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5Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 4, pp. 607-623. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200406
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 70E55, 70H09, 70H14
On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance
A. P. Markeev, T. N. Chekhovskaya
The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincare, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra.
The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).
Keywords: nonlinear oscillations, resonance, stability, canonical transformations
Received October 31, 2020 Accepted November 16, 2020
This work was carried out under the grant of the Russian Science Foundation (project No. 19-11-00116) at the Moscow Aviation Institute (National Research University) and at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Science.
Anatoly P. Markeev [email protected]
Ishlinsky Institute for Problems in Mechanics RAS prosp. Vernadskogo 101-1, Moscow, 119526 Russia Moscow Aviation Institute (National Research University) Volokolamskoe shosse 4, Moscow, 125080 Russia
Tatiana N. Chekhovskaya [email protected]
Moscow Aviation Institute (National Research University) Volokolamskoe shosse 4, Moscow, 125080 Russia
1. Introduction
Consider a system of two identical pendulums moving in a homogeneous gravitational field. Each of the pendulums is a weightless rod of length £. Point masses weighing mg each are fastened to the ends A1 and A2 of these rods. The distance between the suspension points O1 and O2 of the pendulums is constant and equal to 5. The suspension points themselves are located on a horizontal beam which performs harmonic oscillations of small amplitude a along a horizontal straight line (the point O shown in Fig. 1 is a fixed point of this straight line). The pendulums are connected to each other by a linear elastic spring of low stiffness k. The distances O1B1 and O2B2 of the points of attachment of the spring to the pendulums from their suspension points are the same and equal to b. In an unstressed state the length of the spring is equal to the distance 5 between the suspension points. It is assumed that, the oscillation frequency uj of the beam is exactly equal to the partial frequency \fgji of small linear oscillations of each pendulum. The motion of the pendulums occurs in a fixed vertical plane passing through points O1 and O2.
Fig. 1. Coupled pendulums suspended from a beam that oscillates along a horizontal straight line (OlAi = 02A2 = a, 01B1 = 02B2 = b, 0i02 = 5, OOi = acoswi, w = yffii).
Let y1 and y2 be the angles made by the rods O1A1 and O2A2 with the vertical. For the kinetic and potential energies of the system of two pendulums we have the following expressions (the dot denotes differentiation with respect to time t):
T = T2 + T1 + To, 1
II = —mgt{ cos y>\ + cos cp2) + ^ fcA2,
(1.1)
T2 = ^ m£2(Lpi2 + ip22), T\ = —matuj sin wi(cos tpitpi + cos (p2(p2),
T0 = mw2a2 sin2 wt, A = \/[6(cos Lp2 — cos Lpi)]2 + + 6(sin Lp2 — sint^i)]2 — 5.
The system of differential equations given by the Lagrangian function L = T — n admits, for any parameter values, a family of partial solutions in which, for any t, the pendulums deviate from the vertical by equal angles = = y(t), where y(t) satisfies the second-order differential equation:
d2 y g a 2 . ,
—jjp + j sm ip = —uj cosujtcostp. (1.2)
This equation describes the motion of a mathematical pendulum in the presence of harmonic horizontal oscillations of its suspension point with amplitude a and frequency w.
The case w = sjg/i studied in this paper corresponds to the multiple 1:1:1 resonance, which has the property that the frequency of external periodic action on the system coincides with each of its two frequencies of small natural oscillations.
Nonlinear resonant oscillations of coupled pendulums without external periodic actions were investigated earlier in [1-3]. Forced oscillations at the 1:1:1 resonance were considered in the report [4], which contains an analysis of the problem of the existence and linear stability of possible types of periodic motions. Below we consider one of these motions, namely, that for which p1 (t) = p2(t). A procedure for constructing periodic motion is presented, its analytic representation is obtained and the nonlinear problem of stability is considered. The investigation relies on the classical and modern methods of nonlinear dynamics [5-11].
