UDC 517.925: 517.916

STABILITY ON A FINITE TIME INTERVAL IN THE CASE OF ONE ZERO AND A PAIR OF PURELY IMAGINARY ROOTS

R. M. NASYROV

1. Consider a system of differential equations of perturbed motion of the form

\[ \begin{aligned} \frac{dx}{dt} &= X(t, x, x_1, y_1, \eta_1, \ldots, \eta_n),\\ \frac{dx_1}{dt} &= -\lambda y_1 + X_1(t, x, x_1, y_1, \eta_1, \ldots, \eta_n),\\ \frac{dy_1}{dt} &= \lambda x_1 + Y_1(t, x, x_1, y_1, \eta_1, \ldots, \eta_n),\\ \frac{d\eta_s}{dt} &= p_{s1}\eta_1 + \ldots + p_{sn}\eta_n + p_s x + q_s x_1 + r_s y_1 +{}\\ &\quad + \Phi_s(t, x, x_1, y_1, \eta_1, \ldots, \eta_n), \end{aligned} \tag{1.1} \]

\(s=1, 2, \ldots, n\), where the coefficients of the functions \(X, X_1, Y_1, \Phi_s\), nonlinear and holomorphic with respect to \(x, x_1, y_1, \eta_1, \ldots, \eta_n\), are real continuous functions on the interval \([t_0, T]\).

Let the system

\[ \frac{d\eta_s}{dt}=p_{s1}\eta_1+\ldots+p_{sn}\eta_n \quad (s=1,2,\ldots,n) \tag{1.2} \]

with continuous coefficients on the interval \([t_0, T]\) be such that its characteristic equation at the initial moment of time \(t_0\)

\[ \left|p_{sj}(t_0)-\delta_{sj}\varkappa\right|=0 \tag{1.3} \]

has only roots with negative real parts. Then, among the roots of the characteristic equation of system (1.1) at the moment \(t_0\), there will be one zero root and two purely imaginary roots, while the remaining ones have negative real parts. In this case the question of stability on a finite time interval is not decided by the equations of the first approximation and requires consideration of the nonlinear terms [2]. In connection with this there arises the following

Problem. From the structure of the functions \(X, X_1, Y_1, \Phi_s\), establish sufficient conditions for stability of the unperturbed motion on the finite time interval \([t_0, t_0+\tau]\) and determine \(\tau\), assuming \(t_0+\tau \leq T\).

1. Let us first consider a system of the 3rd order

\[ \begin{aligned} \frac{d x}{d t} &= X(t,x,x_1,y_1),\\ \frac{d x_1}{d t} &= -\lambda y_1+X_1(t,x,x_1,y_1), \tag{2.1}\\ \frac{d y_1}{d t} &= \lambda x_1+Y_1(t,x,x_1,y_1). \end{aligned} \]

Introducing the complex variable \(z=x_1+i y_1\), we write

\[ \begin{aligned} \frac{d x}{d t} &= X(t,x,z,\bar z),\\ \frac{d z}{d t} &= i\lambda z+Z(t,x,z,\bar z). \end{aligned} \tag{2.2} \]

By the substitution

\[ \begin{aligned} x&=\xi+u^{(2)}(t,\xi,\zeta,\bar\zeta)+\ldots+u^{(N)}(t,\xi,\zeta,\bar\zeta),\\ z&=\zeta+v^{(2)}(t,\xi,\zeta,\bar\zeta)+\ldots+v^{(N)}(t,\xi,\zeta,\bar\zeta), \end{aligned} \tag{2.3} \]

where \(u^{(k)}, v^{(k)}\) are forms of the \(k\)-th order in the variables \(\xi,\zeta,\bar\zeta\), to be determined, with coefficients continuous on \([t_0,T]\), we bring (2.2) to the form

\[ \begin{aligned} \frac{d\xi}{dt} &=\sum_{n+2m=2}^{N} A_1^{(n,m,m)}(t)\xi^n\zeta^m\bar\zeta^m +F_1(t,\xi,\zeta,\bar\zeta),\\ \frac{d\zeta}{dt} &=i\lambda\zeta+ \sum_{n+2m=1}^{N-1} A_2^{(n,m+1,m)}(t)\xi^n\zeta^{m+1}\bar\zeta^m +F_2(t,\xi,\zeta,\bar\zeta), \end{aligned} \tag{2.4} \]

where \(N\) is a sufficiently large integer; \(F_j\) are functions whose expansions in \(\xi,\zeta,\bar\zeta\) begin with terms no lower than order \(N+1\); \(A_1^{(n,m,m)}(t)\) and \(A_2^{(n,m+1,m)}(t)\) are continuous on \([t_0,T]\).

