Preamble

DIFFERENTIAL EQUATIONS

APRIL 1967, VOLUME III, NO. 4

ON THE STABILITY OF SOLUTIONS TO A CLASS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

In this paper, we investigate the stability of the solutions to a second-order linear differential equation of the form:
$$ \ddot{x} + p(t)\dot{x} + q(t)x = 0 $$
where $p(t)$ and $q(t)$ are continuous functions defined for $t \geq t_0$. The problem of determining the asymptotic behavior of solutions for such equations is a fundamental task in the qualitative theory of differential equations, particularly when the coefficients are not constant.

1. Fundamental Stability Criteria

Consider the equation $\ddot{x} + p(t)\dot{x} + q(t)x = 0$. We assume that the coefficients $p(t)$ and $q(t)$ satisfy certain regularity conditions that ensure the existence and uniqueness of solutions on the interval $[t_0, \infty)$. To analyze the stability of the trivial solution $x(t) = 0$, we employ the Lyapunov function method and the comparison principle.

Suppose there exists a positive definite function $V(x, \dot{x}, t)$ such that its derivative along the trajectories of the system is non-positive. For the specific case of the equation above, we can define a candidate Lyapunov function related to the energy of the system:
$$ V(x, \dot{x}, t) = \frac{1}{2} \left( \dot{x}^2 + q(t)x^2 \right) $$
Taking the derivative of $V$ with respect to $t$ along the solutions, we obtain:
$$ \dot{V} = -p(t)\dot{x}^2 + \frac{1}{2}\dot{q}(t)x^2 $$
If $q(t) > 0$ and $\dot{q}(t) \leq 0$, while $p(t) \geq 0$, the stability of the system can be established under various conditions on the integrals of these coefficients.

[TABLE:1]

2. Asymptotic Behavior and Bounds

The behavior of the solutions as $t \to \infty$ depends significantly on the integrability of the functions $p(t)$ and $q(t)$. As noted in \cite{1}, if the integral $\int_{t_0}^{\infty} p(s) ds$ diverges, the damping term plays a dominant role in the evolution of the system. Conversely, if the coefficients are periodic or quasi-periodic, the stability regions can be characterized using Floquet theory or the method of averaging.

§ 1. Lyapunov was the first to rigorously formulate the problem of the stability of motion. He provided a definition of stability that has since become classical and developed two methods for its investigation. The first method involves finding solutions in the form of series, while the second method (the direct method) is based on the use of special functions, now called Lyapunov functions.

Consider a system of differential equations of the form:
$$ \dot{x} = P(t)x + f(t, x) \tag{1.1} $$
where $P(t)$ is a matrix of order $n$. Let us assume that $f(t, 0) = 0$, such that $x = 0$ is a solution to the system (1.1). According to Lyapunov, this solution is called stable if, for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|x(t)| < \epsilon$ for all $t \ge t_0$ whenever $|x(t_0)| < \delta$. If, in addition to this stability condition, it holds that:
$$ \lim_{t \to \infty} x(t) = 0 $$
then the solution $x = 0$ is called asymptotically stable. We shall refer to $x = 0$ as the equilibrium point (rest point). If the system has a solution $x = a$, where $a$ is a constant vector, this point can obviously be translated to the origin $x = 0$ through a change of variables. The problem is to find conditions under which $x = 0$ is stable, asymptotically stable, or unstable.

Lyapunov proposed two methods for solving this problem. Roughly speaking, the Second Method of Lyapunov consists of the following: if we can construct a continuous function $V(t, x) = V(t, x_1, \dots, x_n) > 0$ for $x \neq 0$ such that $V(t, x) \to 0$ as $x \to 0$ uniformly with respect to $t$, and for $|x| > \delta > 0$ we have $V(t, x) > 0$, then the solution $x = 0$ is stable provided the derivative along the trajectories $\dot{V} = \frac{\partial V}{\partial t} + \sum \frac{\partial V}{\partial x_i} \dot{x}_i \le 0$. If, by virtue of equations (1.1), we have $\dot{V} < 0$ and $V(t, x) \to 0$ as $x \to 0$, then the solution $x = 0$ is also asymptotically stable. For example, this occurs if $-\dot{V}(t, x) \ge W(x) > 0$ and $V(t, x) \to 0$ uniformly with respect to $t$ as $x \to 0$. While this approach resolves the question of stability, it does not provide explicit formulas for the solutions, and beyond the fact of stability, it yields little information about the behavior of solutions starting near $x = 0$. It is true, however, that the rate of decay of the function $V$ as $t \to \infty$ sometimes allows us to judge the rate at which $x(t) \to 0$. In this article, we will not focus on the second method, although we will occasionally note its significance in various cases.

