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The non-linear oscillations of a pendulum of variable length on a vibrating base
Journal of Applied Mathematics and Mechanics 76 (2012) 25–35
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Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The non-linear oscillations of a pendulum of variable length on a vibrating base夽 P.S. Krasil’nikov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 10 February 2011
a b s t r a c t A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Averaging of the standard system. The number of small parameters in many applied problems can be fairly large, as, for example, in the problem of the motion of the planets of the solar system, and in problems of the dynamics of aviation systems and various types of spacecraft. Since the methods of perturbation theory were developed for the case of a single small parameter, reduction is employed, which reduces the initial problem to the case of a single parameter. Thus, if the number of small parameters j is equal to three, we assume, for example, 1 = k1 3 , 2 = k2 3 , taking 3 as the independent quantity, while k1 and k2 are free parameters, of the order of unity (we can also consider any two-parameter family of curves, covering the three-dimensional space of the parameters j ). The equations of motion contain k1 and k2 as additional quantities, by changing which in a continuous way one can investigate the effect of small parameters on the nature of the motion of the system, using classical methods of perturbation theory. However, there is a class of problems for which reduction is not very effective, since it leads to a family of truncated equations, when the change from one region of phase space to another involves problems of joining the solutions of these equations. Using reduction it is impossible to investigate the bifurcations of the solutions of the averaged equations, with the exception of the special case when the family of reduction curves covers the bifurcation surface. We will describe an averaging method for the case when the system of differential equations contains m small parameters 1 , 2 , . . ., m . Without loss of generality we will assume that all the small parameters take positive values. We will also assume that the equations can be reduced to the standard form
(1.1) We introduce the following notation: ␥ = (␥1 ,. . .␥m ), = (1 ,. . .,m ) are row vectors with integral nonnegative components, |␥| = ␥1 +. . .+ ␥m , || = 1 +. . .+ m (|␥| = 1 and || = 2 are the first and second orders of infinitesimals of the corresponding forms, which 1 . . . ␥m , v = v1 . . . vm are symbolic notations of the product of powers of the small parameters (of the first and depend on j ), ␥ = ␥ m m 1 1
夽 Prikl. Mat. Mekh. Vol. 76, No. 1, pp. 36–51, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.03.003
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second orders of infinitesimals respectively), and X␥ (x,t), X (x,t) are vector functions of the phase variable x and the time t. Summation is carried out over all values of ␥ and from the regions ␥ and . Hence, in the first part of Eq. (1.1) we have written linear and quadratic terms (in the small parameters j ), whereas terms of higher order of infinitesimals are represented by the dots. For example, we will consider the case m = 3, x ∈ R1 , when Eq. (1.1) has the form
The vectors and ␥ take the set of values = {200, 020, 110, 011} and ␥ = {100, 001} respectively; these sets do not contain the values 010, 002 and 101, since there are no terms on the right-hand side corresponding to them. We will seek the replacement of variables x → y in the form of a generalized Krylov–Bogolyubov series
(1.2) with the condition that the transformed equations have the autonomous form
(1.3) Note that, in equalities (1.2) and (1.3), unlike Eq. (1.1), the vectors ␥ and can take values which do not belong to the sets ␥ and . Compared with the classical case of one small parameter,1,2 the number of unknown functions uk (x, t) (like the functions Yk (y)) increases sharply, and hence, the volume of calculations also increases. Nevertheless, the construction of generalized series of solutions does not lead to any fundamental complications, but has specific properties related to estimating the accuracy of the approximation and constructing asymptotic time intervals. We differentiate Eq. (1.2) with respect to time:
We have used the equality
which follows from Cauchy’s formula of the multiplication of series and the equality ␥1 + ␥2 = * . The summation is carried out over all the vectors * , which allow of an expansion in the terms ␥1 and ␥2 . Note that in the formulae containing the index * , summation over ␥ (for fixed * ) is carried out over all the vectors ␥ for which an additional vector ␥␣ exists, such that ␥ + ␥␣ = * . If the formulae contain terms ␥ X␥ (y, t) and their derivatives, it is necessary to require that the additional condition ␥ ∈ / ␥ must be satisfied since Eq. (1.1) superimposes these limitations on the terms ␥ X␥ (y, t) and their derivatives. This also applies to expressions of the form X (y, t), for which ∈ . The expression for x˙ is the left-hand side of Eq. (1.1) in the new variables. We will obtain an expression for the right-hand side. To do this we substitute expression (1.2) instead of x and expand the right-hand side in series in the neighbourhood of the point x = y:
The prime on the summation sign denotes that the summation is carried out over the vectors ␥ or , satisfying the conditions |␥| = 1, ␥ ∈ ␥ and || = 2, ∈ . The zero subscript denotes that the expression is calculated for = 0. Equating the left- and right-hand sides we obtain, after identifying the coefficients of like powers of the small parameter, a system of equations of the first and second approximation in u␥ (y, t), Y␥ (y, t) and u (y, t), Y (y, t), respectively:
(1.4)
(1.5)
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Note that in these equations, the vectors ␥ and can take arbitrary values, but with certain reservations. Thus, the right-hand side of Eq. (1.4) is written formally for any ␥, but it actually is non-zero only when ␥ ∈ ␥ . When evaluating the terms of the right-hand side of Eq. (1.5) it is necessary to take into account the fact that the function X (y, t) is non-zero only when ∈ , and summation over ␥ is carried out for any fixed values of , but provided that =␥ + ␥␣ . In this case for the first sum the vector ␥ must belong to the set ␥ , whereas for the second sum there are no such limitations. If the vector function x␥ (y, t) depends on time almost periodically uniformly in y and is represented by a Fourier series, the required function u␥ (y, t) can be written in the form
(1.6) where ␥ (y) is the initial function and {m }(m = 1,2,. . .) is the spectrum of the function X␥ (y, t). Hence
where
(1.7) We proceed similarly with Eq. (1.5): we expand its right-hand side in a Fourier series in the explicitly occurring time, equating the oscillating part of the series of the partial derivative with respect to time of the function u (y, t). We will have
(1.8) Thus, the first approximation has the form