Substituting (2.3) into (2.2) and using (2.4), we find the relations that must be satisfied by the unknown forms

\[ u^{(k)}=\sum B_1^{(n,m_1,m_2)}(t)\xi^n\zeta^{m_1}\bar\zeta^{m_2},\qquad v^{(k)}=\sum B_2^{(n,m_1,m_2)}(t)\xi^n\zeta^{m_1}\bar\zeta^{m_2} \]

\[ (n+m_1+m_2=k;\quad k=2,3,\ldots,N). \]

Having fixed an arbitrary number \(q\leq N\) and equating in these relations the coefficients of \(\xi^n\zeta^{m_1}\bar\zeta^{m_2}\) \((n+m_1+m_2=q)\), we obtain the following equations:

\[ A_1^{(n,m,m)}(t)=C_1^{(n,m,m)}(t),\qquad B_1^{(n,m,m)}(t)=0\quad \text{when } m_1=m_2, \tag{2.5} \]

\[ \frac{dB_1^{(n,m_1,m_2)}}{dt} +i\lambda(m_1-m_2)B_1^{(n,m_1,m_2)} = C_1^{(n,m_1,m_2)}(t) \quad \text{when } m_1\ne m_2, \]

\[ A_2^{(n,m+1,m)}(t)=C_2^{(n,m+1,m)}(t),\qquad B_2^{(n,m+1,m)}(t)=0\quad \text{for } m_1=m_2+1, \]

\[ \frac{dB_2^{(n,m_1,m_2)}}{dt} +i\lambda(m_1-m_2-1)B_2^{(n,m_1,m_2)} =C_2^{(n,m_1,m_2)}(t) \tag{2.6} \]

\[ \text{for } m_1\ne m_2+1. \]

Here \(C_j^{(n,m_1,m_2)}(t)\) are integral rational functions of those coefficients \(A_j^{(n,m_1,m_2)}\), \(B_j^{(n,m_1,m_2)}\) for which \(n+m_1+m_2<q\).

To determine the forms \(u^{(q)}\), \(v^{(q)}\), it is sufficient to know some particular solution of the systems (2.5), (2.6). This particular solution, defined by the initial conditions \(B_j^{(n,m_1,m_2)}(t_0)=0\), on the basis of the existence theorem [3], will be a continuously differentiable function of time in the domain of continuity of the coefficients \(C_j^{(n,m_1,m_2)}(t)\). We note that for \(n+m_1+m_2=2\), \(C_1^{(n,m_1,m_2)}(t)\) and \(C_2^{(n,m_1,m_2)}(t)\) are the coefficients in the expansions of the functions \(X(t,\xi+\ldots,\zeta+\ldots,\bar{\zeta}+\ldots)\), \(Z(t,\xi+\ldots,\zeta+\ldots,\bar{\zeta}+\ldots)\), continuous in \(t\) on the interval \([t_0,T]\); consequently, the transformation (2.3) with the indicated properties exists. Setting successively \(q=2,3,\ldots\), one can determine the forms \(u^{(2)}\), \(v^{(2)}\), \(u^{(3)}\), \(v^{(3)}\), \(\ldots\) and the coefficients \(A_j^{(n,m_1,m_2)}(t)\). The computations are carried out until one arrives at some number \(N\) for which not all of the quantities \(A_1^{(n,m_1,m_2)}(t)\) and \(\operatorname{Re} A_2^{(n,m_1,m_2)}(t)\) with \(n+m_1+m_2=N\) are equal to zero.

Putting \(\zeta=\rho\exp\theta i\), \(A_1^{(n,m,m)}=a_1^{(n,m)}(t)\), \(\operatorname{Re}A_2^{(n,m+1,m)}(t)=a_2^{(n,m)}(t)\), and separating the real part of the second equation (2.4), we shall have

\[ \frac{d\xi}{dt} =\sum a_1^{(n,m)}(t)\xi^n\rho^{2m} +\varphi_1(t,\xi,\rho,\theta),\qquad n+2m=N, \]

\[ \frac{d\rho}{dt} =\sum a_2^{(n,m)}(t)\xi^n\rho^{2m+1} +\varphi_2(t,\xi,\rho,\theta),\qquad n+2m=N-1, \tag{2.7} \]

where, for all values of \(\theta\) and for sufficiently small \(\xi,\rho\), on \([t_0,T]\) the inequalities

\[ |\varphi_j(t,\xi,\rho,\theta)|<C_j\left[|\xi|+|\rho|\right]^{N+1}\qquad (j=1,2), \]

hold, where \(C_j\) is a positive constant.

