), at which the tangent

\[ U(x,y)\equiv A(x-x_1)+B(y-y_1)=0 \tag{37} \]

has an ordinary (not multiple) point of tangency and has no other common points with the curve \(l\). At a distance \(\nu>0\) from the straight line (37), draw the parallel straight line

\[ U_1\equiv Ax+By+D=0. \tag{38} \]

It may be assumed that this straight line does not intersect the cycle for all \(\nu>0\), while the coefficients \(A\) and \(B\) are determined from the equations

\[ A^2+B^2=1,\qquad (AP+BQ)_{M_1}=0. \tag{39} \]

Then the nonnegative periodic function \(W=(U_1(\varphi,\psi))^{-1}\) is the reciprocal of the distance from the point \(x=\varphi(t),\ y=\psi(t)\) of the cycle to the straight line (37). The greatest value of \(W\), equal to \(\dfrac{1}{\nu}\), is attained only at one point of the cycle, namely at the point \(M_1\), corresponding to \(t=t_1\).

Lemma 3. Whatever the function \(F(x,y,\nu)\) of the variables \(x,y\) and the parameter \(\nu\), analytic in the domain \(S(\varepsilon,l)\), single-valued on \(l\), regular with respect to \(\nu\) for \(|\nu|\le \nu_0\), and not equal to zero for \(\nu=0,\ x=x_1,\ y=y_1\), the integral

\[ D_k(\nu)=\int_0^T F(\varphi,\psi,\nu)\,W(\varphi,\psi)^{k+2}\,dt \tag{40} \]

is a function of the parameter \(\nu\), regular for all \(\nu>0\), and as \(\nu\to 0\) has the asymptotic representation

\[ D_k=\delta_k \nu^{-\left(k+\frac{3}{2}\right)}+O\!\left(\nu^{-\left(k+\frac{3}{2}\right)}\right), \tag{41} \]

where \(\delta_k\) is a nonzero constant.

The proof is analogous to Lemma 4 (see [6]).

We apply Lemma 3 to the integrals (35), putting in them

\[ p=\sigma W^2,\qquad q=\delta W^2, \tag{42} \]

where \(\sigma\) and \(\delta\) are constants. From the method of constructing the fractional-linear function \(W\) it follows that, whatever \(\nu>0\), there will always be a neighborhood \(S(\varepsilon,l)\) of the curve \(l\) in which the functions (42) will be continuous. After substituting (42) into (35), the integrals become functions of the parameter \(\nu\), regular for all real \(\nu>0\).

Lemma 4. If the constants \(\sigma\) and \(\delta\) are chosen so that

\[ (\sigma Q-\delta P)_{M_1}\ne 0,\qquad (\sigma A+\delta B)\ne 0, \tag{43} \]

then the corresponding integrals (35), by virtue of (42), have as \(\nu\to 0\) the asymptotic expression

Дифференциальные уравнения № 2

\[ g_k(\nu) \simeq \delta_k \nu^{-\left(k+\frac{3}{2}\right)} \quad (k = 0, 1, \ldots, n-2). \tag{44} \]

Proof. We have

\[ g_0(\nu)=\int_0^T(\sigma Q-\delta P)\mu W^2\,dt. \tag{45} \]

Since the integrating factor on the cycle does not vanish, the integrand at the point \(t=t_1\) is nonzero. Applying Lemma 3 to the integral, we find a representation of \(g_0(\nu)\) in the form in which it follows from (44) for \(k=0\). Direct computations further show that

\[ g_1(\nu)=\int_0^T\left[(\mu p)'_x+(\mu q)'_y\right]\mu^{-1}\,dt =\int_0^T\left[-2(\sigma A+\delta B)+\frac{R_{12}}{W}\right]W^3dt. \tag{46} \]