(1.9) The second approximation includes terms of the second order of infinitesimals, which leads to an increase in the approximation accuracy. The equations of the second approximation have the form
(1.10) provided that the vector function Y (y) is evaluated using formula (1.8). Reverting to the initial variable x, we will write approximate solutions of Eqs (1.1). The first approximation to the required solution has the form
where y(t, ) is the solution of Eqs (1.9). The second approximation to the required solution can be represented in the form
(1.11) assuming that the function u␥ (y, t) was calculated in advance at the stage when investigating the first approximation. Here y(t, ) is the solution of Eqs (1.10). 2. The accuracy of the approximation The generalized Bogolyubov theorem. The problem of the accuracy of the approximation is central in the averaging method. It is inseparably connected with the problem of the value of the time interval in which the solutions of the averaged and rigorous systems of differential equations are compared. In the classical case of a single small parameter, the accuracy of the k-th approximation is k (k is the maximum order of the retained terms of the right-hand side), while the time interval is a quantity of the order 1/, since the quantity t ∼ 1 is the characteristic time for the system of equations in the Bogolyubov standard form. The situation is different when Eqs (1.1) contain several parameters. Here there is no unique characteristic dimension of time, and there is no a priori clarity as regards the accuracy of the approximation, since the order of smallness of the function, which depends on several parameters and the time, is not defined. We will fill this gap by assuming that the comparison interval T* is asymptotically long, i.e., having the form T* = L/␣() provided that ␣() → 0 as → 0. We know that the scalar function q() is of the k-th order of smallness in , if the following limit equality is satisfied
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Corollary.3 If q() = O(k ), then for any ␦ > 0 as small as desired, constants A1 (␦) > 0, A2 (␦) > 0 exist such that when 0 < ≤ ␦ the function q() satisfies the condition of the band
We will generalize this result to the case of a vector function, which depends on m small parameters and the time. To do this we will introduce the additional vector ␦ = (␦1 ,..., ␦m ) with positive components ␦j and the vector s = (s1 ,..., sm ) with non-negative integral arguments sj . Definition 1. We will call the function q(t, ) a function of the k-th order of smallness in in the time interval [0,L/␣()], if for any small vector ␦ with positive components ␦j > 0, constants A1 (␦) ≥ 0, A2 (␦) > 0 and two k-th order forms
exist that are positive definite in the positive cone {:j > 0, j = 1,2,. . .,m}, such that the band condition (2.1) is satisfied when
We will call the function q(t, ), which satisfies Definition 1, O-large of the k-th order of smallness in in the time interval [0,L/␣()] and denote it by Ok (). The following assertion also holds. Lemma.
We will assume that the function q(t, ) is such that the following non-zero limit, bounded and uniform in t, exists
Then q(t, ) is a quantity of the k-th order of smallness in the time interval [0, L/␣()]. We will assume that the functions X␥ (x, t) (|␥| = 1, ␥ ∈ ␥ ), linear in j , are defined in a certain open connected region D’ of the variable x, when t ∈ [0,∞]. We will assume that the functions X␥ (x, t) satisfy the following conditions in a certain subregion D × [0, ∞), D ∈ D’]: 1) the condition of continuity in t, 2) the Lipschitz condition in x with constants ␥ , independent of x and t:
3) the condition of uniform limitation in the region D × [0, ∞), i.e.
and the conditions, in addition to these, 4) the condition of boundedness, uniform in u ∈ D, of the integrals
in the case of almost periodic dependence of the functions X␥ (x, t) on time; 5) the condition that limits (1.7) exists, uniform in u ∈ D, if the functions X␥ (x, t) depend arbitrarily on t. ¯ (u) is an almost Note that condition 4 is equivalent, by Bohr’s theorem, to the condition that the integral of the function X (u, t) − X periodic function of t. It is satisfied for the quasi-periodic function X␥ (u, t) if its Fourier series contains a finite number of terms and there are no resonances. In the more general case when the Fourier series of the function X␥ (u, t) contains an infinite number of terms, the behaviour of the integral depends very much on the conditions for the following series to converge
Thus, if this series converges uniformly in u and t, the integral is uniformly bounded. Non-uniform convergence with respect to t gives examples of an unbounded increase of the integral.4–6 It also follows from Hilden’s papers that the integral of the scalar function f(1 t,
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2 t), which is 2-periodic, depends on each argument j = j t and has zero mean, is a quasi-periodic function of t (consequently, bounded) for almost all irrational values of ␣ = 1 /2 . The case of the divergence of the series has only been investigated to a small extent. We will introduce the following notation
(t,) = ||x(t, )–u(t, )|| is the approximation error, and * is a vector, the components of which are the small parameters j occurring in the first approximation of Eqs (1.1). The generalized Bogolyubov theorem. If conditions (1-3) are satisfied for differential equations (1.1), then for any L > 0, > 0 specified in advance one can obtain a positive scalar number 0 (L, ), such that if u = u(t) is a solution of Cauchy’s problem for the averaged equations
defined in the interval 0 ≤ t < ∞ and belonging to the region D together with its -neighbourhood, then for any , ||||<0 and for any t from the asymptotic time interval (2.2) the following estimates of the accuracy of the approximation hold. 1◦ . X␥ (x, t), ␥ ∈ ␥ , are T-periodic functions of the time t, then (2.3) 2◦ .
X␥ (x, t), ␥ ∈ ␥ are almost periodic functions of t, uniform in x, and condition 4 is satisfied, then for any as small as desired (2.4)
where () is the general -almost period of the functions X␥ (x, t). 3◦ . X␥ (x, t), ␥ ∈ ␥ are arbitrary functions of time and condition 5 is satisfied, then (2.5) where
Inequalities (2.3) and (2.5) are a generalization of the well-known estimates of the classical case of a single small parameter.7 Estimate (2.4) is unknown; we will prove it for the classical case, since the generalization to the case of m small parameters is obvious. We fix . Suppose t* = L*/ is the instant when the phase curve intersects the boundary of the region D and () is the almost common period of the functions X␥ (x, t). The following estimate of the accuracy of the approximation is known7
where x(t, ) and u(t, ) are the solutions of the rigorous and averaged equations, respectively,
We will divide the current time interval [0, t] into intervals of length (). Suppose N is the number of intervals which lie wholly inside the section [0, t]. We have
(2.6) (2.7) Consider Cauchy’s problem
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Then
(2.8) Hence
(2.9) N() ≈ L* /.