System (2.7) is a special case of a system describing perturbed motion in the critical case of two zero roots with two groups of solutions, when the center-focus problem does not arise.

If \(N\) is an even number, then the unperturbed motion is certainly unstable. For \(N\) odd, we consider the plane of the variables \(\xi,\rho\) and pass to polar coordinates by putting \(\xi=r\beta_1\), \(\rho=r\beta_2\), \(\beta_1^2+\beta_2^2=1\). In the new variables we shall have

\[ \frac{dr}{dt}=r^N R(t,\beta_1,\beta_2)+O(r^{N+1}), \]

\[ \frac{d\beta_1}{dt}=r^{N-1}\beta_1\beta_2^2 P(t,\beta_1,\beta_2)+O(r^N), \tag{2.8} \]

where

\[ R(t,\beta_1,\beta_2)=a_1^{(N,0)}(t)\beta_1^{N+1} +\bigl(a_1^{(N-2,2)}(t)-a_2^{(N-1,1)}(t)\bigr)\times \]

\[ \times \beta_1^{N-1}\beta_2^2+\ldots+a_2^{(0,N)}(t)\beta_2^{N+1}, \]

\[ P(t,\beta_1,\beta_2)= \bigl(a_1^{(N,0)}(t)-a_2^{(N-1,1)}(t)\bigr)\beta_1^{N-1}+ \]

\[ +\bigl(a_1^{(N-2,2)}(t)-a_2^{(N-3,3)}(t)\bigr)\beta_1^{N-3}\beta_2^2+\ldots+ \]

\[ +\bigl(a_1^{(1,N-1)}(t)-a_2^{(0,N)}(t)\bigr)\beta_2^{N-1}, \]

\(O(r^k)\) are terms having, with respect to \(r\), order not lower than \(k\).

Introduce the Lyapunov function \(V=\dfrac{1}{2}r^2\exp(-2Mu(\beta_1,\beta_2))\), where \(M\) is some constant; \(u(\beta_1,\beta_2)\) is a bounded function of its arguments. For the derivative of the Lyapunov function by virtue of (2.8) we shall have

\[ \frac{dV}{dt} =r^{N+1}\left[ R(t,\beta_1,\beta_2)-M\beta_1\beta_2P(t,\beta_1,\beta_2)\times \right. \]

\[ \left. \times\left(\beta_2\frac{\partial u}{\partial\beta_1} -\beta_1\frac{\partial u}{\partial\beta_2}\right)\right]\exp(-2Mu)+O(r^{N+2}) \]

or

\[ \frac{dV}{dt} =r^{N+1}\left\{ \left[ R(t_0,\beta_1,\beta_2)-M\beta_1\beta_2P(t_0,\beta_1,\beta_2)\times \right.\right. \]

\[ \left. \times\left(\beta_2\frac{\partial u}{\partial\beta_1} -\beta_1\frac{\partial u}{\partial\beta_2}\right)\right] +\tau\left[ R'_t(t_0+\xi\tau,\beta_1,\beta_2)-\right. \]

\[ \left.\left. -M\beta_1\beta_2P'_t(t_0+\eta\tau,\beta_1,\beta_2) \left(\beta_2\frac{\partial u}{\partial\beta_1} -\beta_1\frac{\partial u}{\partial\beta_2}\right)\right]\right\}\times \]

\[ \times\exp(-2Mu)+O(r^{N+2}), \tag{2.9} \]

\[ 0<\xi,\eta<1,\qquad \tau=t-t_0. \]

At the initial instant of time \(\tau=0\), and

\[ \left.\frac{dV}{dt}\right|_{t=t_0} =r^{N+1}\left[ R(t_0,\beta_1,\beta_2)-M\beta_1\beta_2P(t_0,\beta_1,\beta_2)\times \right. \]

\[ \left. \times\left(\beta_2\frac{\partial u}{\partial\beta_1} -\beta_1\frac{\partial u}{\partial\beta_2}\right)\right]\exp(-2Mu)+O(r^{N+2})= \]

\[ =r^{N+1}\Phi(t_0,\beta_1,\beta_2)\exp(-2Mu)+O(r^{N+2}). \]

