MATHEMATICS

V. M. VOLOSOV

ON SOLUTIONS OF NONLINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS

(Presented by Academician I. G. Petrovskii on 17 I 1957)

§ 1. Statement of the problem. In the paper \((^4)\), nonlinear oscillations described by an equation of the form

\[ \frac{d}{dt}\,[m(\varepsilon t)\dot{x}] +\varepsilon f(\varepsilon t,x,\dot{x})+Q(\varepsilon t,x)=0, \tag{1} \]

were studied, where \(\varepsilon\) is a small parameter characterizing the slowness of variation of the functions \(m, f, Q\) with time \(t\). The function \(Q\) is interpreted as the principal force causing the oscillatory motion, while \(m>0\) and \(\varepsilon f\) are, respectively, the slowly varying mass and the small friction force. It was shown that the solution of (1) satisfying the initial conditions \(x(0)=x_0,\ \dot{x}(0)=\dot{x}_0\), oscillates on a large time interval \(t\sim 1/\varepsilon\) about the \(t\)-axis with slowly varying amplitude and period, if the condition \(\operatorname{sign} Q=\operatorname{sign} x\) and a number of other restrictions are satisfied. For the amplitude and period, formulas of the zeroth approximation were derived, giving for these quantities an error of order \(\varepsilon\) on the interval \(t\sim 1/\varepsilon\).

In a number of practical problems such accuracy is insufficient. In connection with this there arises the question of computing the next approximations of the amplitude and period, which give corrections of order \(\varepsilon\) to the zeroth approximation.

Consider a more general equation of the form

\[ \frac{d}{dt}\,[m(\varepsilon t)\dot{x}] +q(\varepsilon t,x) +\varepsilon\varphi_1(\varepsilon t,x,\dot{x}) +\varepsilon^2\varphi_2(\varepsilon t,x,\dot{x}) +\varepsilon^3\varphi_3(\varepsilon t,x,\dot{x})+\cdots=0. \tag{2} \]

Since in the present paper the amplitude and period are computed with accuracy up to quantities of order \(\varepsilon\) inclusive on the interval \(t\sim 1/\varepsilon\), terms of order \(\varepsilon^3\) and higher entering into (2), as can be shown, play no role in this approximation and may be discarded. Moreover,

\[ \frac{d}{dt}\,[m(\varepsilon t)\dot{x}]=m\ddot{x}+\varepsilon m_{\varepsilon t}\dot{x}, \]

and the term \(\varepsilon m_{\varepsilon t}\dot{x}\) may be included in the expression \(\varepsilon\varphi_1\) entering into (2). Therefore, dividing (2) by \(m\) and taking the above into account, we arrive at an equation of the form

\[ \ddot{x}+\varepsilon f_1(\varepsilon t,x,\dot{x}) +\varepsilon^2 f_2(\varepsilon t,x,\dot{x}) +Q(\varepsilon t,x)=0, \tag{3} \]

which we shall study in this paper.

For (3) it is shown here that, under the condition \(\operatorname{sign} Q=\operatorname{sign} x\) and certain other restrictions, which are stated below, the solution of (3) satisfying the initial conditions \(x(0)=x_0,\ \dot{x}(0)=\dot{x}_0\), where \(x_0^2+\dot{x}_0^2\ne0\), for sufficiently small \(\varepsilon\) oscillates with slowly varying amplitude and period on the interval \(t\sim 1/\varepsilon\), with positive maxima and negative minima alternating, and between them the solution changes monotonically. For the amplitude and period, formulas are derived which determine these quantities with accuracy up to order \(\varepsilon\) inclusive in the interval \(t\sim 1/\varepsilon\).

In the paper \((^4)\) and in the present paper, the method developed by the author in the paper \((^3)\) for equations with a small parameter multiplying the highest derivative is used. The possibility of applying this method to other types of equations was indicated by A. N. Tikhonov.

