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UDC 517.934
RESONANCE PHENOMENA IN NONLINEAR SYSTEMS
A. S. BAKAI
INTRODUCTION
The study of the behavior of nonlinear systems under the influence of external forces is of great interest both from a theoretical and from an applied standpoint. Substantial progress in this question has been achieved by means of asymptotic and qualitative methods [1–3], which in many cases make it possible to find asymptotically stable solutions. The latter problem is often reduced to determining the position and the character of singular points in phase space. The character of a singular point is determined by the behavior of phase trajectories in its neighborhood, so that, knowing the position and character of the singular points, one may establish a qualitative picture in phase space in the neighborhoods of the singular points. In this case the behavior of phase trajectories in the remaining part of phase space remains unclear, and for a quantitative description of the motion one has to use numerical methods. All this makes it desirable to investigate systems admitting a detailed and simple description and making it possible to study effects caused by nonlinearity.
As such a system in the present work we choose an oscillator whose frequency \(\omega\) depends on its energy \(E\). The oscillator is subjected to a sinusoidal external force with slowly varying frequency \(\nu\) and amplitude \(f\). The amplitude \(f\) of the forcing is such that
\[ \Delta \omega / \omega \ll 1, \tag{0.1} \]
where \(\Delta \omega\) is the change of the frequency \(\omega\) during one period of oscillation. Such an oscillator is remarkable in that, in the absence of an external force and force of friction, it performs harmonic oscillations regardless of the magnitude of the amplitude. Owing to this, the assumption of smallness of the nonlinear terms becomes unnecessary. The absence of multiple harmonics also substantially simplifies the consideration. At the same time, the system under study makes it possible to reveal effects caused by nonlinearity.
If at the initial instant the oscillator frequency \(\omega\) and the frequency of the forcing \(\nu\) are far apart \(|\nu-\omega| \sim \nu+\omega\), then, according to (0.1), there is a small parameter
\[ \frac{\omega'(E) f^{2}\nu^{2}} {\omega(E)\left[\omega^{2}(E)-\nu^{2}\right]^{2}}, \]
which permits the use of approximate methods. If \(\nu\) and \(\omega\) are close (the system is in resonance), but the change in the oscillator energy during one period of oscillation is small, one may apply the procedure of averaging the equations over one period. As a result of averaging, a system of two nonlinear first-order equations is obtained, the investigation of which presents no great difficulty. The principal attention in the present work is given to the study of the resonance case.
In the first section the basic equations are presented, as well as the equations obtained as a result of averaging. At the same time, a criterion is established for the validity of the averaging procedure and the range of applicability of the equations obtained. As was to be expected, the averaged equations coincide in form with the first approximation of the asymptotic expansion by the Krylov–Bogolyubov method for nonlinear systems under the influence of an external force [1, 3]. Therefore the results of the present work are valid for systems well described by the first approximation according to the Krylov–Bogolyubov method [1] and by its generalization to the case of systems with varying parameters, carried out by Yu. A. Mitropolskii [3].
The second section is devoted to consideration of the behavior of the aforementioned oscillator in the absence of friction and for constant parameters of the forcing. The equations of motion are written in Hamiltonian form and can be integrated. The picture of phase trajectories on the phase surface is established, and some characteristics of the motion are found. It turns out that the energy, and together with it the frequency, periodically cha-
change with time; moreover, in the resonance region the amplitude of the frequency variation is proportional to \((f/2\nu)^{1/2}\), and the period of the frequency variation is proportional to \((2\nu/f)^{1/2}\).
In the third section the oscillator is considered in the presence of friction, as well as the case when the parameters of the driving force vary with time.
If friction is present and the parameters of the driving force are constant, then it is easy to establish the asymptotically stable states of the system. If there are several such states, then it is not always clear into which of them the system will pass for given initial conditions. Because the initial conditions that bring the system to different asymptotically stable states are densely intermingled, it is expedient to restrict oneself to computing the measure of the initial conditions leading to one or another stable state. The relative magnitudes of these measures are naturally interpreted as the probabilities of transition of the system to the corresponding asymptotically stable states.
The behavior of the oscillator under slowly varying parameters of the driving force is considered by means of the method of adiabatic invariants. It is possible to find the limiting states of the system and to determine which of them are stable. The probabilities are also found that the system tends to a given limiting state for selected initial conditions.