The second term under the integral sign vanishes for \(t=t_1,\ \nu=0\). The first term is a constant number and is nonzero by the conditions of the lemma. Consequently, \(g_1(\nu)\) is obtained from (44) for \(k=1\). To prove the same for \(g_2(\nu)\), denote by \(F(x,y)\) the function of two variables \(x\) and \(y\) standing under the integral sign in (46). Following (33), we find two functions \(\alpha_2\) and \(\beta_2\) such that \(\alpha_2 Q-\beta_2 P=F\). For definiteness one may set

\[ \alpha_2=\frac{Q}{P^2+Q^2}F,\qquad \beta_2=-\frac{P}{P^2+Q^2}F. \]

Then

\[ g_2(\nu)=\int_0^T(\alpha'_{2,x}+\beta'_{2,y})\frac{dt}{\mu} =\int_0^T\left[R_{21}(x,y,\nu)(AQ-BP)(\sigma A+\right. \]

\[ \left. +\delta B)+\frac{R_{22}}{W}\right]W^4dt. \]

The second term vanishes for \(t=t_1,\ \nu=0\), whereas at this point \(R_{21}\ne0\). Thus, on the basis of Lemma 3, for \(k=2\) we find \(g_2\). Continuing, for all subsequent integrals, transformations analogous to \(g_2(\nu)\), with the use of Lemma 3, we obtain (44) for all \(k=0,1,\ldots,n-2\).

1. Choose on the cycle \(n-1\) distinct points \(M_i(x_i=\varphi(t_i),\ y_i=\psi(t_i))\) so that each of them, together with the corresponding linear functions \(U_i=A_ix+B_iy+D_i\), satisfies the conditions of Lemma 4; in particular, the constants \(A_i\) and \(B_i\) are found from the equations

\[ A_i^2+B_i^2=1,\qquad (A_iP+B_iQ)_{M_i}=0 \quad (i=1,2,\ldots,n-1). \tag{47} \]

Let \(\nu_i>0\) be the distance from the line \(U_i=0\) to the cycle \(l\). Put

\[ p_i=\sigma_i W_i^2,\qquad q_i=\delta_i W_i^2 \quad (i=1,2,\ldots,n-1), \tag{48} \]

where \(W_i\) are the functions inverse to \(U_i\), and subject \(\sigma_i\) and \(\delta_i\) to the conditions

\[ (\sigma_iA_i+\delta_iB_i)\ne0,\qquad (\sigma_iQ-\delta_iP)_{M_i}\ne0 \quad (i=1,2,\ldots,n-1). \]

On the basis of Lemma 4, for the corresponding functionals we obtain the expressions

\[ g_0(p_i,q_i)\simeq \delta_{0i}\nu_i^{-\frac32},\qquad g_1(p_i,q_i)\simeq \delta_{1i}\nu_i^{-\frac52},\ldots,\quad g_{n-2}(p_i,q_i)\simeq \]
\[ \simeq \delta_{n-2,i}\nu_i^{-\left(n-2+\frac32\right)} \tag{49} \]
\[ (i=1,2,\ldots,n-1). \]

Consider the determinant of order \(n-1\)

\[ D_{n-1}(\nu_1,\ldots,\nu_{n-1}) = \left|g_k(p_i,q_i)\right|_{\substack{i=1,2,\ldots,n-1\\ k=0,1,\ldots,n-2}}, \tag{50} \]

which is a function of the independent variables \(\nu_1,\ldots,\nu_{n-1}\), defined and regular in a neighborhood of every system of positive values. It is not hard to show that this determinant, with the indicated asymptotic values of its elements, cannot be identically equal to zero:

\[ D_{n-1}(\nu_1,\ldots,\nu_{n-1})\not\equiv 0. \tag{51} \]

1. Fixing the parameters \(\nu_i\) so that condition (51) is satisfied, consider the functions

\[ p=\sum_{i=1}^{n-1}\frac{\sigma_i\lambda_i}{U_i^2+r_i^2},\qquad q=\sum_{i=1}^{n-1}\frac{\delta_i\lambda_i}{U_i^2+r_i^2}, \tag{52} \]

where \(\lambda_i\) are small parameters; \(r_i\) are constants not equal to zero.

Theorem 2. Let the closed trajectory \(l\) be a limiting cycle of multiplicity \(n\geqslant 2\), and let the perturbation space of system (3) contain the functions (52). In that case the maximal number of cycles born in a neighborhood of the curve \(l\) is exactly equal to \(n\).