A systematic study of the problem of nonlinear oscillations of Hamiltonian systems in the presence of resonance goes back to the turn of the 20th century [12-14]. A short bibliography of works on the theory and applications of resonant problems in mechanics and physics can be found, for example, in [15]. References [16-19] address several model problems which are described by an autonomous Hamiltonian system at the 1 : 1 resonance and are important for astrophysics and celestial mechanics. In [15, 20] the 1 : 1 resonance was considered in the problem of the motion of a satellite, a rigid body, relative to the center of mass in a circular orbit. Nonlinear oscillations of a dynamically symmetric satellite at the 1:1:1 resonance in an elliptic orbit of small eccentricity were examined in [21-23]. The problem of the stability of an autonomous Hamiltonian system in the degenerate (transcendental) resonant case was studied in [24]. Multiple resonances, different from the 1:1:1 resonance, in a nonautonomous Hamiltonian system with two degrees of freedom were investigated in the recent series of publications [25-32]. General theoretical results obtained in these works were applied in investigating the motion of a satellite, a rigid body, relative to the center of mass and in analyzing the stability of Lagrangian solutions to the planar bounded elliptic three-body problem.
2. The Hamiltonian function and its expansion as a series
As generalized momenta corresponding to generalized coordinates pi, we take the dimen-sionless variables defined by the equations
1 dL .
P<Pi = o m T"7" = 2 •
mlyjgl dpi
From these equations we obtain expressions for generalized velocities:
& = w (pw + j sinwicost^ (i = 1,2). (2.1)
Instead of time t, as an independent variable we take the dimensionless variable
t = ut. (2.2)
The canonically conjugate variables pi, (i = 1,2) correspond to the Hamiltonian function H' which is calculated from the formula
H' = —(T'2 — To + II), (2.3)
on the right-hand side of which the variables yi should be replaced with their expressions (2.1). This yields
2 1 1
tti r 1 2 a. 1 / a \2 . 2 2 i
H = Z^lop<Pi ~COS(A + -j smr cos <pipVi + -(-) sin rcos yi\ +
i=1 2 * 2 * (2.4)
+ (COS <£>2 - COS(/?i)2 + [7 + (Sin (£2 - Sin^i)]2 - ^ } 2,
For ease of the calculations presented below to investigate the motions in which y1 = y2, it is worthwhile to make a canonical transformation (with a valence equal to 1/2) yi,pipi — ypi,i>^i using the formulae
= ), y2 = ^(yi ~y2),
). (2.6)
The new variables correspond to a Hamiltonian function of the form 1 2
H = ^(p2, +V2Pn) ~ cos yi cos y2 +
+ [^/4 sin2 yi sin2 y2 + (7 - 2 cos 971 sin^)2 - 7]2 + (2-7)
a _ ^ ^ „ „ . 1 /an2 2 ~ ~
+ - smr(j)w cos^i cosc£2 -?V2 sin^i siny2) + ^(-^j sin r cos 2y?i cos 2y2.
The motions under study (for which y1(t) = y2(t)) are given by the equation y2 = p>V2 = 0, and y?1(wt) is a solution to Eq. (1.2).
Let us introduce a small parameter e by setting
a = e3£, 3 = e2s (0 <e < 1). (2.8)
and make a canonical transformation yi,pVi — Qi,Pi (with a valence equal to e"2):
y>1 = eQ1, y>2 = eQ2, pV1 = eP1, pV2 = eP2. (2.9)
The new Hamiltonian function G can be represented by a convergent series in powers of e (the terms that do not depend on Q1, Q2, P1, P2 have been discarded):
G = Go + eG1 + e2G2 + e3G3 + e4 G4 + e5 G5 + .... (2.10)
The coefficients of this series are polynomials in Q1, Q2, P1, P2 and are 2^-periodic in t. For the first six of them one can obtain the following expressions:
12
= + 4 Gi = 0, (2.11)
i=1
G2 = -^Qi-\QjQ22-^Q42 + smrPl+sQ22, G3 = 0, (2.12)
G4 = + Ql) + ^QiQKQi + Ql)~\ sinr[PiiQ2 + Q2) + 2Q1Q2P2] -
1 ^2/0,^2 1 n,2\ n — 2s r\2r\i
(2.13)
-^sQ2(3Q2 + Q2), G5 = --Q(Qi2
The motion under study corresponds to the periodic solution Qi = Q*(r), Pi = P*(r) (i = 1,2) of canonical equations with the Hamiltonian function (2.10). In this case, Qï> = P2 = 0 and Q\ satisfies the second-order differential equation
d2Ql +Q\= e2 (cos r + WA -e4(l cos tQ? + ) + 0(e6). (2.14)