It turns out that an adiabatic change of \(\nu\) can lead to an adiabatic change in the oscillator frequency. In this case the energy changes in such a way that the resonance condition is satisfied:
\[ |\omega-\nu|\sim (f/2\nu)^{1/2}. \]
§ 1. BASIC EQUATIONS
The equation under investigation has the form
\[ \ddot{x}+2\gamma(E)\dot{x}+\omega^2(E)x=f\cos \nu t, \tag{1} \]
where
\[ E=\frac{1}{2}\left[\dot{x}^{\,2}+\omega^2(E)x^2\right], \tag{2} \]
and \(\gamma(E)\) and \(\omega(E)\) are prescribed functions of the energy. We shall seek the solution of equation (1) in the form
\[ x=a\cos(\nu t+\theta), \tag{3} \]
where the amplitude \(a\) and the phase \(\theta\) are connected by the relation
\[ \dot{a}\cos\psi=a\dot{\theta}\sin\psi,\qquad \psi=\nu t+\theta. \tag{4} \]
Then \(\dot{x}=-a\nu\sin\psi\). Substituting (3) into (1) and (2), and taking (4) into account, we obtain
\[ \dot{a}\nu=(\omega^2-\nu^2)a\cos\psi\sin\psi -2\gamma\nu a\sin^2\psi -f\cos\nu t\sin\psi, \tag{5} \]
\[ a\nu\dot{\theta}=(\omega^2-\nu^2)a\cos^2\psi -2\gamma\nu a\sin\psi\cos\psi -f\cos\nu t\cos\psi, \]
\[ E=\frac{1}{2}\left[a^2\nu^2\sin^2\psi+a^2\omega^2\cos^2\psi\right]. \tag{6} \]
If the amplitude \(a\) and the phase \(\theta\) change little over the period \(T_\nu=\dfrac{2\pi}{\nu}\), equations (5), (6) may be averaged over the time interval \(T_\nu\). As a result of averaging we arrive at the following system of equations:
\[ \dot{a}=-\gamma a-\varepsilon\sin\theta, \]
\[ \dot{\theta}=\frac{1}{2\nu}(\omega^2-\nu^2)-\frac{\varepsilon}{a}\cos\theta, \tag{7} \]
\[ E=\frac{1}{4}a^2(\omega^2+\nu^2). \tag{8} \]
From equations (7) it is seen that the assumption of slow variation of the amplitude is satisfied under the condition
\[ \gamma a,\ \varepsilon \ll \nu a . \tag{9} \]
A slow change of the phase occurs in those regions of the phase surface \((a,\theta)\) where the inequality
\[ \left|\nu-\left(\omega^2+f\cos\theta/a\right)^{1/2}\right|\ll \nu . \tag{10} \]
is satisfied. Thus, equations (6) are suitable for describing the system in the regions of the phase surface where conditions (9) and (10) are fulfilled.
As follows from (8), the energy \(E\), and together with it the frequency \(\omega\) and the friction force \(\gamma a\), depend only on the amplitude \(a\). We shall first consider the investigation of the system without friction \((\gamma=0)\), and then determine what the inclusion of friction leads to.
§ 2. STATIONARY REGIME
In the absence of friction the system of equations (7) takes the form
\[ \dot a=-\varepsilon\sin\theta,\quad \dot\theta=\frac{1}{2\nu}\left(\omega^2-\nu^2\right)-\frac{\varepsilon}{a}\cos\theta . \tag{11} \]
For the investigation of these equations it is convenient to apply the method of the phase surface. Since two points on this surface whose phases \(\theta\) differ by \(2\pi\) describe one and the same state of the system, it is convenient to take a cylindrical surface as the phase surface.
1. Integral curves. The system of equations (11) can be represented in Hamiltonian form by introducing a new variable \(u=a^2\). The Hamiltonian function then has the form
\[ H(u,\theta)=\frac{1}{2\nu}\int^u\left[\omega^2(x)-\nu^2\right]dx-2\varepsilon\sqrt{u}\cos\theta . \tag{12} \]
The motion of the phase point takes place on the isoenergetic surface
\[ H(u,\theta)=\mathrm{const}, \tag{13} \]
which in the one-dimensional case leads to the equations for the integral curves
\[ \frac{1}{\nu}\int^a\left[\omega^2(x)-\nu^2\right]x\,dx-2\varepsilon a\cos\theta=\mathrm{const}. \tag{14} \]
From (14) it is seen that the integral curves are symmetric with respect to the generators \(\theta=0,\ 2\pi\), and \(a\) is a bounded function if
\[ \lim_{a\to\infty}\int^a\left[\omega^2(x)-\nu^2\right]x\,dx\ne 0 . \]
Fulfillment of this condition means that at large amplitudes the system must not become linear. Since the integral curves (with the exception of those having singular points as their limiting points) do not intersect, they are all closed curves. This means that the system periodically returns to its initial state, i.e. is in a certain stationary regime of motion.
The change of phase during one complete circuit of the phase trajectory may be equal to 0 or to \(2\pi\). In the first case the phase \(\theta\) oscillates; in the second it increases or decreases without bound. The region of the phase surface enclosed within an integral curve of the first type contains a singular point, evidently a center. Regions of the phase space filled with integral curves of the first and second types are separated by the so-called separatrices. A separatrix passes through a saddle singular point. Regions filled with curves of the first type will be called regions of phase oscillation and denoted by the letter \(O\). Sometimes the family of curves of the first type is denoted by the same letter.