§ 2. Equations of the zeroth approximation for the amplitude and the period and their physical interpretation. The term \(\varepsilon^2 f_2\) of equation (3) plays no role in the zeroth approximation, and therefore the formulas of paper \((^4)\) for the amplitude and period of solutions of (1) are retained here. Let us write these formulas as applied to equation (3). The maxima and minima of the oscillating solution of (3) on the interval \(t\sim 1/\varepsilon\) are described, respectively, by the amplitude curves \(F_{10}(\varepsilon t)\) and \(F_{20}(\varepsilon t)\), with an error of order \(\varepsilon\). These curves, which are the zeroth approximation of the slowly varying amplitude of the solution of (3), are related by the relation

\[ \int_{F_{20}(\varepsilon t)}^{F_{10}(\varepsilon t)} Q(\varepsilon t,x)\,dx=0 \tag{4} \]

and satisfy the differential equation

\[ \frac{d}{dt} I\bigl[\varepsilon t,F_{10}(\varepsilon t),F_{20}(\varepsilon t)\bigr] = \varepsilon A\bigl[\varepsilon t,F_{10}(\varepsilon t),F_{20}(\varepsilon t)\bigr], \tag{5} \]

where

\[ I(\varepsilon t,F_{10},F_{20}) \equiv 2\sum_{k=1}^{2}(-1)^{k+1} \int_{0}^{F_{k0}} dx \left(2\int_{x}^{F_{k0}} Q(\varepsilon t,y)\,dy\right)^{1/2}, \tag{6} \]

\[ A(\varepsilon t,F_{10},F_{20}) \equiv \sum_{i,k=1}^{2}(-1)^{k+i} \int_{0}^{F_{k0}} f_1\left[\varepsilon t,x,(-1)^i \left(2\int_{x}^{F_{k0}} Q(\varepsilon t,y)\,dy\right)^{1/2}\right]\,dx, \]

and the initial values \(F_{10}(0)\), \(F_{20}(0)\) are determined by the formula

\[ 2\int_{x_0}^{F_{k0}(0)} Q(0,x)\,dx=\dot{x}_0^{\,2},\qquad \operatorname{sign} F_{k0}(0)=(-1)^{k+1},\qquad k=1,2. \tag{7} \]

In \((^4)\) it was indicated that (6) is an approximate expression for the action integral \(\oint \dot{x}\,dx\) over a period of oscillation. The expression \(\varepsilon A(\varepsilon t,F_{10},F_{20})\) is an approximate expression for the work of the perturbing force \(-\varepsilon f_1\) over a period of oscillation. Thus equation (5) describes the change of the action integral with time: the increment of the action integral is proportional to the work of the perturbing force. For \(f_1\equiv 0\), equation (5) is integrated in finite form, \(I=\mathrm{const}\), i.e. in this case the action integral is an adiabatic invariant over the time interval \(t\sim 1/\varepsilon\). For \(f_1\ne 0\) (presence of friction), the action integral over the time \(t\sim 1/\varepsilon\) may, according to (5), acquire a finite increment, i.e. in this case it is, generally speaking, not an adiabatic invariant. In a number of cases, however, the action integral is invariant also for \(f_1\ne 0\). In \((^4)\) a series of simplest cases of this kind was given. In addition to this, we point out here one more such case: equation (5) has the integral \(I=\mathrm{const}\) if the function \(Q\) is odd in \(x\), and \(f_1(\varepsilon t,x,\dot{x})\equiv f_1(\varepsilon t,-x,-\dot{x})\).

The period of oscillations (we define it as the time interval between two neighboring extrema of the same sign) in the zeroth approximation, i.e. with an error of order \(\varepsilon\) on the interval \(t\sim 1/\varepsilon\), is expressed by the formula

\[ T_0 = 2\sum_{k=1}^{2}(-1)^{k+1} \int_{0}^{F_{k0}(\varepsilon t)} dx \left( 2\int_{x}^{F_{k0}(\varepsilon t)} Q(\varepsilon t,y)\,dy \right)^{-1/2}, \tag{8} \]

where \(t\) is an arbitrary instant of time within the period under consideration.

