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ON THE STABILITY OF THE TRIVIAL SOLUTION OF A CERTAIN SYSTEM OF DIFFERENTIAL EQUATIONS
N. A. LUKASHEVICH
Consider the system
\[ \frac{dx}{dt}=P_n(x,y), \qquad \frac{dy}{dt}=-y+kx+Q_m(x,y), \tag{1} \]
where \(k\ne 0\),
\[ P_n=\sum_{i=0}^{n} a_{n-i,i}x^{\,n-i}y^i,\qquad Q_m=\sum_{i=0}^{m} b_{m-i,i}x^{\,m-i}y^i, \]
and \(a_{ij}, b_{ij}\) are real constants.
We shall show that, for system (1), the question of the character of the stability of \((0,0)\) is always resolved by a finite number of elementary operations.
The following is known [1]: if \(y=\varphi(x)\), \(\varphi(0)=0\), is a solution of the equation
\[ y=kx+Q_m(x,y) \tag{2} \]
and
\[ P_n(x,\varphi(x))=g_lx^l+g_{l+1}x^{l+1}+\ldots \quad (l\le n), \tag{3} \]
then:
a) if \(l=2s\), the unperturbed motion is unstable;
b) if \(l=2s+1\), the unperturbed motion is stable under the condition that \(g_l<0\), and unstable when \(g_l>0\);
c) if all \(g_l=0\) (i.e., \(P_n(x,\varphi(x))\equiv 0\)), then there exists a continuous family of steady motions, to which the unperturbed motion \((0,0)\) under consideration also belongs; all motions of this family that are sufficiently close to \((0,0)\), including \((0,0)\), will be stable.
First, let us find the solution of equation (2). Direct calculations give
\[ y=\varphi(x)=kx+xQ_m(1,k)\bigl[x^{m-1}+Q'_m(1,k)x^{2m-2}+\alpha_3x^{3m-3}+\ldots\bigr], \tag{4} \]
where
\[ \alpha_s=\varepsilon^{(s)}_1 Q_m^{(s-1)}(1,k)Q_m^{s-2}(1,k) +\varepsilon^{(s)}_2 Q_m^{(s-2)}(1,k)Q_m^{s-3}(1,k)Q'_m \]
\[ +\ldots+Q_m^{\prime\,s-1}(1,k); \]
\(\varepsilon_i^{(s)}\ne 0\) are parameters independent of the coefficients of system (1); \(Q_m^{(r)}(1,k)\) is the \(r\)-th derivative with respect to \(k\) of the polynomial \(Q_m(1,k)\).
\[ P_n(x,\varphi(x))=x^n\{P_n(1,k)+P'_n(1,k)Q_m(1,k)x^{m-1}+g_2x^{2m-2}+\ldots\}, \tag{5} \]
where
\[ g_s=Q_m(1,k)\sum_{i=1}^{s}\gamma_i^{(s)}P_n^{(i)}(1,k),\qquad s=2,3,\ldots, \tag{6} \]
\(\gamma_i^{(s)}\) are coefficients depending on \(Q_m^{(\theta)}(1,k)\) \((\theta<i)\), with \(\gamma_s^{(s)}=Q_m^{s-1}(1,k)\); \(P_n^{(i)}(1,k)\) is the \(i\)-th derivative with respect to \(k\) of the polynomial \(P_n(1,k)\).
Thus, if \(P_n(1,k)\ne 0\), the question of the character of stability is completely determined by the number \(n\) and \(\operatorname{sgn} P_n(1,k)\).
Let \(P_n(1,k)=0\). Two cases are possible:
\[ \text{a) } Q_m(1,k)=0 \quad \text{and} \quad \text{b) } Q_m(1,k)\ne 0. \]
If \(Q_m(1,k)=0\), then all \(g_l=0\) \((l=1,2,\ldots)\), and system (1) has the form
\[ \frac{dx}{dt}=(y-kx)P_{n-1}^{*}(x,y)\equiv P_n(x,y), \]
\[ \frac{dy}{dt}=(y-kx)[-1+Q_{m-1}^{*}(x,y)]\equiv -y+kx+Q_m(x,y). \tag{7} \]
The question of stability of the unperturbed motion for system (7) is clear.
Suppose that \(P_n(1,k)=0\), but \(Q_m(1,k)\ne 0\). From (6) it is obvious that all \(g_s=0\) \((s=1,2,\ldots)\) only in the case when
\[ P_n'(1,k)=P_n''(1,k)=\ldots=P_n^{(n)}(1,k)=0. \]
In this case system (1) degenerates into the system
\[ \frac{dx}{dt}=0,\qquad \frac{dy}{dt}=-y+kx+Q_m(x,y). \tag{8} \]
The unperturbed motion is stable.
Let \(P_n^{(s)}(1,k)\ne 0\) \((s<n)\). Then \(g_{s+1}\ne 0\), and the character of stability is completely determined by the sign of the constant \(g_{s+1}\) and by the parity of the number \(\sigma=n+(s+1)(m-1)\), since
\[ P_n(x,\varphi(x))=\frac{1}{(s+1)!}\,P_n^{(s+1)}(1,k)Q_m^{s+1}(1,k)x^\sigma+\ldots \tag{9} \]
We formulate the following rule in the form of a theorem.