2. Singular points. The singular points of the second of equations (11) are the points with coordinates
\[ a=0,\quad \theta=\frac{\pi}{2},\quad \frac{3\pi}{2} \tag{15} \]
The remaining singular points on the phase surface are determined from the conditions
\[ \dot a=\dot\theta=0. \tag{16} \]
When applied to system (11), conditions (14) lead to the following equations:
\[ \varepsilon\sin\theta=0,\quad \varepsilon\cos\theta=-\frac{a}{2\nu}\left(\omega^2-\nu^2\right), \]
i.e. \(\theta=0,\ \pi\), and consequently
\[ \frac{a}{2\nu}\left(\omega^2-\nu^2\right)=\pm\varepsilon, \tag{17} \]
so that the roots of equation (17) with the upper sign lie on the generator \(\theta=0\), and the roots of equation (17) with the lower sign lie on the generator \(\theta=\pi\). Relation (17) defines the so-called resonance curves in the plane \((a,\nu)\) (Fig. 1):
\[ \nu=\nu_{\pm}(a);\quad \nu_{\pm}(a)=\left(\omega^2\pm f/a\right)^{1/2}, \tag{18} \]
with the aid of which it is easy to find the coordinates of the singular points for fixed \(\nu\). The coordinate \(a\) of a singular point is a root of equation (18). The phase \(\theta\) of a singular point is equal to \(\pi\) if the singular point belongs to \(\nu_{+}(a)\), and to 0 if the singular point belongs to \(\nu_{-}(a)\). Applying the usual methods (see, e.g., [1, 2]), it is not difficult to establish the character of the singular points. The singular points determined by the curve \(\nu_{+}(a)\) are centers when \(\nu_{+}'(a)<0\), saddle points when \(\nu_{+}'(a)>0\), and compound points of center-saddle type at extremal points. The singular points determined by the curve \(\nu_{-}(a)\) are centers when \(\nu_{-}'(a)>0\), saddle points when \(\nu_{-}'(a)<0\), and compound singular points of center-saddle type at extremal points.
We shall denote centers by the letter \(P\), saddles by the letter \(S\), and their coordinates by
\[ a(P),\quad \theta(P);\quad a(S),\quad \theta(S). \]
Let us give several consequences following from equations (18) and the subsequent classification of singular points. As \(a\) increases, both resonance curves approach one another, tending asymptotically to the backbone curve defined by the equation
\[ \nu=\omega(a), \tag{19} \]
so that \(\nu_{+}(a)\) always lies above, and \(\nu_{-}(a)\) below, the skeleton curve (see Fig. 1).
The first extremum of \(\nu_{+}(a)\), which appears as \(a\) increases, is a minimum, while for the curve \(\nu_{-}(a)\) it is a maximum. For any value of the frequency \(\nu\) there is at least one simple singular point, which is a center; moreover, the number of centers exceeds the number of saddles by one.
Every multiple singular point under a virtual change either disappears or splits into two simple singular points—a center and a saddle. Through each saddle point, and through the points (15), there passes one separatrix each. In contrast to saddle points, the points (15) are not stationary. It is convenient to renumber the saddles in the order of increasing coordinates \(S_1, S_2, \ldots, S_n\). By \(S_0\) we shall denote the points (15). The regions of phase oscillation and the centers contained in them will be marked by the number of the saddle point through which the separatrix bounding the given region \(O\) passes.
Fig. 1
3. Separatrices. Separatrices bound the regions of phase oscillation \(O_i\) and pass through saddle points. Since the position of the saddle points can be established as a function of the frequency \(\nu\), it is thereby possible to trace the change of the regions of phase oscillation as a function of \(\nu\).
The equation for the separatrix passing through the saddle point \(S_i\) is, evidently, determined by the equation
\[ \frac{1}{\nu}\int^a \left|\omega^2(x)-\nu\right|\,x\,dx - 2\varepsilon a\cos\theta = C(S_i), \tag{20} \]
where
\[ C(S_i)=\frac{1}{\nu}\int^{a(S_i)} \left|\omega^2(x)-\nu^2\right|\,x\,dx -2\varepsilon a(S_i)\cos\theta(S_i). \]
The constants \(C(S_i)\) depend on the parameter \(\nu\), as do the coordinates of the saddle point, and for certain values of \(\nu\) \((\nu=\nu_{\mathrm{cr}})\) it may turn out that two or more constants coincide. This means that the corresponding separatrices have common branches; however, under an arbitrarily small change of \(\nu\) they separate.
Figure 2 shows the regions of phase oscillation for \(\nu\ne\nu_{\mathrm{cr}}\) (Fig. 2, a) and \(\nu=\nu_{\mathrm{cr}}\) (Fig. 2, b).