§ 3. Equations of higher approximations. It has been shown that there exist functions \(F_{11}(\varepsilon t)\) and \(F_{21}(\varepsilon t)\) such that the curves \(F_1=F_{10}+\varepsilon F_{11}\) and \(F_2=F_{20}+\varepsilon F_{21}\) (\(F_{k0}\) are the zeroth approximations of the amplitude, determined ...

from (4), (5), (7)) describe, on the interval \(t\sim 1/\varepsilon\), the maxima and minima of the solution (3) with accuracy up to quantities of order \(\varepsilon\) inclusive. For the functions \(F_{11}, F_{21}\)—the first approximation of the amplitude (\(\varepsilon F_{k1}\) are corrections of order \(\varepsilon\) to the zeroth approximation \(F_{k0}\))—a system of linear differential equations of the form

\[ \frac{d}{dt}\{F_{k1}[Q\varepsilon t,F_{k0}(\varepsilon t)]\} = \varepsilon \left\{ \sum_{i=1}^{2} \left( \frac{\partial}{\partial F_{i0}}\, \psi[\varepsilon t,F_{j0}(\varepsilon t)] \right)_{(j=1,2)} F_{i1} + R_k[\varepsilon t,F_{j0}(\varepsilon t)] \right\}_{(k=1,2)}, \tag{9} \]

where

\[ \psi(\varepsilon t,E_{10},F_{20}) \equiv \frac{1}{T_0(\varepsilon t,F_{10},F_{20})} \times \]

\[ \times \left\{ 2\sum_{k=1}^{2}(-1)^{k+1} \int_{0}^{F_{k0}} Q'_{\varepsilon t}(\varepsilon t,x)\,dx \int_{x}^{F_{k0}} dy \left(2\int_{y}^{F_{k0}} Q(\varepsilon t,z)\,dz\right)^{-1/2} + A(\varepsilon t,F_{10},F_{20}) \right\}, \]

\[ R_k(\varepsilon t,F_{10},F_{20}) \equiv -\frac{1}{2}\frac{d}{d(\varepsilon t)} \sum_{i=1}^{2} \int_{0}^{F_{k0}(\varepsilon t)} f_1 \left[ \varepsilon t,x,(-1)^i \left( 2\int_{x}^{F_{k0}(\varepsilon t)} Q(\varepsilon t,y)\,dy \right)^{1/2} \right]dx + \frac{1}{T_0(\varepsilon t,F_{10},F_{20})} \sum_{i,j=1}^{2} \int_{0}^{F_{j0}} dx \times \]

\[ \times \left\{ (-1)^{i+j} f_2 \left[ \varepsilon t,x,(-1)^i \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dx \right)^{1/2} \right] + (-1)^{j+1}\psi(\varepsilon t,F_{10},F_{20}) \times \right. \]

\[ \times \int_{x}^{F_{j0}} dy \left( 2\int_{y}^{F_{j0}} Q(\varepsilon t,z)\,dz \right)^{-3/2} \int_{y}^{F_{j0}} f_1 \left[ \varepsilon t,z,(-1)^i \left( 2\int_{z}^{F_{j0}} Q(\varepsilon t,u)\,du \right)^{1/2} \right]dz - \]

\[ - (-1)^{i+j} f'_{1\dot x} \left[ \varepsilon t,x,(-1)^i \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dy \right)^{1/2} \right] \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dy \right)^{-1/2} \times \]

\[ \times \int_{x}^{F_{j0}} Q'_{\varepsilon t}(\varepsilon t,y)\,dy \int_{y}^{F_{j0}} dz \left( 2\int_{z}^{F_{j0}} Q(\varepsilon t,u)\,du \right)^{-1/2} + (-1)^j \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dy \right)^{-1/2} \times \]

\[ \times f'_{1x} \left[ \varepsilon t,x,(-1)^i \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dy \right)^{1/2} \right] \times \]

\[ \times \int_{x}^{F_{j0}} f_1 \left[ \varepsilon t,y,(-1)^i \left( 2\int_{y}^{F_{j0}} Q(\varepsilon t,z)\,dz \right)^{1/2} \right]dy + (-1)^j Q'_{\varepsilon t}(\varepsilon t,x) \int_{x}^{F_{j0}} dy \times \]

\[ \times \left( 2\int_{y}^{F_{j0}} Q(\varepsilon t,z)\,dz \right)^{-3/2} \int_{y}^{F_{j0}} f_1 \left[ \varepsilon t,z,(-1)^i \left( 2\int_{z}^{F_{j0}} Q(\varepsilon t,u)\,du \right)^{1/2} \right]dz + \]