Proof. On the basis of Lemma 1 this number is not greater than \(n\), independently of one or another class \(D\). It remains to show that for any \(\varepsilon<\varepsilon_0\), \(\delta<\delta_0\) there exist continuous rational functions of the form (52) on the whole plane, satisfying the condition \(p^2+q^2<\delta\), and such that the corresponding system has exactly \(n\) simple cycles in the domain \(S(\varepsilon,l)\). In view of what was said in § 2 (Theorem 2), for this it is sufficient to show that the linear functionals (35), as a result of substituting (52), will be linearly independent. The determinant \(\overline D_{n-1}(r_1,\ldots,r_{n-1})\) of these forms is a function of the parameters \(r_1,r_2\ldots r_{n-1}\), regular in a neighborhood of every system of values \(\{r_i\}\). For \(r_i=0\) we have

\[ \overline D_{n-1}(0,\ldots,0)=D_{n-1}(\nu_1,\ldots,\nu_{n-1})\ne 0. \]

This inequality shows that, in the domain of analyticity of the function \(\overline D_{n-1}\), a point has been found at which \(\overline D_{n-1}\ne 0\). Hence it follows that \(\overline D_{n-1}\not\equiv 0\). In other words, whatever the point \(M_0(r_1^{(0)},\ldots,r_{n-1}^{(0)})\) in the parameter space \(r_i\) may be, there exists another point \(M\subset S(\delta,M_0)\) at which \(\overline D_{n-1}\ne 0\). This is what was required to prove.

It should be noted that the splitting properties of the system of functions (52) do not depend on the analytic structure of the original equations, and consequently, neither on the cycles nor on their location in the plane.

1. Under some additional restrictions, a partial decomposition of a complex cycle by polynomials of degree not exceeding a fixed natural number is possible.

Lemma 5. Let

\[ x=\varphi(s),\qquad y=\psi(s) \tag{53} \]

is a simple closed analytic curve satisfying the conditions:

1) In the domain \(D\) bounded by this curve, there exists a point \(A(x_1,y_1)\) with respect to which the domain \(D\) is star-shaped.

2) The curve (53) is situated in the circular annulus

\[ \left(\frac{2}{m+2}\right)^{\frac{1}{m}} R<r\leq R,\qquad r=\left[(x-x_1)^2+(y-y_1)^2\right]^{\frac12}. \tag{54} \]

In that case

\[ \frac{1}{S}\iint\limits_D r^m(x,y)\,dxdy < \frac{1}{T}\int\limits_0^T r^m(\varphi(s),\psi(s))\,ds, \tag{55} \]

where \(S\) is the area of the domain \(D\); \(m\) is a positive number.

Proof. Passing in the double integral to polar coordinates with center at the point \(A\), and applying the known mean-value theorems to both integrals, we shall have

\[ \frac{1}{S}\iint\limits_D r^m(x,y)\,dxdy = \frac{1}{m+2}\,\frac{1}{S}\int\limits_0^{2\pi} r^m(\varphi) r^2(\varphi)\,d\varphi = \]

\[ = \frac{r^m(c_1)}{m+2}\,\frac{1}{S}\int\limits_0^{2\pi} r^2\,d\varphi = \frac{2}{m+2}\,r^m(c_1), \tag{56} \]

\[ \frac{1}{T}\int\limits_0^T r^2(\varphi(s),\psi(s))\,ds = r^m(c_2), \tag{57} \]

where \(c_1\) and \(c_2\) are points on the curve (53). From (54) it follows that

\[ \frac{2}{m+2}R^m<r^m(c_2),\qquad r^m(c_1)\leq R^m. \tag{58} \]

Hence, and from (54), we have the inequalities

\[ \frac{2}{m+2}r^m(c_1)<\frac{2}{m+2}R^m<r^m(c_2), \tag{59} \]

from which, taking (56) and (57) into account, (55) follows.