4. Some characteristics of the motion. In this section we shall give some characteristics of the motion of the system under investigation: the period, and the magnitudes of the change of amplitude and phase along the phase trajectory. By the period \(T(C)\) we shall mean the interval of time after which the system returns to its initial state. Obviously,
\[ T(C)=\oint_{L(C)} \frac{d\theta}{\dot{\theta}}, \tag{21} \]
where the integration is carried out along the integral curve \(L(C)\). The magnitude of the period varies from trajectory to trajectory, but since we con-
consider regions close to singular points, then the magnitude of the period can be estimated, in order of magnitude, in the following way. We linearize equations (11) in a neighborhood of the center \(P_i\):
\[ \dot a=-\varepsilon\theta,\qquad \dot\theta=-\alpha_i[a-a(P_i)],\qquad (\theta(P_i)=0), \]
where
\[ \alpha_i=\frac{\omega_i'\omega}{\nu}+\frac{\varepsilon}{a^2},\qquad \omega_i=\omega(a(P_i)),\qquad \omega_i'=\omega'(a(P_i)). \tag{22} \]
Fig. 2
Or, instead of (22),
\[ \ddot\xi+\varepsilon\alpha_i\xi=0,\qquad \xi=a-a(P_i). \]
Thus the frequency of oscillation of the amplitude in the limit \(C\to C(P_i)\) is equal to \(\lim \Omega(C)=(\varepsilon\alpha_i)^{1/2}\), and consequently for the period we have
\[ T(C(P_i))=\lim_{C\to C(P_i)}T(C)=\frac{2\pi}{(\varepsilon\alpha_i)^{1/2}}. \tag{23} \]
If the trajectory is not very close to the center, then the integral (21) must be evaluated exactly, which may prove difficult. However, in a number of practically important cases one can obtain simple approximate expressions. For example, in the region \(a\gg\varepsilon\), the equations of motion (11), with accuracy up to terms of order \(O(\varepsilon/a)\), have the form
\[ \dot\xi=-\varepsilon\sin\theta,\qquad \dot\theta=\alpha_i\xi, \]
and the integral curves are determined by the equation
\[ \frac{\alpha_i}{2\varepsilon}\xi^2-\cos\theta=C_1. \tag{24} \]
Therefore from (21), (23), (24) we have
\[ T(C)=\oint_{L(C)}[2\varepsilon\alpha_i(C_1+\cos\theta)]^{-1/2}\,d\theta= \]
\[ = \begin{cases} 2(\varepsilon\alpha_i)^{-1/2}K\!\left(\sqrt{\dfrac{1+C_1}{2}}\right), & (1>C_1>-1),\\[1.2em] 2\left[\varepsilon\alpha_i(1+C_1)/2\right]^{-1/2}K\!\left(\sqrt{\dfrac{2}{1+C_1}}\right), & (C_1>1), \end{cases} \qquad (a_i>0), \tag{25} \]
where \(K(\varphi)\) is the complete elliptic integral of the first kind. The value \(C_1=-1\) corresponds to the center, and the value \(C_1=1\) to the separatrix. As \(C_1\to -1\), (25) coincides with the expression (23) obtained earlier. As \(C_1\to 1\), \(T(C_1)\) tends logarithmically to infinity. This shows that the phase point, moving along the separatrix, asymptotically approaches the saddle point.
The range of variation of the phase \(\theta\) along the phase trajectory is bounded by the turning points \(\theta_1(C)\) and \(\theta_2(C)\), which are determined from the condition
\[
\frac{\partial}{\partial a}\,\theta(a,C)=0.
\]
In doing so, one must choose that interval \(\Delta\theta(C)\), bounded by turning points, which contains \(\theta(P)\). Turning points exist only for trajectories belonging to some family \(O_i\). If the phase trajectory does not belong to any family \(O_i\), the phase \(\theta\) does not change without bound along it.
We define the magnitude of the change in amplitude along an integral curve as follows:
\[
\Delta a(C)=\max a(C,\theta)-\min a(C,\theta).
\]
It is not difficult to verify that
\[
\Delta a(C)=a_+(C,\theta(P))-a_-(C,\theta(P)),
\tag{26}
\]
if the curve belongs to the region \(O\). Here \(a_+\) and \(a_-\) denote the branches of the integral curve lying above and below the curve \(\dot\theta=0\). If the phase trajectory does not belong to any family \(O_i\), then
\[
\Delta a(C)=a_+(C,\theta(P))-a_+(C,\theta(S))
\tag{27}
\]
or
\[
\Delta a(C)=a_-(C,\theta(S))-a_-(C,\theta(P)).
\]
In particular, for the integral curves (24) we have
\[
\Delta a(C_1)=2\left|\,2\varepsilon(C_1+1)/\alpha_i\,\right|^{1/2}\quad (C_1<1),
\tag{28}
\]
\[
\Delta a(C_1)=(2\varepsilon/\alpha_i)^{1/2}
\left|(C_1+1)^{1/2}-(C_1-1)^{1/2}\right|\quad (C_1>1).
\]
The range of variation of the phase in this case is determined by the inequality \(C_1>-\cos\theta\). It follows from (28) that the change in amplitude increases with increasing \(C_1\), reaches a maximum as \(C_1\) approaches 1 from the left, and then decreases discontinuously by a factor of two, which corresponds to a jump-like change in the regime of motion of the system upon crossing the boundary of the region \(O\).