For stability on a finite time interval it is sufficient to require that

\[ \left.\frac{dV}{dt}\right|_{t=t_0}<0. \]

For a sufficiently small region of initial perturbations, the sign of

\[ \left.\frac{dV}{dt}\right|_{t=t_0} \]

is determined by the sign of the function \(\Phi(t_0,\beta_1,\beta_2)\). On the line

\[ \beta_1=0 \]

\[ \Phi(t_0,\beta_1,\beta_2)=R(t_0,0,\beta_2)=a_2^{(0,N)}(t_0)\beta_2^{N+1}, \]

on the line \(\beta_2=0\)

\[ \Phi(t_0,\beta_1,\beta_2)=R(t_0,\beta_1,0)=a_1^{(N,0)}(t_0)\beta_1^{N+1}. \]

Consequently, for stability to be present it is necessary to require

\[ a_1^{(N,0)}(t_0)<0,\qquad a_2^{(0,N)}(t_0)<0. \tag{2.10} \]

On the lines with angular coefficients determined by the equation \(P(t_0,1,k)=0\), it is necessary to require that

\[ R(t_0,\beta_1,\beta_2)<0. \tag{2.11} \]

If \(P(t_0,1,k)\) has no real roots, then condition (2.11) is absent.

In all the remaining region of variation of \(\beta_1,\beta_2\) one can arrange that

\[ \left.\frac{dV}{dt}\right|_{t=t_0}<0. \]

Indeed, put

\[ \beta_2\frac{\partial u}{\partial \beta_1} -\beta_1\frac{\partial u}{\partial \beta_2} =\beta_1\beta_2 P(t_0,\beta_1,\beta_2), \]

then

\[ \left.\frac{dV}{dt}\right|_{t=t_0} =r^{N+1}\left[R(t_0,\beta_1,\beta_2)-M\beta_1^2\beta_2^2P^2(t_0,\beta_1,\beta_2)\right]\times \]

\[ {}\times \exp(-2M u)+O(r^{N+2}), \]

and by choosing a sufficiently large positive number \(M\) one can achieve the desired result.

Since \(P(t_0,\beta_1,\beta_2)\) is a form of order \(N-1\) with respect to \(\beta_1,\beta_2\), \(u(\beta_1,\beta_2)\) can be sought in the form of a form of order \(N+1\), and its coefficients are expressed directly in terms of the coefficients of \(P(t_0,\beta_1,\beta_2)\).

If conditions (2.10), (2.11) are satisfied, then, by continuity, there will exist a time interval \(\tau\) guaranteeing the stability of the unperturbed motion. This interval is found from the condition

\[ R(t_0,\beta_1,\beta_2)-M\beta_1^2\beta_2^2P^2(t_0,\beta_1,\beta_2) +\tau\left[R_t'(t_0+\xi\tau,\beta_1,\beta_2)-\right. \]

\[ \left.{}-M\beta_1^2\beta_2^2P_t'(t_0+\eta\tau,\beta_1,\beta_2)P(t_0,\beta_1,\beta_2)\right]<0,\qquad 0<\xi,\eta<1. \]

It is inexpedient to give an explicit estimate of \(\tau\) in the general case, since in each concrete problem it will be obtained considerably more accurately.

1. Let us now consider system (1.1). Introducing the complex variable \(z=x_1+iy_1\), we shall have

\[ \frac{dx}{dt}=X(t,x,z,\bar z,\eta_1,\ldots,\eta_n), \]

\[ \frac{dz}{dt}=i\lambda z+Z(t,x,z,\bar z,\eta_1,\ldots,\eta_n), \tag{3.1} \]

\[ \frac{d\eta_s}{dt}=p_{s1}\eta_1+\cdots+p_{sn}\eta_n+p_sx+p_{s*}z+\bar p_{s*}\bar z+ \]

\[ + \Phi_s(t, x, z, \bar z, \eta_1, \ldots, \eta_n), \tag{3.2} \]