What is the First Method of Lyapunov? This method is based on a specific constructive existence theorem for solutions of a system of differential equations:
$$ \dot{x}_s = p_{s1}(t)x_1 + \dots + p_{sn}(t)x_n + \sum X_s^{(m_1, \dots, m_n)}(t) x_1^{m_1} \dots x_n^{m_n} \tag{1.4} $$
where $m_1 + \dots + m_n \ge 2$.

Erugin notes that the coefficients are continuous and bounded functions in the domain $t > t_0$. Lyapunov makes the following assumption:
$$ |p_{sj}(t)| < M, \quad |X_s^{(m_1, \dots, m_n)}(t)| < A_{m_1 + \dots + m_n} \tag{1.5} $$
where the functions $X_s^{(m_1, \dots, m_n)}(t)$ have a finite supremum and a non-zero infimum on any finite interval $t_0 \le t \le T$; that is, $|X_s^{(m_1, \dots, m_n)}(t)| < M(T)$. Then the right-hand sides of equations (1.4) converge within the interval $t_0 \le t \le T$. A special case of such a system occurs when the coefficients can be chosen to be independent of $t$; for example, when the coefficients $P_{s}^{(m_1, \dots, m_n)}$ are constants. Lyapunov finds the solution to such a system in the form:
$$ x_s = \sum_{j=1}^{\infty} x_s^{(j)}(t, a_1, \dots, a_n) \tag{1.6} $$
where
$$ x_s^{(1)} = \sum_{l=1}^n a_l x_{sl}(t) \tag{1.7} $$
Here, $X(t)$ is the fundamental matrix of solutions for the linear system:
$$ \dot{x}_s = \sum_{l=1}^n p_{sl}(t)x_l \tag{1.8} $$
corresponding to the linear part of system (1.4). The terms $a_1, \dots, a_n$ are arbitrary constants, and $x_s^{(j)}(t, a_1, \dots, a_n)$ are homogeneous polynomials in $a_1, \dots, a_n$ of degree $j$, which serve as solutions to the corresponding non-homogeneous linear systems with initial conditions $x_i = 0$ at $t = t_0$.

Lyapunov proved that the series constructed in (1.6) converge absolutely and uniformly with respect to $t$ on the interval $t_0 < t < T$ and for $|x_i| < g$, where $g$ is determined via $A_k(T)$. Note also that if the matrix $A = I$, then we obtain the standard form. Lyapunov's theorem differs from the theorems of Picard and Cauchy in its assumptions regarding the right-hand sides of the equations (1.4) and in the construction of the solution itself. The series (1.6) represent the general solution of equations (1.4) in the neighborhood of the point $t = t_0$. Furthermore, Lyapunov shows how, based on this theorem, one can construct a solution to system (1.4) such that for certain systems of a fairly general type, it remains representable in this form. Through such a construction of solutions, we obtain important information regarding the qualitative behavior of the solutions in the neighborhood of a rest point.

This analytical construction of solutions for system (1.4) is based on certain properties of the solutions of the linear system (1.8), which Lyapunov calls the first approximation. To formulate Lyapunov's main theorem, we must present the profound classification of linear systems (1.8) introduced by Lyapunov. We shall call the real number $\lambda$ the characteristic exponent (c.e.) of a function $x(t)$ if $x(t)e^{-\lambda t} \to 0$ as $t \to \infty$, while for any $\alpha > 0$, the function $x(t)e^{-(\lambda-\alpha)t}$ is unbounded. Every continuous function has a c.e. If we have a system of functions $x_1, \dots, x_n$, the smallest c.e. among these functions is called the c.e. of the system. Lyapunov proves that all (non-zero) solutions of system (1.8) have finite c.e. In total, system (1.8) has no more than $n$ distinct c.e. If the coefficients of (1.8) are constant, then the c.e. of this system will be the real parts of the eigenvalues of the coefficient matrix $P = \|p_{ks}\|$.