§ 3. NONSTATIONARY REGIME
As was shown, for constant parameters in the absence of friction all phase trajectories on the phase surface are closed, so that to each initial state of the system there corresponds some stationary regime. In the presence of friction or of parameters \(\lambda(t)\) varying in time, the phase trajectories cease to be closed. However, if the values of the system parameters tend to certain limiting values as \(t\to\infty\), \(\lim_{t\to\infty}\lambda(t)=\lambda_+\), then, irrespective of the presence of friction, each phase trajectory tends to a closed trajectory or to a fixed point. For this reason, in the study—
in nonlinear systems it is important to establish the position of these limiting trajectories and points on the phase surface and to determine to which of them the phase trajectory tends for given initial conditions. The present paragraph is devoted to the clarification of these questions as applied to the system under investigation.
1. A dissipative system with constant parameters. In this subsection we shall consider the motion of system (1) in the presence of friction and with fixed parameters. The equations of motion have the form
\[ \dot a=-\gamma a-\varepsilon\sin\theta,\qquad \dot\theta=\frac{1}{2\nu}(\omega^2-\nu^2)-\frac{\varepsilon}{a}\cos\theta. \tag{29} \]
The determination of the integral curves in general form is difficult, but the reconstruction of the qualitative picture on the phase surface presents no great difficulties.
Equations (16) now determine a resonance curve having two branches
\[ \nu=\nu_{\pm}(a), \tag{30} \]
where equations (16) have real solutions if the condition \(|\gamma a|<\varepsilon\) is satisfied. The explicit form of equations (30) is easy to obtain (see, for example, [1]); \(\nu_{+}(a)\) lies above, and \(\nu_{-}(a)\) below, the skeleton curve. The coordinate \(\theta\) of the singular point is now not constant along the curves (30). If the singular point is determined by the curve \(\nu_{+}(a)\), the phase changes while remaining in the interval \(\left[\frac{\pi}{2},\,\frac{3\pi}{2}\right]\). If the singular point is determined by the curve \(\nu_{-}(a)\), the phase changes while remaining in the interval \(\left[-\frac{\pi}{2},\,\frac{\pi}{2}\right]\). The branches \(\nu_{+}(a)\) and \(\nu_{-}(a)\) merge on the skeleton curve at \(a=a_k\), satisfying the equation \(\gamma a=\varepsilon\). In the presence of friction, the character of the singular points changes. The singular points determined by \(\nu_{+}(a)\) \([\nu_{-}(a)]\) are foci for \(\nu'_{+}(a)<0\) \([\nu'_{-}(a)>0]\), saddles for \(\nu'_{+}(a)>0\) \([\nu'_{-}(a)<0]\), and complex singular points at the extremal points of \(\nu_{+}(a)\) and \(\nu_{-}(a)\). The foci are stable (unstable) for
\[ \frac{d}{da}(\gamma a)>0 \]
\[ \left[\frac{d}{da}(\gamma a)<0\right]. \]
We shall denote a focus by the letter \(\Phi\), and its coordinates by \(a(\Phi)\) and \(\theta(\Phi)\).
Each phase trajectory approaches one of the stable foci or stable limit cycles. Phase trajectories for which the saddle point \(S_i\) is a limiting singular point separate the set of trajectories \(O_i\), which asymptotically tend to the focus \(\Phi_i\) or to the limiting cycle surrounding this focus (as \(t\to\infty\) to the stable one, as \(t\to-\infty\) to the unstable one).
The appearance of two or more stable limiting points on the phase surface and the hysteresis phenomena associated with this in nonlinear systems is a thoroughly studied phenomenon (see, for example, [1]). Of special interest is the question of which stable limiting regime the system will enter for given initial conditions.
Let the state of the system at the initial instant be characterized by the point \(y_0(a_0,\theta_0)\), where
\[ a(\Phi_{i+1})-2(\varepsilon/a_{i+1})^{1/2}>a_0>a(\Phi_i)+2(\varepsilon/a_i)^{1/2}. \tag{31} \]
The amplitude \(a\), owing to the presence of friction, on the average slowly decreases with time, so that the phase trajectory makes a large number of revolutions around the cylinder before it first intersects the line \(a=a(\Phi_i)\), after which it will either asymptotically tend to \(\Phi_i\), or, passing by \(\Phi_i\), will tend to one of the stable foci situated below (Fig. 3). Without an exact solution of the equations of motion, one cannot make a reliable assertion about the limiting regime of the system. However, one can determine the probabilities \(W_i^+\) and \(W_i^-\) that a phase trajectory passing through a point \(y_0\) lying in the region (31) will tend to \(\Phi_i\) or will pass it by. For small friction such an approach is justified, since the phase trajectories tending to the focus \(\Phi_i\) and those passing it by are densely intermingled in the region (31).