since The second term on the right-hand side of inequality (2.6), by virtue of condition 4, allows of the estimate
Here u() is the piecewise-constant function of :
Hence we have the inequality
It is easy to see that in this formula we can replace L* by as large a value of L as desired, if < 0 , where
In the case of m independent small parameters, we have the inequality
Assuming ␣() = ||* ||, we will have
Hence accuracy estimate (2.4) follows. Therefore, the generalized Bogolyubov theorem guarantees that, for periodic and almost periodic cases in the asymptotic time interval (2.2), the accuracy of the approximation has a value of the first order of smallness in , by virtue of Definition 1. It has been shown,8 that the construction of the function ␣() based on physical considerations is not always correct, and hence one must choose the function ␣() in the form ||*||. Note that, along different curves of the space of small parameters, the value of the accuracy of the first approximation and the value of the asymptotic time interval may vary considerably. Thus, for the case of two independent parameters and the accuracy of the √ approximation along the curve = , = 2 reaches values of the order of and falls sharply to a value of the order of on the curve = , √ √ = . The asymptotic time intervals on these curves are of the order of 1/ and 1 , respectively. 3. A pendulum of variable length on a vibrating base We will consider the problem of the oscillations of a mathematical pendulum of variable length on a vibrating base. As is well known, vibration has a considerable effect on the dynamics of a pendulum of constant length, stabilizing its upper equilibrium position at a high vibration frequency.9,10 Some conditions for the stability of the upper equilibrium position for a pendulum of variable length have also been obtained.11,12 Note that pendulum models find applications in problems of investigating the wave motions of a fluid in the fuel tanks of carrier rockets.13 Formulation of the problem. The equations of motion. We will assume that the suspension point of the pendulum executes harmonic oscillations along the vertical y axis: y = asint. In movable axes, connected with the vibrating base, the equation of the pendulum oscillations has the form
Here l(t) is the length of the pendulum, which varies harmonically with time, l0 is the mean value of the pendulum length and ae is the translational acceleration.
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We will introduce three small parameters ␥, and and a new time by the formulae
Here ω0 = g/l0 is the frequency of natural oscillations. Hence, the vibration frequency is assumed to be large, whereas the amplitudes of the oscillations of the suspension point of the pendulum and its length are assumed to be fairly small. We will change to a new time and expand the right-hand side of the equation obtained in series in small parameters. We have
A derivative with respect to is denoted by a prime. In order to convert the equation of the oscillations to a standard system, we will use the following non-linear replacement of variables1
The converted equations take the form
(3.2) where
The parameter = ␥/ takes values of the order of unity if the region being investigated is limited by the inequality ␥ ≤ k, since in this case satisfies the band condition
The equations of the first approximation of the averaging method. We will obtain the averaged equations of the pendulum oscillations, assuming x = (,) . We reduce system of equations (3.2) to the form (1.1) (3.3) Here
The averaged equations of the first approximation have the form
(3.4) in system (3.4)). of the classical case (the parameter does not occur They are identical with the averaged equations √ √ It is well known,1,10 that when > 1/ 2 the equilibrium = is unstable, while when < 1/ 2 it becomes stable and two unstable equilibria occur in its neighbourhood, namely,
√ The bifurcation surface has the form ␥ = 2 and is a ruled surface in three-dimensional space of the small parameters , and ␥, and hence the classical averaging scheme, based on linear reduction to a single parameter, was used in Ref. 1. In the initial variables and ’ the upper equilibrium position of the pendulum = , ’ = 0 corresponds to the stationary value = , whereas the periodic motions of the form (3.5) of the upper equilibrium position of the pendulum occurs through correspond to the equilibria = 1 and = 2 . Hence, stabilization √ bifurcation of the stationary point = , ’ = 0 when = 1 2, which is accompanied by the occurrence of two unstable periodic motions (3.5).
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Bearing in mind that = 0 l0 /(a), we also arrive at the conclusion that stabilization of the equilibrium is observed for fairly large values of , which satisfy the inequality
Note that the parameter has no effect on the stability of the equilibrium in the first approximation of the averaging method, but in the second approximation its effect is considerable. The equations of the second approximation of the averaging method. The non-resonance case. We will now construct and analyse the second approximation. According to formulae (1.10) and (1.11) in the second approximation we have (3.6) The vector y must be determined from the following system of equations
Here we have used a notation
The expressions for Y ( = 110, 101, 011) are calculated from formulae (1.8), where for each we carry out summation over two values of the index ␥: ␥ = 100 and 010, if = 110, ␥ = 100 and 001 in the case when = 101, and ␥ = 010 and 001 when = 011. Calculations show that
Here the average of the function X110 undergoes a discontinuity, if = 2. Hence, when = 2 parametric resonance occurs in the system. For a start we will consider the non-resonance case = / 2. Calculations using formulae (1.8) show that all the vector functions Y (y) are equal to zero:
Hence, the averaged equations of the second approximation are identical with the averages of the first approximation (3.4). Consequently, the previous conclusions remain in force, and only the values of the variable j , corresponding to the stationary solutions = j , = 0 of Eqs (3.4) are made more precise. Thus, if k = / 1, we have, on the basis of formulae (1.6) and (3.6)
In the case when k = 1
The equations of the second approximation in the neighbourhood of resonance. Oscillations at strict resonance = 2. We will assume that the resonance equality is satisfied approximately: / – 2 = , where is a small quantity of the first order of infinitesimals, i.e., = a, where a is a constant quantity of the order of unity. Introducing the additional slow variable , which satisfies the equality
we obtain, after averaging, equations of the second approximation, which describe oscillations in the neighbourhood of the resonance = 2:
Hence it is clear that (3.7) In the case when the resonance relation = 2 is satisfied rigorously, the parameter is equal to zero, and hence Eq. (3.7) takes the autonomous form (cos = 1), and we have two types of steady motions, described by the equalities
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The second type of equilibrium depends on the parameter , which leads to the occurrence of new effects in the behaviour of the system. Thus, if varies in the limits
the first type of equilibrium predominates, completely suppressing the second type. The equilibrium = 0 is then stable while the equilibrium = is unstable. The situation changes as soon as reaches the boundary values of this interval. A complex equilibrium appears, consisting of the solution = of family 2 and of the solution = of family 1, if = 2 . A further reduction in to zero leads to the occurrence of two unstable equilibria