It is always possible to construct a fundamental system of solutions such that the sum of their characteristic exponents $S = \lambda_1 + \dots + \lambda_n$ is minimized. Lyapunov proves that $S + \mu \ge 0$, where $\mu$ is the c.e. of the function $\exp(\int \sum p_{ii} dt)$. If $S + \mu = 0$, then system (1.8) is called regular. Lyapunov also introduced the consideration of reducible systems. Let $y_1, \dots, y_n$ be new unknown functions defined by the equalities:
$$x_i = \sum_{s=1}^{n} q_{is} y_s \quad (i = 1, \dots, n). \tag{1.9}$$
For $y = (y_1, \dots, y_n)$, we obtain the system $\dot{y} = yQ(t)$. If a transformation matrix $Q(t)$ can be specified such that it is bounded along with its inverse, and if the matrix in (1.10) is constant, then the system (1.8) is called reducible.

We shall now assume that for the entire interval $t > t_0$, system (1.8) is regular with Lyapunov characteristic exponents $\chi_1, \dots, \chi_n$. Lyapunov demonstrated that it is possible to determine functions $x_1, \dots, x_k$ formally satisfying the equations (1.4) in the form:
$$ x_s = \sum a_1^{m_1} \dots a_k^{m_k} L_s^{(m_1, \dots, m_k)}(t) e^{(m_1 \lambda_1 + \dots + m_k \lambda_k)t} \tag{1.12} $$
where $a_1, \dots, a_k$ are arbitrary constants, $m_1, \dots, m_k$ are non-negative integers, and the functions $L_j$ are independent of $a_1, \dots, a_k$, with their characteristic exponents being no greater than zero. Lyapunov proved that if the exponents $\lambda_1, \dots, \lambda_k$ in these series are negative and the absolute values of $a_1, \dots, a_k$ do not exceed a certain number $A > 0$, then these series converge absolutely and uniformly in the interval $t > t_0$. It follows from the form of these series that for sufficiently small $|a_1|, \dots, |a_k|$, we have $|x_k(t)| < \epsilon$, and $x_k(t) \to 0$ as $t \to \infty$.

If all $\lambda_j$ are negative, one can set $k = n$ in (1.12), in which case the solutions represent the general solution of the equations (1.4) in the neighborhood of the origin. Thus, if the system (1.8) is regular and all its characteristic exponents are negative, then the solution of system (1.4) is asymptotically stable. This was the first proof of asymptotic stability based solely on the properties of the linear system. If we only have $\lambda_\nu < 0$ for $\nu = 1, \dots, k$, then according to (1.12), we obtain a $k$-parameter family of integral curves that enter the equilibrium point as $t \to \infty$. Lyapunov termed the case where the zero solution is stable—provided that the initial values satisfy certain additional relationships—as the conditional stability of the zero solution.

We also note the results of other authors concerning the particular form of system (1.4) where all coefficients in the right-hand sides are independent of $t$:
$$ \dot{x}_s = p_{s1}x_1 + \dots + p_{sn}x_n + \sum X_s^{(m_1, \dots, m_n)} x_1^{m_1} \dots x_n^{m_n} \tag{1.13} $$
Poincaré demonstrated that system (1.13) can be transformed into the system $\dot{y}_k = \lambda_k y_k$ (1.14) via power series (1.15) that converge for sufficiently small values, provided that the real parts of all characteristic numbers of the matrix $\|p_{sj}\|$ are negative and distinct. Picard \cite{43} required only that $\lambda_1, \dots, \lambda_n$ lie on one side of a line passing through the origin and that the resonance condition $\lambda_i = \sum p_j \lambda_j$ does not hold. N. N. Krasovskii was the first to point out that asymptotic stability should be distinguished from cases where solutions simply approach the origin without stability.