dT2 \ 6^1 J \2 120
If the function Q1 is found, then P£ is calculated according to the equation
p* = dQl _ .2 gin T + Q(£4y (215)
dT
3. Construction of a periodic solution
The periodic solution Q^(t) to Eq. (2.14) can be found using an algorithm based on the Poincare method for quasi-linear systems with one degree of freedom [6]. Set
Q\ = C1 sin T + C2 cos T + e2/2(t) + e4 f4(T) + ..., (3.1)
where c1,c2 are constants and /2,/4,... are 2^- periodic functions which need to be found. Substituting this expression for Ql into Eq. (2.14) and equating the coefficients at e2 and e4 in series expansions of its left- and right-hand sides, we obtain
d2 f2 1
+ /2 = cost + -(ci sinr + C2COST)3, (3.2)
dr2 " 6
d2U . ? 1 / , \2 + /4 = —-(ci smr + c2cost)
13
cos r - /2 + —(ci sin r + c2 cos r) 60
(3.3)
In order for Eq. (3.2) to admit a periodic solution, it is necessary and sufficient that the coefficients at sin t and cos t on its right-hand side be equal. This condition allows a unique definition of the constants c\ and c2. We find
ci =0, C2 = -2. (3.4)
For this choice of c1 and c2 the system of equations (3.2)-(3.3) can be written as
+ /2 = -^cos3r, (3.5)
^§ + /4 = 2 cos2 r (f2 - cos r + ^ cos3 r^ . (3.6)
From (3.5) we have
/2 = C3 sin r + C4 cos r + ^j- cos 3r, (3.7)
where c3 and c4 are constants. For such a function /2 the right-hand side of Eq. (3.6) can be written as
-^-[40c3 sin r + (I2OC4 — 105) cos r + A3 sin 3r + B3 cos 3r + sin 5r + B5 cos 5r]. 80
We do not present the expressions for the constant coefficients A3, B3, A5, B5 since they will not be required in what follows. From the condition that there are no harmonics sin t and cos t in the last expression, we obtain
c3 = 0, c4 = I. (3.8)
8
From (3.1), (3.4), (3.7) and (3.10) we have the following expression for the periodic solution QI(t) to equation (2.14) calculated with an error of order e4:
Q* = -2cosr + ^t2(21cosr + cos3r) + 0(s4). (3.9)
The function P£(t) corresponding to this solution is found from (2.15) and (3.9):
P* = 2 sin r — t2 (15 sin r + sin 3r) + 0(s4). (3.10)
18
We note that in the terms through degree three in e the constructed periodic solution does not depend on the parameter y contained in the fifth-degree terms of the expansion of the Hamiltonian function (2.10).
4. The Hamiltonian function of the perturbed motion and its preliminary transformation
To investigate the stability of the pendulums' motion described by the found periodic solution Q1 = Ql (t), P1 = Pl (t), Q2 = 0, P2 = 0, we introduce perturbations qj, pj (j = 1,2) by setting
Q1 = Q1 + q1, Q2 = q2, P1 = Pl + P1, P2 = P2. (4.1)
The equations of the perturbed motion are given by the Hamiltonian function $, which is calculated by the formula
* = G + (4.2)
dT dT
where G is the function (2.10), in which Qj, Pj (j = 1,2) have been replaced by their expressions (4.1).
The function $ is analytic in the variables qj, pj (j = 1,2) and 2^-periodic in t. Its expansion as a series contains no first-degree terms in qj, pj (j = 1,2). Neglecting terms that are insignificant for the equations of motion and do not depend on qj, pj (j = 1,2), one can write this expansion as
$ = $2 + $3 + $4 + o( (q2 + P2 + q2 + P2)5/2) , (4.3)
1
$2 = + Pi + 92 + pi) + A- cos2 T(li + (s - cos2 + (4-4)
= e2 COSTqf + COSTqiq22j +t4$3, (4.5)
$4 = e2
-¿(tfi + 02) - ^0102
+ e4$l. (4.6)
Here $£ (k = 2,3,4) is the form of degree k in q1, q2, p1, p2 with coefficients 2^-periodic in t.
For further analysis it is worthwhile to make the transformation qj,pj — Uj,Vj (j = 1,2), which eliminates in the form (4.4) its first four terms independent of the parameter e. This transformation is a rotation by angle t in the planes q\p\ and q2p2, is canonical and given by
qj = Uj cos t + Vj sint, pj = -Uj sint + Vj cos t (j = 1,2). (4.7)
The new Hamiltonian function F is obtained from the function (4.3) by discarding the first four terms from the form $2 and by subsequently replacing qj, pj with their expressions (4.7). The function F is of order two in e.