in the neighbourhood of the stable point = . Consequently, when = − bifurcation of the equilibrium = , akin to the bifurcation described above, is observed, and the resonance oscillations of the pendulum when < − are similar to the non-resonance oscillations in the case when 2 < 1/2. If = + , a complex equilibrium = 0 appears. It splits into two stable equilibria
in the neighbourhood of the unstable equilibrium = 0, if > + . It follows from a comparison with the results of previous investigations that the region of stability 2 < 1/2 of the upper equilibrium position = of the non-resonance case is preserved during resonance provided < − . In the opposite case the upper equilibrium position is unstable. Moreover, the lower equilibrium position is unstable when > − . Hence, periodic resonance oscillations of the length l(t) of the pendulum have a destabilizing influence on the upper and lower equilibrium states. In , ’ variables, periodic oscillations of the form
where
correspond to the equilibria 1 and 2 . Note that the bifurcation equality
describes a two-dimensional surface in three-dimensional space of the small parameters , and ␥. The surface is not ruled, and hence it is impossible to obtain it if we use the averaging method, based on linear reductions of the problem to a single parameter. Such a surface cannot be seen a priori, and hence any attempt to construct it using non-linear reduction is unsound. / 0). Oscillations in the neighbourhood of the resonance = 2. In conclusion we will briefly consider a case close to resonance ( = Equation (3.7), which describes near-resonance oscillations, contains a slowly varying parameter = . In this case the oscillations of the pendulum are close to non-resonance oscillations, if ∼ , which follows directly from Eq. (3.7): after averaging over “fast” time we obtain equation (3.4). In the case ∼ 2 this averaging does not occur. We must distinguish five cases:
The simplest cases to investigate are 2 and 3, since in these cases the resonance and non-resonance oscillations are similar, and a change in leads to a small shift in the stationary points 1 and 2 on the axis. A more complex evolution of the oscillations is observed in the remaining cases. In case 1 we have a transition from a strict resonance case, in which the motion of the pendulum is similar to the oscillations of an ordinary mathematical pendulum, to a non-resonance case, when its oscillations have a more complex form due to the presence of two unstable equilibria in the neighbourhood of = . Replotting of the phase portrait for a continuous increase in of quantities of the order of n (n ≥ 2) to occurs by splitting the separatrice, and the occurrence of a chaotic zone of oscillations in the neighbourhood of the equilibrium = , ’ = 0 and in the neighbourhood of the separatrice, following the evolution of this zone. The behaviour of the phase curves of a chaotic layer is extremely complex. It depends very much on the value of the parameter and on the accuracy with which the initial conditions are specified. Without pretending to absolute reliability of the description, we will give an approximate description of the reorganization of the oscillations, based on numerical calculations, which were carried out for different values of and different initial conditions.
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Fig. 1.
Consider case 1. The accuracy with which numbers with floating points can be represented is twenty significant figures, and the error of the method is 10−14 . We will fix the following values of the parameters
When = 10−4 the phase portrait of the oscillations is represented in the upper part of Fig. 1 (here and below we do not show trajectories corresponding to purely rotational motions and, in view of the symmetry, we only show the region ’ ≥ 0). The phase portrait of the oscillations practically coincides with the phase portrait of a mathematical pendulum. The width of the chaotic layer is small, and it cannot be seen in the scale of the figure. An increase in leads to the appearance of a complex lattice of loops of trajectories in the neighbourhood of the point = and in the neighbourhood of the separatrice (see the case = 0.0359; the chaotic layer in the region of the separatrice is not shown). When is increased further the width of the chaotic layer in the region of = increases to a value of the order of 0.02 ( = 0.104); in addition to this, the phase point, which performs oscillations in the neighbourhood of the point
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= , leaves it as time passes. Beginning with a value of the parameter = 0.11, the oscillations in the neighbourhood of = slow down so much that the phase point pertains to the region = during the whole time of motion, forming a dense network of trajectories. The value = 0.11 corresponds to this case, and additional curves are also represented, the initial data for which approximate to the values = 1 and = 2 , respectively (1 = 2.4981, 2 = 3.7851 are additional equilibrium positions in the non-resonance case). It can be seen that these additional curves form a zone of oscillations in the region of the points = 1 and = 2 , leaving them as time passes. A further increase in leads to a sharp reduction in the zone of chaotic oscillations in the region of the point = : already when = 0.112 the size of this zone is of the order of 10−7 , while when = 0.113 its dimensions are so small that it is practically indistinguishable from the geometrical point = , ’ = 0 (the contraction of a chaotic web to a point). The lower part of Fig. 1 corresponds to the case = 0.2. A new zone of oscillations is observed in the neighbourhood of the stable equilibrium = , ’ = 0 and there are trajectories similar to heterocyclic trajectories which connect the unstable “equilibria” = 1 and = 2 . These “equilibria” are small zones of oscillations, which the phase curves rapidly leave, making several rotations around the points = 1 and = 2 . Hence, the phase portrait of the oscillations takes a form similar to the phase portrait in the non-resonance case, for which the equilibrium = is stable, while the adjacent equilibria 1 and 2 are unstable. These results can be made more accurate by constructing the asymptotic approximation of the solutions of Eqs (3.2) for the case of large values of . References 1. Bogolyubov NN, Mitropol’skii YuA. Asymptotic Methods in the Theory of Non-linear Oscillations. New York: Gordon & Breach; 1963. 2. Volosov VM. Averaging in systems of ordinary differential equations. Usp Mat Nauk 1962; 17(6)(108):3–126. 3. Krasil’nikov PS. The averaging method as a procedure for separating the principal terms in series of solutions. In: Theoretical Mechanics. Collection of Scientific - Methodological Papers. Moscow: Izd MGU;2009:56–68. 4. Poincaré H. Sur les corbes définies par les équations differentielles. J Math Pures et Appl Ser 4 1886;2(2):151–217. 5. Poincaré H. Sur les series trigonometriques. Comptes Rendus Acad Sci 1885;101(2):1131–4. 6. Kozlov VV. Methods of Qualitative Analysis in Rigid Body Dynamics. Moscow; Izhevsk: RKhD; 2000. 7. Zhuravlev VF, Klimov DM. Applied Methods in the Oscillation Theory. Moscow: Nauka; 1988. 8. Krasil’nikov PS. The averaging of differential equations with two independent small parameters. Dokl Ross Akad Nauk 2001;436(3):332–5. 9. Jeffereys H, Jeffereys BS. Methods of Mathematical Physics. Cambridge: University Press; 1950. 10. Kapitsa PL. A pendulum with a vibrating suspension. Usp Fiz Nauk 1951;44(1):7–20. 11. Strizhak TG. Methods of Investigating “Pendulum”-Type Dynamical Systems. Nauka: Alma-Ata; 1981. 12. Vershinin BA, Litvin-Sedoi MZ. The stability of a pendulum of variable length with motion of the point of support. Izv Akad Nauk SSSR MTT 1973;1:40–2. 13. Rabinovich BI. Introduction to Spacecraft Carrier Rocket Dynamics. Moscow: Mashinostroyeniye; 1975.