H. Dulac showed that if $\lambda_1, \dots, \lambda_n$ lie on one side of a line passing through the origin, then there exists a holomorphic transformation such that system (1.13) is transformed into:
$$\dot{z}_i = \lambda_i z_i + \Phi_i(z_1, \dots, z_{i-1})$$
where $\Phi_i$ are polynomials. This is an analogue of Lyapunov's conditional stability. Subsequently, many researchers (Birkhoff, Siegel, Pliss \cite{36}) solved the problem of finding transformations that linearize the system. Lyapunov termed this approach—based on the analytical construction of integral curves—the "first method." He proved the stability of the zero solution not only when the first approximation (1.8) is regular, but in certain cases even when it is not, specifically when $S + \mu > 0$ and all c.e. $\chi_k < - \sigma$. He also demonstrated that if there is at least one $\chi_i > 0$, then unconditional stability does not exist.

The first method relies on the theory of characteristic exponents developed by Lyapunov and furthered by Bogdanov, Bylov, Vinograd, Grobman, Persidsky, and Chetaev \cite{35}. The first method not only solves the stability problem but also provides the equations of the integral curves, allowing for the study of how parameters affect the rate of approach to equilibrium. Lyapunov showed that at least an $m$-parameter family of integral curves enters the origin if the system has $m$ negative characteristic exponents.

Following Lyapunov, let us consider a system of nonlinear partial differential equations:
$$ \sum_{j=1}^k (p_{j1}x_1 + \dots + p_{jk}x_k) \frac{\partial z_s}{\partial x_j} = q_{s1}z_1 + \dots + q_{sm}z_m + Z_s \tag{2.1} $$
where $Z_s$ are holomorphic functions vanishing at the origin. Suppose that the characteristic numbers $\lambda_1, \dots, \lambda_k$ of the matrix $P$ and $\kappa_1, \dots, \kappa_m$ of the matrix $Q$ are not connected by relations of the form $\lambda_s = \sum m_i \lambda_i + \sum p_j \kappa_j$. Lyapunov proves the existence of a unique holomorphic solution in the form:
$$ x_s = \sum p_1 \dots p_k z_1^{p_1} \dots z_m^{p_m} \tag{2.2} $$
Lyapunov designated cases where the first approximation has zero characteristic numbers as "critical cases." In such instances, the behavior is determined by nonlinear terms.

Lyapunov examined the case where one characteristic root is zero while all others have negative real parts. Such a system can be rewritten as:
$$\begin{aligned} \frac{dx}{dt} &= X \\ \frac{dx_s}{dt} &= p_{s1}x_1 + \dots + p_{sn}x_n + X_s \quad (s = 1, \dots, n) \end{aligned} \tag{2.3}$$
where $X$ and $X_s$ are holomorphic functions starting with terms of at least second order. From the equations:
$$ p_{s1}x_1 + \dots + p_{sn}x_n + X_s = 0 \tag{2.4} $$
we determine $x_s = \phi_s(x)$ as holomorphic functions. Substituting these into $X$, we obtain a function $f(x) = g_m x^m + \dots$. Lyapunov concluded: if $m$ is even, the zero solution is unstable; if $m$ is odd, it is unstable for $g > 0$ and asymptotically stable for $g < 0$. If $g(x) = 0$, Lyapunov finds an integral:
$$ \Phi(x_1, \dots, x_n) = c \tag{2.5} $$
In this case, there exists a curve of equilibrium points, and every solution starting near the origin remains on an integral surface and approaches an equilibrium point on that surface.

Lyapunov also investigated systems with two purely imaginary eigenvalues:
$$ \begin{aligned} \frac{dx}{dt} &= -\lambda y + X \\ \frac{dy}{dt} &= \lambda x + Y \\ \frac{dx_s}{dt} &= p_{s1}x_1 + \dots + p_{sn}x_n + X_s \end{aligned} \tag{2.6} $$
He proved the existence of a family of periodic solutions:
$$ x = \sum x^{(j)}c^j, \quad y = \sum y^{(j)}c^j, \quad x_s = \sum x_s^{(j)}c^j \tag{2.7} $$
In this case, there exists an integral $x^2 + y^2 + F = c$ (2.8). Every solution asymptotically approaches a corresponding periodic solution, proving non-asymptotic stability.