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

We transform equations (3.2) so that the expansion of their right-hand sides with respect to the critical variables \(x, z\) begins with terms of arbitrarily high order. By the substitution

\[ \xi_s=\eta_s-v_s(t, x, z, \bar z) \quad (s=1, 2, \ldots, n), \tag{3.3} \]

where \(v_s\) are holomorphic functions of \(x, z, \bar z\) with coefficients continuous in \(t\), we reduce (3.1), (3.2) to the form

\[ \frac{dx}{dt}=X(t, x, z, \bar z, \xi_1+v_1, \ldots, \xi_n+v_n)= \]

\[ = X^*(t, x, z, \bar z, \xi_1, \ldots, \xi_n), \tag{3.4} \]

\[ \frac{dz}{dt}=i\lambda z+Z(t, x, z, \bar z, \xi_1+v_1, \ldots, \xi_n+v_n)= \]

\[ = i\lambda z+Z^*(t, x, z, \bar z, \xi_1, \ldots, \xi_n), \]

\[ \frac{d\xi_s}{dt}=p_{s1}\xi_1+\cdots+p_{sn}\xi_n+\Phi^*_{s0}(t, x, z, \bar z)+ \]

\[ +\Phi^*_s(t, x, z, \bar z, \xi_1, \ldots, \xi_n), \tag{3.5} \]

where

\[ \Phi^*_{s0}=p_{s1}v_1+\cdots+p_{sn}v_n+p_sx+p_{s^*}z+p_{s^*}\bar z+ \]

\[ +\Phi_s(t, x, z, \bar z, v_1, \ldots, v_n)-\frac{\partial v_s}{\partial t}-\frac{\partial v_s}{\partial x}\times \]

\[ \times X(t, x, z, \bar z, v_1, \ldots, v_n)-\frac{\partial v_s}{\partial z}\bigl(i\lambda z+Z(t, x, z, \bar z, v_1, \ldots, v_n)\bigr)- \]

\[ -\frac{\partial v_s}{\partial \bar z}\bigl(-i\lambda \bar z+\bar Z(t, x, z, \bar z, v_1, \ldots, v_n)\bigr), \]

\[ \Phi^*_s=\Phi_s(t, x, z, \bar z, \xi_1+v_1, \ldots, \xi_n+v_n)- \]

\[ -\Phi_s(t, x, z, \bar z, v_1, \ldots, v_n)- \]

\[ -\frac{\partial v_s}{\partial x}\bigl(X(t, x, z, \bar z, \xi_1+v_1, \ldots, \xi_n+v_n)- \]

\[ -X(t, x, z, \bar z, v_1, \ldots, v_n)\bigr)- \]

\[ -\frac{\partial v_s}{\partial z}\bigl(Z(t, x, z, \bar z, \xi_1+v_1, \ldots, \xi_n+v_n)- \]

\[ -Z(t, x, z, \bar z, v_1, \ldots, v_n)\bigr)- \]

\[ -\frac{\partial v_s}{\partial z}\left(\overline Z'(t,x,z,\overline z,\xi_1+v_1,\ldots,\xi_n+v_n)-\right. \]

\[ \left.-\overline Z(t,x,z,\overline z,v_1,\ldots,v_n)\right). \]

To annihilate in \(\Phi_{s0}^{*}\) all terms up to order \(N\), put

\[ v_s(t,x,z,\overline z)=v_s^{(1)}(t,x,z,\overline z)+ \]

\[ +v_s^{(2)}(t,x,z,\overline z)+\ldots+v_s^{(N)}(t,x,z,\overline z), \]

where \(v_s^{(k)}\) are forms of the \(k\)-th order in the variables \(x,z,\overline z\), and require that the relations

\[ \frac{\partial v_s^{(1)}}{\partial t} +i\lambda\left(\frac{\partial v_s^{(1)}}{\partial z}z- \frac{\partial v_s^{(1)}}{\partial \overline z}\overline z\right) =p_{s1}v_1^{(1)}+\ldots+ \]

\[ +p_{sn}v_n^{(1)}+p_sx+p_{s*}z+\overline p_{s*}\overline z, \]

\[ \frac{\partial v_s^{(k)}}{\partial t} +i\lambda\left(\frac{\partial v_s^{(k)}}{\partial z}z- \frac{\partial v_s^{(k)}}{\partial \overline z}\overline z\right) =p_{s1}v_1^{(k)}+\ldots+ \]

\[ +p_{sn}v_n^{(k)}+U_s^{(k)}(t,x,z,\overline z) \]

\[ (s=1,2,\ldots,n;\quad k=2,3,\ldots,N) \]

be satisfied. Here \(U_s^{(k)}\) are forms of the \(k\)-th order in the variables \(x,z,\overline z\), depending on those forms \(v_s^{(m)}\) for which \(m<k\).