Translated by R.C.G.
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The non-linear oscillations of a pendulum of variable length on a vibrating base夽 P.S. Krasil’nikov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 10 February 2011
a b s t r a c t A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Averaging of the standard system. The number of small parameters in many applied problems can be fairly large, as, for example, in the problem of the motion of the planets of the solar system, and in problems of the dynamics of aviation systems and various types of spacecraft. Since the methods of perturbation theory were developed for the case of a single small parameter, reduction is employed, which reduces the initial problem to the case of a single parameter. Thus, if the number of small parameters j is equal to three, we assume, for example, 1 = k1 3 , 2 = k2 3 , taking 3 as the independent quantity, while k1 and k2 are free parameters, of the order of unity (we can also consider any two-parameter family of curves, covering the three-dimensional space of the parameters j ). The equations of motion contain k1 and k2 as additional quantities, by changing which in a continuous way one can investigate the effect of small parameters on the nature of the motion of the system, using classical methods of perturbation theory. However, there is a class of problems for which reduction is not very effective, since it leads to a family of truncated equations, when the change from one region of phase space to another involves problems of joining the solutions of these equations. Using reduction it is impossible to investigate the bifurcations of the solutions of the averaged equations, with the exception of the special case when the family of reduction curves covers the bifurcation surface. We will describe an averaging method for the case when the system of differential equations contains m small parameters 1 , 2 , . . ., m . Without loss of generality we will assume that all the small parameters take positive values. We will also assume that the equations can be reduced to the standard form
(1.1) We introduce the following notation: ␥ = (␥1 ,. . .␥m ), = (1 ,. . .,m ) are row vectors with integral nonnegative components, |␥| = ␥1 +. . .+ ␥m , || = 1 +. . .+ m (|␥| = 1 and || = 2 are the first and second orders of infinitesimals of the corresponding forms, which 1 . . . ␥m , v = v1 . . . vm are symbolic notations of the product of powers of the small parameters (of the first and depend on j ), ␥ = ␥ m m 1 1
夽 Prikl. Mat. Mekh. Vol. 76, No. 1, pp. 36–51, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.03.003
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second orders of infinitesimals respectively), and X␥ (x,t), X (x,t) are vector functions of the phase variable x and the time t. Summation is carried out over all values of ␥ and from the regions ␥ and . Hence, in the first part of Eq. (1.1) we have written linear and quadratic terms (in the small parameters j ), whereas terms of higher order of infinitesimals are represented by the dots. For example, we will consider the case m = 3, x ∈ R1 , when Eq. (1.1) has the form
The vectors and ␥ take the set of values = {200, 020, 110, 011} and ␥ = {100, 001} respectively; these sets do not contain the values 010, 002 and 101, since there are no terms on the right-hand side corresponding to them. We will seek the replacement of variables x → y in the form of a generalized Krylov–Bogolyubov series
(1.2) with the condition that the transformed equations have the autonomous form
(1.3) Note that, in equalities (1.2) and (1.3), unlike Eq. (1.1), the vectors ␥ and can take values which do not belong to the sets ␥ and . Compared with the classical case of one small parameter,1,2 the number of unknown functions uk (x, t) (like the functions Yk (y)) increases sharply, and hence, the volume of calculations also increases. Nevertheless, the construction of generalized series of solutions does not lead to any fundamental complications, but has specific properties related to estimating the accuracy of the approximation and constructing asymptotic time intervals. We differentiate Eq. (1.2) with respect to time:
We have used the equality
which follows from Cauchy’s formula of the multiplication of series and the equality ␥1 + ␥2 = * . The summation is carried out over all the vectors * , which allow of an expansion in the terms ␥1 and ␥2 . Note that in the formulae containing the index * , summation over ␥ (for fixed * ) is carried out over all the vectors ␥ for which an additional vector ␥␣ exists, such that ␥ + ␥␣ = * . If the formulae contain terms ␥ X␥ (y, t) and their derivatives, it is necessary to require that the additional condition ␥ ∈ / ␥ must be satisfied since Eq. (1.1) superimposes these limitations on the terms ␥ X␥ (y, t) and their derivatives. This also applies to expressions of the form X (y, t), for which ∈ . The expression for x˙ is the left-hand side of Eq. (1.1) in the new variables. We will obtain an expression for the right-hand side. To do this we substitute expression (1.2) instead of x and expand the right-hand side in series in the neighbourhood of the point x = y:
The prime on the summation sign denotes that the summation is carried out over the vectors ␥ or , satisfying the conditions |␥| = 1, ␥ ∈ ␥ and || = 2, ∈ . The zero subscript denotes that the expression is calculated for = 0. Equating the left- and right-hand sides we obtain, after identifying the coefficients of like powers of the small parameter, a system of equations of the first and second approximation in u␥ (y, t), Y␥ (y, t) and u (y, t), Y (y, t), respectively:
(1.4)
(1.5)
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Note that in these equations, the vectors ␥ and can take arbitrary values, but with certain reservations. Thus, the right-hand side of Eq. (1.4) is written formally for any ␥, but it actually is non-zero only when ␥ ∈ ␥ . When evaluating the terms of the right-hand side of Eq. (1.5) it is necessary to take into account the fact that the function X (y, t) is non-zero only when ∈ , and summation over ␥ is carried out for any fixed values of , but provided that =␥ + ␥␣ . In this case for the first sum the vector ␥ must belong to the set ␥ , whereas for the second sum there are no such limitations. If the vector function x␥ (y, t) depends on time almost periodically uniformly in y and is represented by a Fourier series, the required function u␥ (y, t) can be written in the form
(1.6) where ␥ (y) is the initial function and {m }(m = 1,2,. . .) is the spectrum of the function X␥ (y, t). Hence
where
(1.7) We proceed similarly with Eq. (1.5): we expand its right-hand side in a Fourier series in the explicitly occurring time, equating the oscillating part of the series of the partial derivative with respect to time of the function u (y, t). We will have
(1.8) Thus, the first approximation has the form
(1.9) The second approximation includes terms of the second order of infinitesimals, which leads to an increase in the approximation accuracy. The equations of the second approximation have the form