\[ \left. + (-1)^{j+1} f'_{1\varepsilon t} \left[ \varepsilon t,x,(-1)^i \left( 2\int_{x}^{F_{j0}} Q(\varepsilon t,y)\,dy \right)^{1/2} \right] \int_{x}^{F_{j0}} dy \left( 2\int_{y}^{F_{j0}} Q(\varepsilon t,z)\,dz \right)^{-1/2} \right\}, \]

and the initial values \(F_{11}(0), F_{21}(0)\) are determined by the formula

\[ F_{k1}(0)Q[0,F_{k0}(0)] = -\operatorname{sign}\dot x_0 \left[ \frac{d}{d(\varepsilon t)} \int_{x_0}^{F_{k0}(\varepsilon t)} dx \left( 2\int_{x}^{F_{k0}(\varepsilon t)} Q(\varepsilon t,y)\,dy \right)^{1/2} \right]_{t=0} - \]

\[ - \int_{x_0}^{F_{k0}(0)} f_1 \left[ 0,x,\operatorname{sign}\dot x_0 \left( 2\int_{x}^{F_{k0}(0)} Q(0,y)\,dy \right)^{1/2} \right]dx . \tag{10} \]

We note that system (9) has the integral

\[ \sum_{k=1}^{2}(-1)^{k+1}F_{k1}Q[\varepsilon t,F_{k0}(\varepsilon t)]= \]

\[ =\frac{1}{2}\sum_{i,k=1}^{2}(-1)^k \int_{0}^{F_{k0}(\varepsilon t)} f_1\left[\varepsilon t,x,(-1)^i\left(2\int_x^{F_{k0}(\varepsilon t)}Q(\varepsilon t,y)\,dy\right)^{1/2}\right]dx. \tag{11} \]

With the aid of (11) one unknown function can be eliminated from (9), and therefore system (9) can always be integrated in finite form.

The method of paper (4) makes it possible to compute approximations of arbitrary order for the amplitudes of oscillations. The equations of higher approximations are also linear and have a structure analogous to (9). These equations are not given here because of their complexity.

With the aid of the functions \(F_{k1}(\varepsilon t)\) \((k=1,2)\), determined from (9), (10), the correction of order \(\varepsilon\) to the zeroth approximation of the period (8) is computed. Since the refined formula for the period of oscillations, taking into account terms of order \(\varepsilon\), was given in (4), we do not write it out here.

§ 4. Conditions for the applicability of the formulas of the zeroth and first approximations. Sufficient conditions under which the results of § 2, concerning the zeroth approximation, are valid coincide with the conditions of the theorems of paper (4), and we do not repeat them here. For the validity of the results of § 3, concerning the first approximation, the conditions of theorem (4) should be supplemented by the following requirements: 1) \(Q\) and \(f_1\) must have continuous derivatives up to order one higher than that indicated in (4); 2) \(f_2\) must be continuous together with its first-order partial derivatives.

§ 5. Summary. Oscillatory solutions of equation (3) have been studied, and formulas have been derived for the zeroth and first approximations of the amplitude and the period.

A particular case of (3) is the quasiharmonic equation, in which \(Q=k^2(\varepsilon t)x\). For such equations the results of §§ 2 and 3 coincide with the results of the method of N. M. Krylov—N. N. Bogoliubov (1).

Yu. A. Mitropolsky (2) extended the Krylov—Bogoliubov method to a broader class of nonlinear equations, which includes (3). In (2) the general solution of a certain unperturbed nonlinear equation of special form is assumed known in the form of an expansion in a trigonometric Fourier series, which is used for constructing asymptotic approximations to the solution of the perturbed equation.

In the present paper, for the basic characteristics of an oscillatory solution of equation (3)—the amplitude and the period—formulas are derived which express these quantities directly in terms of the functions \(Q\), \(f_1\), \(f_2\) entering into equation (3).

The author takes the opportunity to express gratitude to L. S. Solov'ev for interesting discussions.

Moscow State University
named after M. V. Lomonosov

Received
15 I 1957

REFERENCES

¹ N. N. Bogoliubov, Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, 1955.
² Yu. A. Mitropolsky, Nonstationary Processes in Nonlinear Oscillatory Systems, 1955.
³ V. M. Volosov, Matem. sbornik, 30(72), 2 (1952); 31(73), 3 (1952); 36(78), 3 (1955); DAN, 105, No. 3 (1955).
⁴ V. M. Volosov, DAN, 106, No. 1 (1956).