If we write

\[ v_s^{(k)}=\sum b_s^{(p,q,r)}(t)x^p z^q\overline z^{\,r},\qquad U_s^{(k)}=\sum C_s^{(p,q,r)}(t)x^p z^q\overline z^{\,r}, \]

\[ p+q+r=k, \]

then, for determining \(b_s^{(p,q,r)}(t)\), we obtain the equations

\[ \frac{db_s^{(p,q,r)}}{dt} =p_{s1}b_1^{(p,q,r)}+\ldots+ \]

\[ +\bigl[p_{ss}-i\lambda(q-r)\bigr]b_s^{(p,q,r)} +\ldots+p_{sn}b_n^{(p,q,r)}+C_s^{(p,q,r)}(t) \tag{3.6} \]

\[ (s=1,2,\ldots,n;\quad p+q+r=k;\quad k=1,2,\ldots,N). \]

The initial conditions for (3.6) are: \(b_s^{(p,q,r)}(t_0)=0\).

System (3.6) makes it possible, beginning with \(k=1\), to determine all \(b_s^{(p,q,r)}\), and thereby also \(v_s^{(k)}\). In the region of continuity of the coefficients \(C_s^{(p,q,r)}(t)\), system (3.6) admits a unique continuously differentiable solution [3]. We note that for \(k=1\), \(C_s^{(p,q,r)}(t)\) are defined and continuous for \(t_0\leq t\leq T\).

Thus, by the substitution (3.3), the system of order \((n+3)\) is reduced to the form (3.4), (3.5), and the expansion of the functions \(\Phi^*_{s0}\) in \(x, z, \bar z\) will begin with terms of order not lower than \((N+1)\).

1. Let, for the “shortened” system

\[ \frac{dx}{dt}=X(t,x,z,\bar z,v_1,\ldots,v_n), \]

\[ \frac{dz}{dt}=i\lambda z+Z(t,x,z,\bar z,v_1,\ldots,v_n), \]

obtained from (3.4) for \(\xi_1=\xi_2=\cdots=\xi_n=0\), the stability problem be solved by a finite number of terms, and let \(N\) be the highest order of the terms on which the solution of the problem depends. Assuming that the transformation (3.3) has been carried out, and after it the change of variables (2.3), we write the transformed system of equations (3.4), (3.5) in the form

\[ \frac{d\xi}{dt} = \sum_{n+2m=N} a_1^{(n,m)}(t)\xi^n\rho^{2m} +\varphi_1(t,\xi,\rho,\theta) + \sum_{s=1}^{n}\xi_s r_{s1}(t,\xi,\rho,\theta) + \sum_{l,m=1}^{n}\xi_l\xi_m R_{lm}^{(1)}(t,\xi,\rho,\theta,\xi_1,\ldots,\xi_n), \tag{4.1} \]

\[ \frac{d\rho}{dt} = \sum_{n+2m=N} a_2^{(n,m)}(t)\xi^n\rho^{2m+1} +\varphi_2(t,\xi,\rho,\theta) + \sum_{s=1}^{n}\xi_s r_{s2}(t,\xi,\rho,\theta) + \sum_{l,m=1}^{n}\xi_l\xi_m R_{lm}^{(2)}(t,\xi,\rho,\theta,\xi_1,\ldots,\xi_n), \]

\[ \frac{d\xi_s}{dt} = p_{s1}\xi_1+\cdots+p_{sn}\xi_n+\Phi^*_{s0}(t,\xi,\rho,\theta) + \Phi^*_s(t,\xi,\rho,\theta,\xi_1,\ldots,\xi_n). \tag{4.2} \]

Here

\[ r_{sj}(t,0,0,\theta)=0,\qquad R_{lm}^{(j)}(t,0,0,\theta,0,\ldots,0)=0 \quad (j=1,2), \]

\[ \Phi^*_{s0}(t,0,0,\theta)=0,\qquad \Phi^*_s(t,\xi,\rho,\theta,0,\ldots,0)=0, \]

and the expansions of \(\varphi_j,\ \Phi^*_{s0}\) in powers of \(\xi,\rho\) begin with terms of order not lower than \((N+1)\).