(1.10) provided that the vector function Y (y) is evaluated using formula (1.8). Reverting to the initial variable x, we will write approximate solutions of Eqs (1.1). The first approximation to the required solution has the form
where y(t, ) is the solution of Eqs (1.9). The second approximation to the required solution can be represented in the form
(1.11) assuming that the function u␥ (y, t) was calculated in advance at the stage when investigating the first approximation. Here y(t, ) is the solution of Eqs (1.10). 2. The accuracy of the approximation The generalized Bogolyubov theorem. The problem of the accuracy of the approximation is central in the averaging method. It is inseparably connected with the problem of the value of the time interval in which the solutions of the averaged and rigorous systems of differential equations are compared. In the classical case of a single small parameter, the accuracy of the k-th approximation is k (k is the maximum order of the retained terms of the right-hand side), while the time interval is a quantity of the order 1/, since the quantity t ∼ 1 is the characteristic time for the system of equations in the Bogolyubov standard form. The situation is different when Eqs (1.1) contain several parameters. Here there is no unique characteristic dimension of time, and there is no a priori clarity as regards the accuracy of the approximation, since the order of smallness of the function, which depends on several parameters and the time, is not defined. We will fill this gap by assuming that the comparison interval T* is asymptotically long, i.e., having the form T* = L/␣() provided that ␣() → 0 as → 0. We know that the scalar function q() is of the k-th order of smallness in , if the following limit equality is satisfied
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Corollary.3 If q() = O(k ), then for any ␦ > 0 as small as desired, constants A1 (␦) > 0, A2 (␦) > 0 exist such that when 0 < ≤ ␦ the function q() satisfies the condition of the band
We will generalize this result to the case of a vector function, which depends on m small parameters and the time. To do this we will introduce the additional vector ␦ = (␦1 ,..., ␦m ) with positive components ␦j and the vector s = (s1 ,..., sm ) with non-negative integral arguments sj . Definition 1. We will call the function q(t, ) a function of the k-th order of smallness in in the time interval [0,L/␣()], if for any small vector ␦ with positive components ␦j > 0, constants A1 (␦) ≥ 0, A2 (␦) > 0 and two k-th order forms
exist that are positive definite in the positive cone {:j > 0, j = 1,2,. . .,m}, such that the band condition (2.1) is satisfied when
We will call the function q(t, ), which satisfies Definition 1, O-large of the k-th order of smallness in in the time interval [0,L/␣()] and denote it by Ok (). The following assertion also holds. Lemma.
We will assume that the function q(t, ) is such that the following non-zero limit, bounded and uniform in t, exists
Then q(t, ) is a quantity of the k-th order of smallness in the time interval [0, L/␣()]. We will assume that the functions X␥ (x, t) (|␥| = 1, ␥ ∈ ␥ ), linear in j , are defined in a certain open connected region D’ of the variable x, when t ∈ [0,∞]. We will assume that the functions X␥ (x, t) satisfy the following conditions in a certain subregion D × [0, ∞), D ∈ D’]: 1) the condition of continuity in t, 2) the Lipschitz condition in x with constants ␥ , independent of x and t:
3) the condition of uniform limitation in the region D × [0, ∞), i.e.
and the conditions, in addition to these, 4) the condition of boundedness, uniform in u ∈ D, of the integrals
in the case of almost periodic dependence of the functions X␥ (x, t) on time; 5) the condition that limits (1.7) exists, uniform in u ∈ D, if the functions X␥ (x, t) depend arbitrarily on t. ¯ (u) is an almost Note that condition 4 is equivalent, by Bohr’s theorem, to the condition that the integral of the function X (u, t) − X periodic function of t. It is satisfied for the quasi-periodic function X␥ (u, t) if its Fourier series contains a finite number of terms and there are no resonances. In the more general case when the Fourier series of the function X␥ (u, t) contains an infinite number of terms, the behaviour of the integral depends very much on the conditions for the following series to converge
Thus, if this series converges uniformly in u and t, the integral is uniformly bounded. Non-uniform convergence with respect to t gives examples of an unbounded increase of the integral.4–6 It also follows from Hilden’s papers that the integral of the scalar function f(1 t,
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2 t), which is 2-periodic, depends on each argument j = j t and has zero mean, is a quasi-periodic function of t (consequently, bounded) for almost all irrational values of ␣ = 1 /2 . The case of the divergence of the series has only been investigated to a small extent. We will introduce the following notation
(t,) = ||x(t, )–u(t, )|| is the approximation error, and * is a vector, the components of which are the small parameters j occurring in the first approximation of Eqs (1.1). The generalized Bogolyubov theorem. If conditions (1-3) are satisfied for differential equations (1.1), then for any L > 0, > 0 specified in advance one can obtain a positive scalar number 0 (L, ), such that if u = u(t) is a solution of Cauchy’s problem for the averaged equations
defined in the interval 0 ≤ t < ∞ and belonging to the region D together with its -neighbourhood, then for any , ||||<0 and for any t from the asymptotic time interval (2.2) the following estimates of the accuracy of the approximation hold. 1◦ . X␥ (x, t), ␥ ∈ ␥ , are T-periodic functions of the time t, then (2.3) 2◦ .
X␥ (x, t), ␥ ∈ ␥ are almost periodic functions of t, uniform in x, and condition 4 is satisfied, then for any as small as desired (2.4)
where () is the general -almost period of the functions X␥ (x, t). 3◦ . X␥ (x, t), ␥ ∈ ␥ are arbitrary functions of time and condition 5 is satisfied, then (2.5) where
Inequalities (2.3) and (2.5) are a generalization of the well-known estimates of the classical case of a single small parameter.7 Estimate (2.4) is unknown; we will prove it for the classical case, since the generalization to the case of m small parameters is obvious. We fix . Suppose t* = L*/ is the instant when the phase curve intersects the boundary of the region D and () is the almost common period of the functions X␥ (x, t). The following estimate of the accuracy of the approximation is known7
where x(t, ) and u(t, ) are the solutions of the rigorous and averaged equations, respectively,
We will divide the current time interval [0, t] into intervals of length (). Suppose N is the number of intervals which lie wholly inside the section [0, t]. We have
(2.6) (2.7) Consider Cauchy’s problem
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Then
(2.8) Hence
(2.9) N() ≈ L* /.