Theorem 3. Let \(l\) be a periodic solution of system (2) satisfying the following conditions:

1) \(l\) is a limit cycle of multiplicity \(n\geq 3\).

2) The domain \(D\) bounded by the cycle is star-shaped with respect to some interior point \(A_1(x_1,y_1)\).

3) The curve \(l\) is situated in the circular annulus with center at the point \(A_1\)

\[ \left(\frac{2}{m+2}\right)^{\frac{1}{m}} R<r\leq R,\qquad r=\left((x-x_1)^2+(y-y_1)^2\right)^{\frac12}, \tag{59_1} \]

where \(m\) is a positive even number.

4) On the cycle \(l\) the identities

\[ (\chi)_l \equiv (P'_x+Q'_y)_l \equiv 0,\qquad (\chi_x)_l \equiv 0. \tag{60} \]

hold.

In this case, whatever positive numbers \(\varepsilon\) and \(\delta\) may be, there always exist polynomials \(p\) and \(q\) of degree not exceeding \(m+1\), satisfying in the domain \(S(\varepsilon,l)\) the smallness conditions \(p^2+q^2<\delta\), and such that the corresponding perturbed system (3) will have in \(S(\varepsilon,l)\) at least two cycles for even \(n\) and at least three for odd \(n\).

Proof. On the basis of Theorem 2, § 2, it suffices to prove that the functionals

\[ g_0=\int_0^T \mu(Qp-Pq)\,dt,\qquad g_1=\int_0^T \left[(\mu p)'_x+(\mu q)'_y\right]\frac{dt}{\mu} \tag{61} \]

are linearly independent in the Euclidean space of the coefficients of the polynomials

\[ p=\sum_{i+k=0}^{m+1}\alpha_{ik}x^i y^k,\qquad q=\sum_{i+k=0}^{m+1}\beta_{ik}x^i y^k. \tag{62} \]

Just as in Theorem 6 (see [5]), condition (60) permits one to transform the integrals (61) to the form

\[ g_0=-\iint_D (p'_x+q'_y)\,dxdy,\qquad g_1=\int_0^T (p'_x+q'_y)\,dt+Cg_0, \tag{63} \]

where \(C\) is a constant. Hence it follows that the problem of the linear independence of the integrals (61) reduces to the same problem for the linear forms

\[ h_0=\iint_D (p'_x+q'_y)\,dxdy,\qquad h_1=\int_0^T (p'_x+q'_y)\,dt. \tag{64} \]

Putting here

\[ q=0,\qquad p=\lambda_1 x+\lambda_2\int_0^x r^m dx, \]

where \(\lambda_1\) and \(\lambda_2\) are independent small parameters, we shall have

\[ h_0=\lambda_1 S+\lambda_2\iint_D r^m dxdy,\qquad h_1=\lambda_1 T+\lambda_2\int_0^T r^m(\varphi,\psi)\,dt. \]

The coefficients of \(\lambda_1\) and \(\lambda_2\) satisfy the conditions of Lemma 5. By virtue of this lemma, the integrals (64), and consequently also (61), are linearly independent.

References

1. Andronov A. A., Pontryagin L. S. DAN SSSR, 14, 247, 1937.
2. Bogolyubov N. N., Mitropolsky Yu. A. Asymptotic Methods in the Theory of Nonlinear Oscillations. Moscow, GIFML, 1958.
3. Goursat É. A Course in Mathematical Analysis, 2, part 2, GTTI, 1933.
4. Leontovich E. A., Andronov A. A. The birth of limit cycles from a rough focus or center and from a rough limit cycle. Matem. sb., 40, 1956, pp. 179–224.
5. Otrokov N. F. Multiple limit cycles, Matem. sb., 41, issue 4, 1957.
6. Otrokov N. F. Vestn. LGU, No. 19, issue 4, 1961, pp. 23–44.
7. Sibirsky K. S. Differential Equations, 1, No. 1, 53–66, 1965.
8. Duff. Limit Cycles and rotated vector fields. Ann. P. Math., 57, 1, 15–31, 1953.

Received by the editors
March 3, 1965

Gorky State University
named after N. I. Lobachevsky