The functions \(r_{sj}\) in the general case have the form

\[ r_{sj}=\sum_{l=P}^{\infty} r_{sj}^{(l)}(t,\xi,\rho,\theta), \]

where \(r_{sj}^{(l)}\) are forms of the \(l\)-th order in the variables \(\xi,\rho\). If \(P\geq N\), then the system (4.1), (4.2) requires no further transformations and, as will be shown below, the stability problem is solved directly by considering it. If, however, \(P<N\), then, transforming (4.1) by the substitution [4]

\[ \eta=\xi+\sum_{s=1}^{n}\xi_s q_{s1}^{(l)}(t,\xi,\rho,\theta), \qquad \sigma=\rho+\sum_{s=1}^{n}\xi_s q_{s2}^{(l)}(t,\xi,\rho,\theta), \tag{4.3} \]

we obtain a system of the same form, but in which the functions \(r_{sj}^{(l)}\) have already been eliminated by the choice of the forms \(q_{sj}^{(l)}\) \((l=1,2,\ldots,N-1)\). We shall assume that, in the necessary case, the transformations (4.3) have been performed and that in (4.1) \(P \geqslant N\).

Passing to the new variables \(\xi=r\beta_1,\ \rho=r\beta_2,\ \beta_1^2+\beta_2^2=1\), we shall have

\[ \begin{aligned} \frac{dr}{dt}={}&r^N R(t,\beta_1,\beta_2)+O(r^{N+1}) +r^N\sum_{s=1}^n \xi_s f_s(t,r,\beta_1,\beta_2,\theta) \\ &+\sum_{l,m=1}^n \xi_l\xi_m f_{lm}(t,r,\beta_1,\beta_2,\theta,\xi_1,\ldots,\xi_n), \\[1.2em] \frac{d\beta_1}{dt}={}&r^{N-1}\beta_1\beta_2^2 P(t,\beta_1,\beta_2)+O(r^N) \\ &+r^{N-1}\sum_{s=1}^n \xi_s g_s(t,r,\beta_1,\beta_2,\theta) \\ &+\frac{1}{r}\sum_{l,m=1}^n \xi_l\xi_m g_{lm}(t,r,\beta_1,\beta_2,\theta,\xi_1,\ldots,\xi_n), \\[1.2em] \frac{d\xi_s}{dt}={}&p_{s1}^0\xi_1+\cdots+p_{sn}^0\xi_n+\sum_{k=1}^n \Delta p_{sk}(t)\xi_k \\ &+r^{N+1}F_{s0}(t,r,\beta_1,\beta_2,\theta) +F_s(t,r,\beta_1,\beta_2,\theta,\xi_1,\ldots,\xi_n). \tag{4.4} \end{aligned} \]

Here, for functions depending only on \(t,\beta_1,\beta_2\), the notation of Section 2 is adopted;
\(p_{sk}^0=p_{sk}(t_0)\), \(\Delta p_{sk}(t)=p_{sk}(t)-p_{sk}(t_0)\),
\(f_s(t,0,\beta_1,\beta_2,\theta)=g_s(t,0,\beta_1,\beta_2,\theta)=F_{s0}(t,0,\beta_1,\beta_2,\theta)=0\),
\(f_{lm}(t,0,\beta_1,\beta_2,\theta,0,\ldots,0)=g_{lm}(t,0,\beta_1,\beta_2,\theta,0,\ldots,0)=0\),
\(F_s(t,r,\beta_1,\beta_2,\theta,0,\ldots,0)\equiv0\).

Introduce the Lyapunov function

\[ V=\frac{1}{2}r^2\exp\bigl(-2Mu(\beta_1,\beta_2)\bigr)+W(\xi_1,\ldots,\xi_n), \tag{4.5} \]

where \(M\) is some positive number, \(u(\beta_1,\beta_2)\) is a certain bounded function; the positive-definite function \(W\) is such that

\[ \sum_{s=1}^n \frac{\partial W}{\partial \xi_s} \left(p_{s1}^0\xi_1+\cdots+p_{sn}^0\xi_n\right) =-\sum_{s=1}^n \xi_s^2. \tag{4.6} \]