since The second term on the right-hand side of inequality (2.6), by virtue of condition 4, allows of the estimate
Here u() is the piecewise-constant function of :
Hence we have the inequality
It is easy to see that in this formula we can replace L* by as large a value of L as desired, if < 0 , where
In the case of m independent small parameters, we have the inequality
Assuming ␣() = ||* ||, we will have
Hence accuracy estimate (2.4) follows. Therefore, the generalized Bogolyubov theorem guarantees that, for periodic and almost periodic cases in the asymptotic time interval (2.2), the accuracy of the approximation has a value of the first order of smallness in , by virtue of Definition 1. It has been shown,8 that the construction of the function ␣() based on physical considerations is not always correct, and hence one must choose the function ␣() in the form ||*||. Note that, along different curves of the space of small parameters, the value of the accuracy of the first approximation and the value of the asymptotic time interval may vary considerably. Thus, for the case of two independent parameters and the accuracy of the √ approximation along the curve = , = 2 reaches values of the order of and falls sharply to a value of the order of on the curve = , √ √ = . The asymptotic time intervals on these curves are of the order of 1/ and 1 , respectively. 3. A pendulum of variable length on a vibrating base We will consider the problem of the oscillations of a mathematical pendulum of variable length on a vibrating base. As is well known, vibration has a considerable effect on the dynamics of a pendulum of constant length, stabilizing its upper equilibrium position at a high vibration frequency.9,10 Some conditions for the stability of the upper equilibrium position for a pendulum of variable length have also been obtained.11,12 Note that pendulum models find applications in problems of investigating the wave motions of a fluid in the fuel tanks of carrier rockets.13 Formulation of the problem. The equations of motion. We will assume that the suspension point of the pendulum executes harmonic oscillations along the vertical y axis: y = asint. In movable axes, connected with the vibrating base, the equation of the pendulum oscillations has the form
Here l(t) is the length of the pendulum, which varies harmonically with time, l0 is the mean value of the pendulum length and ae is the translational acceleration.
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We will introduce three small parameters ␥, and and a new time by the formulae
Here ω0 = g/l0 is the frequency of natural oscillations. Hence, the vibration frequency is assumed to be large, whereas the amplitudes of the oscillations of the suspension point of the pendulum and its length are assumed to be fairly small. We will change to a new time and expand the right-hand side of the equation obtained in series in small parameters. We have
A derivative with respect to is denoted by a prime. In order to convert the equation of the oscillations to a standard system, we will use the following non-linear replacement of variables1
The converted equations take the form
(3.2) where
The parameter = ␥/ takes values of the order of unity if the region being investigated is limited by the inequality ␥ ≤ k, since in this case satisfies the band condition
The equations of the first approximation of the averaging method. We will obtain the averaged equations of the pendulum oscillations, assuming x = (,) . We reduce system of equations (3.2) to the form (1.1) (3.3) Here
The averaged equations of the first approximation have the form
(3.4) in system (3.4)). of the classical case (the parameter does not occur They are identical with the averaged equations √ √ It is well known,1,10 that when > 1/ 2 the equilibrium = is unstable, while when < 1/ 2 it becomes stable and two unstable equilibria occur in its neighbourhood, namely,
√ The bifurcation surface has the form ␥ = 2 and is a ruled surface in three-dimensional space of the small parameters , and ␥, and hence the classical averaging scheme, based on linear reduction to a single parameter, was used in Ref. 1. In the initial variables and ’ the upper equilibrium position of the pendulum = , ’ = 0 corresponds to the stationary value = , whereas the periodic motions of the form (3.5) of the upper equilibrium position of the pendulum occurs through correspond to the equilibria = 1 and = 2 . Hence, stabilization √ bifurcation of the stationary point = , ’ = 0 when = 1 2, which is accompanied by the occurrence of two unstable periodic motions (3.5).
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Bearing in mind that = 0 l0 /(a), we also arrive at the conclusion that stabilization of the equilibrium is observed for fairly large values of , which satisfy the inequality
Note that the parameter has no effect on the stability of the equilibrium in the first approximation of the averaging method, but in the second approximation its effect is considerable. The equations of the second approximation of the averaging method. The non-resonance case. We will now construct and analyse the second approximation. According to formulae (1.10) and (1.11) in the second approximation we have (3.6) The vector y must be determined from the following system of equations
Here we have used a notation
The expressions for Y ( = 110, 101, 011) are calculated from formulae (1.8), where for each we carry out summation over two values of the index ␥: ␥ = 100 and 010, if = 110, ␥ = 100 and 001 in the case when = 101, and ␥ = 010 and 001 when = 011. Calculations show that
Here the average of the function X110 undergoes a discontinuity, if = 2. Hence, when = 2 parametric resonance occurs in the system. For a start we will consider the non-resonance case = / 2. Calculations using formulae (1.8) show that all the vector functions Y (y) are equal to zero:
Hence, the averaged equations of the second approximation are identical with the averages of the first approximation (3.4). Consequently, the previous conclusions remain in force, and only the values of the variable j , corresponding to the stationary solutions = j , = 0 of Eqs (3.4) are made more precise. Thus, if k = / 1, we have, on the basis of formulae (1.6) and (3.6)
In the case when k = 1
The equations of the second approximation in the neighbourhood of resonance. Oscillations at strict resonance = 2. We will assume that the resonance equality is satisfied approximately: / – 2 = , where is a small quantity of the first order of infinitesimals, i.e., = a, where a is a constant quantity of the order of unity. Introducing the additional slow variable , which satisfies the equality
we obtain, after averaging, equations of the second approximation, which describe oscillations in the neighbourhood of the resonance = 2:
Hence it is clear that (3.7) In the case when the resonance relation = 2 is satisfied rigorously, the parameter is equal to zero, and hence Eq. (3.7) takes the autonomous form (cos = 1), and we have two types of steady motions, described by the equalities
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The second type of equilibrium depends on the parameter , which leads to the occurrence of new effects in the behaviour of the system. Thus, if varies in the limits
the first type of equilibrium predominates, completely suppressing the second type. The equilibrium = 0 is then stable while the equilibrium = is unstable. The situation changes as soon as reaches the boundary values of this interval. A complex equilibrium appears, consisting of the solution = of family 2 and of the solution = of family 1, if = 2 . A further reduction in to zero leads to the occurrence of two unstable equilibria