Replacing in (29) \(\gamma\) by \(\mu\gamma\) and, following V. I. Arnol’d [4], we define \(W_i^+\) as follows:
\[ W_i^+ = \lim_{d\to 0}\lim_{\mu\to 0} \frac{\operatorname{mes}\Omega_i(d,\mu)}{\operatorname{mes}\Omega(d)}, \]
where \(\Omega(d)\) is the \(d\)-neighborhood of the point \(y_0\); \(\Omega_i(d)\) is the set of initial conditions from \(\Omega(d)\) which bring the phase trajectory to \(\Phi_i\). As \(\mu\to 0\), the friction is switched off. Obviously,
\[ W_i^+ + W_i^- = 1. \]
For the lower focus the answer is trivial: since all trajectories may end either at the zero stable focus* (for \(\gamma_0' > 0\)) or on the line \(a=0\) (for \(\gamma_0' < 0\)), we have
\[ W_i^+ = 1 \quad \text{for } \gamma_0' > 0, \]
\[ W_i^+ = 0 \quad \text{for } \gamma_0' < 0. \]
Here
\[ \gamma_i' = (\gamma a)'_{a=a(\Phi_0)}. \]
Fig. 3
Let us proceed to consider the case \(i\ne 0\). It is not difficult to show [4] that in this case the required probability is given by the expression
\[ W_i^+ = \oint_{L(C(S_i))} \gamma a\,d\theta \bigg/ \int_{L(C(S_i))} \gamma a\,d\theta, \tag{32} \]
where in the denominator the integration is carried out along the upper branch of the separatrix of the nondissipative system \((\mu=0)\), and in the numerator along the whole separatrix. For what follows, however, it is convenient to express \(W_i^+\) in another form. For this purpose, note that in a neighborhood of \(\Phi_i\) the equations of motion (29), to accuracy \(O(\varepsilon/a)\), can be represented in the form
\[ \dot{\xi} = -\gamma_i - \gamma_i'\xi - \varepsilon\sin\theta, \qquad \dot{\theta} = -\alpha_i\xi, \tag{33} \]
where \(\xi = a-a(\Phi_i)\) has been put.
\[ \text{* Here the case of the existence of a stable limiting cycle surrounding } \Phi_0 \text{ is excluded from consideration.} \]
Introduce a new variable
\[ \sigma(\xi,\theta)=\frac{1}{2}a_i\xi^2-\varepsilon\cos\theta-\gamma_i\theta . \tag{34} \]
Then, by virtue of (33), we have
\[ \dot{\sigma}(\xi,\theta)=-\gamma_i' a_i \xi^2 . \tag{35} \]
As a result of simple calculations, expression (32), using (34), (35), can be represented in the following form:
\[ W_i^+=\frac{2\Delta\sigma}{2\pi\gamma_i+\Delta\sigma}, \tag{36} \]
where
\[ \Delta\sigma=\left|\int_{L(C(S_i))}\dot{\sigma}(\xi,\theta)\frac{d\theta}{\dot{\theta}}\right|. \tag{37} \]
The integration in (37) is carried out along the upper branch of the separatrix.
2. Behavior of the system under slow variation of the parameters. The behavior of nonlinear systems under slow variation of the parameters is of considerable interest. Usually the word “slow” is understood in the sense that the parameters change little over one period of the oscillator \(T_\omega=\dfrac{2\pi}{\omega}\). This makes it possible to regard the motion of the system as quasiperiodic, with slowly varying amplitude and frequency. The solution may be sought in the form of asymptotic expansions. It is precisely in this way that the basic formulas for describing nonstationary processes in nonlinear systems were obtained by Yu. A. Mitropolsky [3].
From the preceding discussion it follows that a nonlinear system, in addition to the period \(T_\omega\), possesses another characteristic time—the period of oscillation of the amplitude \(T(C)\) (21). If the characteristic time of variation of the parameter \(\lambda\), \(t_\lambda\), is large in comparison with \(T(C)\), then the character of the motion of the system can be established by knowing the solution for the system with constant parameters and by using the existence of an adiabatic invariant.