The derivative of the Lyapunov function, by virtue of (4.4), (4.6), will have the form

\[ \begin{aligned} \frac{dV}{dt} &= r^{N+1}\left[ R(t,\beta_1,\beta_2)-M\beta_1\beta_2 P(t,\beta_1,\beta_2)\times \right.\\ &\qquad\left. {}\times \left(\beta_2\frac{\partial u}{\partial \beta_1} -\beta_1\frac{\partial u}{\partial \beta_2}\right)\right]\exp(-2Mu)+\\ &\quad + r^{N+1}\sum_{s=1}^{n}\xi_s f_s^{*}(t,r,\beta_1,\beta_2,\theta)+\\ &\quad + \sum_{l,m=1}^{n}\xi_l\xi_m f_{lm}^{*}(t,r,\beta_1,\beta_2,\theta,\xi_1,\ldots,\xi_n)-\\ &\quad - \sum_{s=1}^{n}\xi_s^2+\sum_{j,k=1}^{n}\Delta p_{jk}^{*}\xi_j\xi_k+O(r^{N+2}). \end{aligned} \tag{4.7} \]

Here \(\Delta p_{jk}^{*}=\sum_{s=1}^{n} b_{js}\Delta p_{sk}(t)\), where \(b_{js}\) are the constant coefficients of the form \(W\);

\[ f_{lm}^{*}(t,0,\beta_1,\beta_2,\theta,0,\ldots,0) \equiv f_s^{*}(t,0,\beta_1,\beta_2,\theta)=0. \]

For sufficiently small \(r\) and \(\xi_s\), the sign of the derivative \(\dfrac{dV}{dt}\) is determined by the principal terms

\[ \begin{aligned} &r^{N+1}\left[ R(t,\beta_1,\beta_2)-M\beta_1\beta_2 P(t,\beta_1,\beta_2) \left(\beta_2\frac{\partial u}{\partial \beta_1} -\beta_1\frac{\partial u}{\partial \beta_2}\right)\right]\times\\ &\qquad {}\times \exp(-2Mu)-\sum_{s=1}^{n}\xi_s^2+\sum_{j,k=1}^{n}\Delta p_{jk}\xi_j\xi_k, \end{aligned} \tag{4.8} \]

the remaining terms entering into (4.7) will be of higher order of smallness. At the initial instant of time all \(\Delta p_{jk}^{*}(t)=0\), and, as follows from (4.8), for stability at the instant \(t_0\) it is sufficient to require that

\[ R(t_0,\beta_1,\beta_2)-M\beta_1\beta_2 P(t_0,\beta_1,\beta_2) \left(\beta_2\frac{\partial u}{\partial \beta_1} -\beta_1\frac{\partial u}{\partial \beta_2}\right)<0. \tag{4.9} \]

The stability conditions following from (4.8) are written out in detail in Sec. 2. If (4.9) is satisfied, then, by continuity, there will always exist a time interval guaranteeing the stability of the unperturbed motion. This interval is found from the joint consideration of the inequalities

\[ \begin{aligned} &R(t_0,\beta_1,\beta_2)-M\beta_1^2\beta_2^2 P^2(t_0,\beta_1,\beta_2)+\\ &\quad +\tau\left[R_t'(t_0+\xi\tau,\beta_1,\beta_2) -M\beta_1^2\beta_2^2 P_t'(t_0+\eta\tau,\beta_1,\beta_2)\times \right.\\ &\qquad\left. {}\times P(t_0,\beta_1,\beta_2)\right]<0,\qquad 0<\xi,\eta<1, \end{aligned} \tag{4.10} \]

\[ -\sum_{s=1}^{n}\xi_s^2+\sum_{j,k=1}^{n}\Delta p_{jk}^{*}\xi_j\xi_k<0 . \tag{4.11} \]

If \(t_1, t_2\) are the instants of time at which inequalities (4.10), (4.11), respectively, cease to hold, then the smaller of \(t_1, t_2\) will determine the instant of time up to which the stability of the unperturbed motion can be guaranteed. A method for determining \(t_2\) is indicated in [4].

References

1. Malkin I. G. Theory of Stability of Motion. Moscow, GITTL, 1952.
2. Kamenkov G. V. PMM, vol. XVII, no. 5, 1953.
3. Petrovskii I. G. Lectures on the Theory of Ordinary Differential Equations. Moscow, GITTL, 1952.
4. Nasyrov R. M. Stability on a finite time interval in the critical case of a pair of purely imaginary roots. Izv. vuzov, Matematika, no. 5 (42), 1964.

Received by the editors
June 26, 1965

Kazan State University
named after V. I. Ulyanov-Lenin