in the neighbourhood of the stable point = . Consequently, when = − bifurcation of the equilibrium = , akin to the bifurcation described above, is observed, and the resonance oscillations of the pendulum when < − are similar to the non-resonance oscillations in the case when 2 < 1/2. If = + , a complex equilibrium = 0 appears. It splits into two stable equilibria
in the neighbourhood of the unstable equilibrium = 0, if > + . It follows from a comparison with the results of previous investigations that the region of stability 2 < 1/2 of the upper equilibrium position = of the non-resonance case is preserved during resonance provided < − . In the opposite case the upper equilibrium position is unstable. Moreover, the lower equilibrium position is unstable when > − . Hence, periodic resonance oscillations of the length l(t) of the pendulum have a destabilizing influence on the upper and lower equilibrium states. In , ’ variables, periodic oscillations of the form
where
correspond to the equilibria 1 and 2 . Note that the bifurcation equality
describes a two-dimensional surface in three-dimensional space of the small parameters , and ␥. The surface is not ruled, and hence it is impossible to obtain it if we use the averaging method, based on linear reductions of the problem to a single parameter. Such a surface cannot be seen a priori, and hence any attempt to construct it using non-linear reduction is unsound. / 0). Oscillations in the neighbourhood of the resonance = 2. In conclusion we will briefly consider a case close to resonance ( = Equation (3.7), which describes near-resonance oscillations, contains a slowly varying parameter = . In this case the oscillations of the pendulum are close to non-resonance oscillations, if ∼ , which follows directly from Eq. (3.7): after averaging over “fast” time we obtain equation (3.4). In the case ∼ 2 this averaging does not occur. We must distinguish five cases:
The simplest cases to investigate are 2 and 3, since in these cases the resonance and non-resonance oscillations are similar, and a change in leads to a small shift in the stationary points 1 and 2 on the axis. A more complex evolution of the oscillations is observed in the remaining cases. In case 1 we have a transition from a strict resonance case, in which the motion of the pendulum is similar to the oscillations of an ordinary mathematical pendulum, to a non-resonance case, when its oscillations have a more complex form due to the presence of two unstable equilibria in the neighbourhood of = . Replotting of the phase portrait for a continuous increase in of quantities of the order of n (n ≥ 2) to occurs by splitting the separatrice, and the occurrence of a chaotic zone of oscillations in the neighbourhood of the equilibrium = , ’ = 0 and in the neighbourhood of the separatrice, following the evolution of this zone. The behaviour of the phase curves of a chaotic layer is extremely complex. It depends very much on the value of the parameter and on the accuracy with which the initial conditions are specified. Without pretending to absolute reliability of the description, we will give an approximate description of the reorganization of the oscillations, based on numerical calculations, which were carried out for different values of and different initial conditions.
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Fig. 1.
Consider case 1. The accuracy with which numbers with floating points can be represented is twenty significant figures, and the error of the method is 10−14 . We will fix the following values of the parameters
When = 10−4 the phase portrait of the oscillations is represented in the upper part of Fig. 1 (here and below we do not show trajectories corresponding to purely rotational motions and, in view of the symmetry, we only show the region ’ ≥ 0). The phase portrait of the oscillations practically coincides with the phase portrait of a mathematical pendulum. The width of the chaotic layer is small, and it cannot be seen in the scale of the figure. An increase in leads to the appearance of a complex lattice of loops of trajectories in the neighbourhood of the point = and in the neighbourhood of the separatrice (see the case = 0.0359; the chaotic layer in the region of the separatrice is not shown). When is increased further the width of the chaotic layer in the region of = increases to a value of the order of 0.02 ( = 0.104); in addition to this, the phase point, which performs oscillations in the neighbourhood of the point
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= , leaves it as time passes. Beginning with a value of the parameter = 0.11, the oscillations in the neighbourhood of = slow down so much that the phase point pertains to the region = during the whole time of motion, forming a dense network of trajectories. The value = 0.11 corresponds to this case, and additional curves are also represented, the initial data for which approximate to the values = 1 and = 2 , respectively (1 = 2.4981, 2 = 3.7851 are additional equilibrium positions in the non-resonance case). It can be seen that these additional curves form a zone of oscillations in the region of the points = 1 and = 2 , leaving them as time passes. A further increase in leads to a sharp reduction in the zone of chaotic oscillations in the region of the point = : already when = 0.112 the size of this zone is of the order of 10−7 , while when = 0.113 its dimensions are so small that it is practically indistinguishable from the geometrical point = , ’ = 0 (the contraction of a chaotic web to a point). The lower part of Fig. 1 corresponds to the case = 0.2. A new zone of oscillations is observed in the neighbourhood of the stable equilibrium = , ’ = 0 and there are trajectories similar to heterocyclic trajectories which connect the unstable “equilibria” = 1 and = 2 . These “equilibria” are small zones of oscillations, which the phase curves rapidly leave, making several rotations around the points = 1 and = 2 . Hence, the phase portrait of the oscillations takes a form similar to the phase portrait in the non-resonance case, for which the equilibrium = is stable, while the adjacent equilibria 1 and 2 are unstable. These results can be made more accurate by constructing the asymptotic approximation of the solutions of Eqs (3.2) for the case of large values of . References 1. Bogolyubov NN, Mitropol’skii YuA. Asymptotic Methods in the Theory of Non-linear Oscillations. New York: Gordon & Breach; 1963. 2. Volosov VM. Averaging in systems of ordinary differential equations. Usp Mat Nauk 1962; 17(6)(108):3–126. 3. Krasil’nikov PS. The averaging method as a procedure for separating the principal terms in series of solutions. In: Theoretical Mechanics. Collection of Scientific - Methodological Papers. Moscow: Izd MGU;2009:56–68. 4. Poincaré H. Sur les corbes définies par les équations differentielles. J Math Pures et Appl Ser 4 1886;2(2):151–217. 5. Poincaré H. Sur les series trigonometriques. Comptes Rendus Acad Sci 1885;101(2):1131–4. 6. Kozlov VV. Methods of Qualitative Analysis in Rigid Body Dynamics. Moscow; Izhevsk: RKhD; 2000. 7. Zhuravlev VF, Klimov DM. Applied Methods in the Oscillation Theory. Moscow: Nauka; 1988. 8. Krasil’nikov PS. The averaging of differential equations with two independent small parameters. Dokl Ross Akad Nauk 2001;436(3):332–5. 9. Jeffereys H, Jeffereys BS. Methods of Mathematical Physics. Cambridge: University Press; 1950. 10. Kapitsa PL. A pendulum with a vibrating suspension. Usp Fiz Nauk 1951;44(1):7–20. 11. Strizhak TG. Methods of Investigating “Pendulum”-Type Dynamical Systems. Nauka: Alma-Ata; 1981. 12. Vershinin BA, Litvin-Sedoi MZ. The stability of a pendulum of variable length with motion of the point of support. Izv Akad Nauk SSSR MTT 1973;1:40–2. 13. Rabinovich BI. Introduction to Spacecraft Carrier Rocket Dynamics. Moscow: Mashinostroyeniye; 1975.
Translated by R.C.G.