As was already indicated above, the equations of motion can be represented in Hamiltonian form (12). A Hamiltonian system, as is known, possesses the adiabatic invariant \(I(C)\):
\[ I(C)=\oint_{L(C)} a^2\,d\theta, \tag{38} \]
where the integration is carried out along a closed phase trajectory. The evaluation of the integral (38) is difficult in the general case; however, for \(a\gg\varepsilon\) it is not hard to compute it by using the equation for the integral curves (24). Indeed,
\[ I(C)=\oint_{L(C)} a^2\,d\theta= \begin{cases} \displaystyle \oint \left|a^2-a^2(P_i)\right|\,d\theta, & L(C)\in O_i,\\[6pt] \displaystyle \oint \left|a^2-a^2(P_i)\right|\,d\theta+2\pi a^2(P_i), & L(C)\,\overline{\in}\,O_i. \end{cases} \]
Hence, using expression (24), we have
\[ I(C)\simeq \begin{cases} 2a(P_i)G(C), & L(C)\in O_i,\\ 2a(P_i)G(C)+2\pi a^2(P_i), & L(C)\,\overline{\in}\,O_i, \end{cases} \tag{39} \]
where
\[ G(C)=\oint_{L(C)}\left|a-a(P_i)\right|\,d\theta . \]
For \(a\gg\varepsilon\), from (24) we have
\[ G_i(C)= \begin{cases} 8\left(\varepsilon/a_i\right)^{1/2} \left[ 2E\left(\sqrt{\dfrac{1+C_1}{2}}\right) -(1-C_1)K\left(\sqrt{\dfrac{1+C_1}{2}}\right) \right], & 1\ge C_1>-1,\\[1.2em] 4\left|2\varepsilon(1+C_1)/a_i\right|\, E\left(\sqrt{\dfrac{2}{1+C_1}}\right), & C_1>1, \end{cases} \tag{40} \]
where \(E(\varphi)\) is the complete elliptic integral of the second kind. The two signs in (40) correspond to different branches of the trajectory \(L(C_1)\).
2a. A frictionless system with slowly varying parameters. The equations of motion in this case have the form (11), where \(\nu\) and \(f\) vary with time. The variation of \(\nu\) and \(f\) leads to a change in the position and sizes of the regions on the phase surface. In this process some of the trajectories may leave the regions or enter them. To follow this process in greater detail, it is convenient to make use of the presence of the adiabatic invariant (39). If at the initial moment the trajectory belongs to the region \(O_i\), then, provided the condition of adiabatic variation of the parameters is satisfied,
\[ \dot{\nu}/\nu,\quad \dot{f}/f \ll 1/T(C) \tag{41} \]
then, owing to conservation of the adiabatic invariant, from (39) we have
\[ \frac{\dot a(P_i)}{a(P_i)}+\frac{\dot G(C_1)}{G(C_1)}=0, \tag{42} \]
or, using (40),
\[ \dot C_1= -\left[ \frac{\dot a(P_i)}{a(P_i)} +\frac{\dot\varepsilon}{2\varepsilon} -\frac{\dot a_i}{2a_i} \right] \left[ \frac{\partial}{\partial C_1}\ln G_1(C_1) \right]^{-1}, \tag{43} \]
where \(G_1(C_1)=(a_i/\varepsilon)^{1/2}G(C_1)\).
Since \(C_1\) decreases when the trajectory is drawn toward the center, the phase trajectories contract for \(\dot C_1<0\) and move away from the center for \(\dot C_1>0\). The departure of a phase trajectory from the region of phase oscillation occurs when it crosses the separatrix, i.e., when \(C_1=1\).
2b. A dissipative system with slowly varying parameters. The equations of motion in this case have the form (29), where \(f\) and \(\nu\) depend on time. For the region where \(a\gg\varepsilon\), the equations of motion may be represented in the following form:
\[ \dot{\xi}=-\dot a(P_i)-\gamma_i-\gamma_i'\xi-\varepsilon\sin\theta,\qquad \dot\theta=\alpha_i\xi,\qquad \xi=a-a(\Phi_i). \tag{44} \]
The limiting points in the case under consideration are, as usual, determined from the conditions
\[ \dot{\xi}=\dot{\theta}=0, \tag{45} \]
which, when applied to (44), makes it possible to find the coordinates of the limiting points:
\[ \xi=0,\qquad \sin\theta=-\frac{\dot a(P_i)+\gamma_i}{\varepsilon}. \tag{46} \]
The limiting points are not stationary on the phase plane; their coordinates vary with time. As follows from (45) and (44),
The condition for the existence of limit points has the form \(|\dot a(P_i)+\gamma_i|<\varepsilon\). Equations (51) define two special points, a focus and a saddle point. To establish the character of the focus, note that in the presence of friction the invariant \(I(C)\), generally speaking, changes with time:
\[ I(C)\simeq \frac{2a(P_i)}{T(C)}\oint_{L(C)}\gamma\,a\,d\theta = -\frac{2a(P_i)}{T(C)}\gamma_i'G(C_1). \]
Then, instead of (46), we have
\[ \dot C_1 = -\left[ \frac{\gamma_i'}{T(C_1)} + \frac{\dot a(P_i)}{a(P_i)} + \frac{\varepsilon}{2\varepsilon} - \frac{\gamma_i}{2a_i} \right] \left[ \frac{\partial}{\partial C_1}\ln G_1(C_1) \right]^{-1}. \tag{47} \]
Thus the focus is stable (unstable) if
\[ V(C_1,\nu)>0\quad [V(C_1,\nu)<0]. \tag{48} \]