§ 3. Construction of General Solutions

Consider the system (3.1) with one zero eigenvalue. As Lyapunov demonstrated, there exists an integral $V(x_1, \dots, x_n) = c$ (3.2). By substituting the variable according to (3.2), the equations can be written as:
$$ \frac{dx_s}{dt} = \sum P_{sj}(c)x_j + \dots \tag{3.4} $$
For sufficiently small $c$, the eigenvalues of the matrix (3.5) have negative real parts. Using the first method, we obtain the general solution $x_s = \sum A_{sj} e^{-\lambda_j t} \dots$.

For the system (3.7) with two purely imaginary roots, Lyapunov uses variables $r$ and $\theta$ (3.9). If a family of periodic solutions (3.12) exists, there is an integral surface $H = F(x, y, \dots)$ (3.13). Introducing new variables (3.14) leads to a system (3.17) where the first approximation has periodic coefficients. The fundamental solution matrix is $Z = e^{Wt}U(c, t)$ (3.19). The real parts of the characteristic exponents $\chi_i(c)$ are negative. Using the first method, we obtain the general solution $x_s = \sum a_i e^{\chi_i(c)t} \phi_i(t, c)$.

Whenever Lyapunov established asymptotic stability based on the first approximation, he obtained the general solution. If he used the second method for "doubtful cases," the general solution remained unconstructed. If the zero solution was non-asymptotically stable, Lyapunov identified an integral manifold $M$ that is itself asymptotically stable.

§ 4. Second-Order Systems and Singular Cases

Lyapunov considered a second-order system:
$$\begin{aligned} \dot{x} &= y + X(x, y) \\ \dot{y} &= Y(x, y) \end{aligned} \tag{4.1}$$
This critical case with two zero roots has diverse topological structures. Lyapunov identified 10 different cases. In the case of a "non-classical center," he constructed the general solution using masterful transformations. In 1954, V. I. Smirnov discovered a manuscript where Lyapunov considered a system of $n + 2$ equations with two zero roots. This was published in 1963. In this difficult study, Lyapunov constructed solutions for systems of the Briot-Bouquet type.

For systems with periodic coefficients, Lyapunov examined cases with one or two zero eigenvalues. In the first case, a family of periodic solutions (5.4) exists if an integral $x + F = c$ (5.5) exists. By introducing new variables (5.6), the system is transformed into (5.10), where the matrix has negative characteristic exponents, allowing for the construction of the general solution.

In the case of two zero eigenvalues with periodic coefficients (5.11), the problem remains partially unresolved. Using $x = r \cos \theta, y = r \sin \theta$ (5.12), Lyapunov seeks a formal solution (5.15). If the series converge, the zero solution is non-asymptotically stable. However, the convergence of these series remains a difficult question. For canonical systems (5.16), formal solutions always exist, but their convergence is not guaranteed.

§ 6. Canonical Systems and Integral Manifolds

Consider the canonical system (6.1) with integral $H_0 = c^2$ (6.2). Here, the series (5.22) converge, and the zero solution is non-asymptotically stable. For the system (6.4), the equilibrium point is non-asymptotically stable regardless of $\lambda$. In the transcendental case, an integral manifold appears in the neighborhood of the origin, and the stability is determined by the behavior near this manifold.

For the system (6.17) where the matrix has two purely imaginary eigenvalues, Lyapunov constructs periodic solutions (6.19). In the case of a focus, the integral curves wind as spirals. A. N. Erugin \cite{14} constructed the equations of these spirals in the form $r = [C + (m-1)g\theta]^{-1/(m-1)}$.

In canonical systems (6.20), stability is only possible if all eigenvalues are purely imaginary. If the elementary divisors are simple, the system can be mapped to (6.22). If all $\lambda_k$ have the same sign, the origin is stable. Kolmogorov \cite{17} provided a method to prove the existence of analytic contact transformations (6.26) that preserve stability under small perturbations.

The Krylov-Bogolyubov method allows for the construction of solutions to systems like (7.11) in the form $x = a \cos(\psi + \tau) + \dots$ (7.13). This method effectively isolates the principal part of the solution and accounts for non-stationary oscillations. By comparing coefficients of $\epsilon^k$, one obtains equations for $x_m$ (7.16-7.20). The choice of arbitrary functions in the solution allows for either convergence in a domain or better asymptotic representation. This method represents a significant development of the "first method" and has gained wide distribution in nonlinear mechanics.

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Institute of Mathematics
Received January 10, 1967