Theorem. In order to determine the character of stability of the unperturbed motion \(x=0,\ y=0\) for system (1), write down the polynomials \(P_n(1,k)\) and \(Q_m(1,k)\).
If \(k\) is not a root of the polynomial \(P_n(1,k)\), then the unperturbed motion is stable provided that \(n=2p+1\), \(\operatorname{sgn}P_n(1,k)<0\), and is unstable if \(n=2p\) or \(n=2p+1\), \(\operatorname{sgn}P_n(1,k)>0\).
If \(k\) is a root of the polynomial \(P_n(1,k)\), but \(Q_m(1,k)\ne 0\), then we compute the derivatives with respect to \(k\) of \(P_n(1,k)\), and then, if there exists a natural number \(j\) such that \(P_n^{(s)}(1,k)=0,\ s<j,\ P_n^{(j)}(1,k)\ne 0\), then:
a) the unperturbed motion is unstable if \(\sigma=n+j(m-1)\) is even, or if \(\sigma\) is odd but \(g_j=P_n^{(j)}(1,k)Q_m^j(1,k)>0\);
b) the unperturbed motion is stable if \(\sigma\) is odd and \(g_j<0\).
Finally, if all \(P_n^{(j)}(1,k)=0\), the unperturbed motion is stable (the system has the form (8)). Stability also occurs in the case when \(k\) is a common root of the polynomials \(P_n(1,k)\) and \(Q_m(1,k)\) (the system has the form (7)).
Example [1]:
\[ \frac{dx}{dy}=ax^2+bxy+cy^2,\qquad \frac{dy}{dt}=-y+kx+lx^2+mxy+ny^2. \]
Writing down the polynomials \(P_2(1,k)\) and \(Q_2(1,k)\), in this case we have
\[ P_2(1,k)=a+bk+ck^2,\qquad Q_2(1,k)=l+mk+nk^2. \]
According to the theorem, we have:
1) \(a+bk+ck^2\ne 0\)—the unperturbed motion is unstable, since \(n=2\);
2) \(a+bk+ck^2=0,\ (b+2ck)(l+mk+nk^2)>0\)—the unperturbed motion is unstable, since \(\sigma=3,\ g_1>0\);
3) \(a+bk+ck^2=0,\ (b+2ck)(l+mk+nk^2)<0\)—the unperturbed motion is stable, since \(\sigma=3,\ g_1<0\);
4) \(a+bk+ck^2=0,\ P_2'=b+2ck=0,\ P_2''=2c\ne 0\)—the unperturbed motion is unstable, since \(\sigma=4\);
5) \(a+bk+ck^2=0,\ l+mk+nk^2=0\)—the unperturbed motion is stable;
6) \(P_2(1,k)=P_2'(1,k)=P_2''(1,k)=0\), i.e. \(a=b=c=0\)—the unperturbed motion is stable.
Remark 1. It is easy to see that completely analogous results concerning the character of stability of \((0,0)\) are obtained also for the system
\[ \frac{dx}{dt}=P_n(\psi(x),y),\qquad \frac{dy}{dt}=-y+\psi(x)+Q_m(\psi(x),y), \tag{10} \]
where the holomorphic function \(\psi(x)\) is such that \(\psi(0)=0,\ \psi'(0)=k\ne 0\), and \(P_n(\psi(x),y)\) and \(Q_m(\psi(x),y)\) are homogeneous polynomials in \(\psi(x)\) and \(y\), of degrees \(n\) and \(m\), respectively. The difference consists only in the fact that for system (10) the constants \(g_j\) have the form
\[ g_j=k^\sigma P_n^{(j)}(1,k)Q_m^j(1,k),\qquad \sigma=n+j(m-1). \]
To verify the latter, it suffices to seek the solution \(y=\varphi(x)\) of equation (2) in the form
\[ \varphi(x)=\psi(x)+A_2\psi^2(x)+A_3\psi^3(x)+\cdots \]
\(A_i\) are constants.
Remark 2. For the system
\[ \frac{dx}{dt}=P_n(x,y)+P(x,y),\qquad \frac{dy}{dt}=-y+kx+Q_m(x,y)+Q(x,y), \tag{11} \]
where \(P(x,y)\) and \(Q(x,y)\) are holomorphic functions whose expansions in powers of \(x\) and \(y\) contain no terms below dimension \(m(n+1)\), \(Q_m(1,k)\ne 0,\ P_n^{(j)}(1,k)\ne 0,\ j\le n\), the character of stability of the unperturbed motion coincides with the character of stability of the unperturbed motion of system (1).
Remark 3. According to the results of [2], the singular point \((0,0)\) for systems (1), (10), (11) will be:
a) a node, if \(n+j(m-1)\) is an odd number and
\[ g_j=P_n^{(j)}(1,k)Q_m^j(1,k)<0; \]
b) a saddle, if \(n+j(m-1)\) is an odd number and \(g_j>0\);
c) a saddle-node, if \(n+j(m-1)\) is an even number.
References
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Lyapunov A. M. The General Problem of the Stability of Motion. Collected Works, 2. Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1956.
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Vorob'ev A. P. Doklady of the Academy of Sciences of the BSSR, 3, No. 8, 1959.
Received by the editors
5 September 1964
Belorussian State University
named after V. I. Lenin