Here \(V(C_1,\nu)\) denotes the expression in square brackets in (47). From equation (47) it is easy to establish the condition for the existence and the character of the limit cycle in \(O_i\). Indeed, \(\dot C_1=0\) for \(C_1=\overline C_1\), if \(\overline C_1\) is a root of the equation
\[ V(C_1,\nu)=0\quad (C_1(S_i)>C_1>-1). \tag{49} \]
Thus, equation (49) is the condition for the existence of a limit cycle in \(O_i\). The limit cycle is stable (unstable) if
\[ \left.\frac{\partial}{\partial C_1}V(C_1,\nu)\right|_{C_1=\overline C_1}<0 \quad \left[ \left.\frac{\partial}{\partial C_1}V(C_1,\nu)\right|_{C_1=\overline C_1}>0 \right], \tag{50} \]
and is transitional when
\[ \left.\frac{\partial}{\partial C_1}V(C_1,\nu)\right|_{C_1=\overline C_1}=0 \quad \left( \left.\frac{\partial^2}{\partial C_1^2}V(C_1,\nu)\right|_{C_1=\overline C_1}\ne 0 \right). \tag{51} \]
If at the initial moment \(t_0\) the system is characterized by a phase point \(y_0(a_0,\theta_0)\) belonging to the region (31), and moreover \(\varepsilon>\gamma_i+\dot a(P_i)>0\), then with time the phase trajectory will approach the line \(\xi=0\) and at some instant \(t_i\) will cross it, after which it will either tend asymptotically to \(\Phi_i\) (the phase trajectory belongs to the family \(O_i\)), or will pass by \(\Phi_i\). The probability \(W_i^{+}\) that the phase trajectory which passes through the point \(y_0(a_0,\theta_0)\) belongs to the family \(O_i\) is not difficult to compute, if the instant \(t_i\) when the phase trajectory crosses the line \(\xi=0\) is known. Indeed, then, by analogy with (36), we have:
\[ W_i^{+} = \frac{\sigma(\theta_2)-\sigma(\theta_1)} {\sigma(0)-\sigma(\theta_1)+\sigma(\theta(S_i))-\sigma(2\pi)}, \qquad \sigma(\theta)=\sigma(0,\theta), \tag{52} \]
where \(\sigma(\theta_1)\) and \(\sigma(\theta_2)\) should be computed from the following formulas:
\[ \sigma(\theta_1) = \sigma\left[ \theta\left( S_i,t_i-\frac{1}{2}T_s \right) \right] + \int_{t_i-\frac{1}{2}T_s}^{t_i} \dot\sigma(\xi,\theta)\,dt, \tag{53} \]
\[ \sigma(\theta_2)=\sigma\left[\theta\left(S_i,\ t_i+\frac{1}{2}T_s\right)\right] -\int_{t_i}^{t_i+\frac{1}{2}T_s}\dot{\sigma}(\xi,\theta)\,d\theta,\quad T_s=T\bigl(C(S_i)\bigr). \tag{53} \]
Moreover, as follows from the definition of \(\sigma(\xi,\theta)\) (34) and equations (44),
\[ \dot{\sigma}(\xi,\theta)= \left(\frac{1}{2}\dot{\alpha}_i-\alpha_i\gamma_i'\right)\xi^2 -\dot{\varepsilon}\cos\theta-\dot{\gamma}_i\theta . \tag{54} \]
When evaluating the integrals in (53) with substitution (54), one should use the equations for the trajectories of the nondissipative system with constant parameters.
The computation of the instant \(t_i\) as a function of the initial conditions \(y_0(a_0,\theta_0,t_0)\) must be carried out with the greater accuracy the more rapidly \(W_i(t)\) changes. Let us note that the computation of \(t_i\) with the required accuracy often presents no great difficulty because of the presence of small parameters.
From the preceding discussion it becomes clear that a periodic forcing force, acting on a nonlinear oscillator, does not lead to a systematic accumulation of energy in it even in the absence of friction. Apparently this is the case for any non-self-excited nonlinear system, provided only that it does not become a linear nondissipative system as \(a\to\infty\). However, a forcing force with an adiabatically varying frequency may lead to an accumulation of energy in the system. Thus, if the focus \(\Phi_i\) is stable for the chosen regime of variation of the parameters of the forcing force, then the phase trajectory belonging at the initial instant to the region \(O_i\) will remain in it as the latter moves over the phase surface.
The author is grateful to G. Ya. Lyubarskii for numerous discussions and useful advice.
References
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N. N. Bogolyubov, Yu. A. Mitropolsky. Asymptotic Methods in the Theory of Nonlinear Oscillations, Fizmatgiz, 1958.
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N. N. Nemytskii, V. V. Stepanov. Qualitative Theory of Differential Equations. Gostekhizdat, 1949. L. Cesari. Asymptotic Behavior and Stability of Solutions of Ordinary Differential Equations. Mir, 1964.
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Yu. A. Mitropolsky. Problems of the Asymptotic Theory of Nonstationary Oscillations. Nauka, 1964.
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V. I. Arnol’d. UMN, vol. XVIII, no. 6, 91, 1963.
Received by the editors
July 6, 1965
Physico-Technical Institute
Academy of Sciences of the Ukrainian SSR