The function F can be simplified by making a canonical, near-identity change of variables Uj, Vj — Uj, Vj which eliminates the independent variable t from the terms of order e2 in the expansion of F as a series in powers of e. The explicit form of the change of variables Uj, Vj — Uj ,Vj will not be required in what follows and is therefore not presented. The new variables Uj, Vj correspond to the Hamiltonian function r of the form
r = r2 + r + r + 0((u2 + vf + u2 + vlf2), (4.8)
r2 = (-fu? - ^ + ^ul + + t~4r*, (4.9)
F3 = t~2 (^U22ui + ^V22UI + ju2v2vi + iuiv? + ^M? J + (4.10)
r4 = e2
+ 3ulu2 + U1V2 + «2^1 + 3vlv2 + V%vl) ~
1 ( ^ 4 i 4 i 4 \ 1
- ^ [Ui + U2 + Vl + V2 ) - ~UlU2ViV2
+ e4rl, (4.11)
where r£ (k = 2,3,4) is the form of degree k in u\, u2, v\, v2 with coefficients 2^-periodic in t. Note that — e4r£ (k = 2,3,4) are averaged (over t) sets of terms of order e2 in the functions (4.4)-(4.6), in which the transformation (4.7) was made and then Uj, Vj were replaced with Uj, Vj.
5. Analysis of the linear problem
The characteristic equation of the linearized system of equations of the perturbed motion is reciprocal, as is that of any linear system of Hamiltonian differential equations with periodic coefficients, and its roots (multipliers) are continuous in e [5, 6]. It immediately follows from this fact and from (4.9) that, if the parameters (4s — 3) and (4s — 1) have opposite signs, then at sufficiently small e the charaacteristic equation has a root whose absolute value is larger than 1. Therefore [5], in the interval
13
4 < * < 4 (5.1)
the periodic motion of the pendulums is Lyapunov unstable (not only by virtue of the linearized equations of the perturbed motion, but also in a complete nonlinear system with the Hamiltonian function (4.8)).
If 1 3
0 < s < - or s ^ -, (5.2)
44
then, at small e, all multipliers have absolute values equal to 1, and a rigorous solution to the problem of stability requires an additional (nonlinear) analysis of the equations of the perturbed motion.
For values of s from the intervals (5.2), the characteristic exponents defined by linear equations with the Hamiltonian function (4.9) will be (for small e) purely imaginary quantities ±fA1, ±i\2, where i is an imaginary unit and A1, \2 are real numbers of any sign. If A1, \2 are such that
k1 A1 + k2 A2 = N, (5.3)
where N, k1, k2 are integer numbers such that \k1 | + |k21 = k, then the system exhibits a resonance of order k. In the stability problem, the most important resonances are those of lower orders (through order four).
To investigate the nonlinear problem, we first need to make a canonical transformation reducing the quadratic part r2 of expansion (4.8) to normal coordinates. For this we make a linear change of variables u1,u2,v1,v2 — {1,{2,n1 ,n2 by the formulae
U1 = -<1, U2 = —[¿{2, V1 = a-1n1, V2 = /3-1 V2, (5.4)
a=i> P= S 1
4s 3
This transformation is canonical with a valence equal to -1. It transforms the function r2 to a function r2 of the form
f 2 = t~2 wi^? + ni) - i aW2(e22 + 4)] + t-4f *2, (5.5)
where
1
uji = -j- ~ 0.433013, uj2 = - s/(4s - l)(4s - 3), (5.6)
and a = —1 in the region 0 < s < 1/4 and a = 1 in the region s > 3/4. In the interval 0 < s < 1/4 the parameter oj2 decreases monotonically from the value \/3/4 to zero, and in the interval s > 3/4 it increases monotonically. Graphs of the function w2(s) are presented in Figs. 2 and 3.
If we eliminate the value s = 1 which lies in the region s > 3/4 and for which w1 = w2 = = \/3/4, then for the other values of s inside the regions (5.2) of linear stability at small t no first- or second-order resonances are possible. For such parameter values, we use a linear real canonical transformation n1,n2 — x1,x2,X1 ,X2, 2^-periodic in t and analytic in e, which can be constructed, for example, by the Deprit-Hori method [8, 33]. Then the Hamiltonian function of the perturbed motion reduces to the form
H = H2 + H3 + H4 + O((x1 + X2 + x2 + X22 )5/2), (5.7)
where
H2 = \ \i{x\ + X2) + i A2(.t2 + X2), (5.8)
A1 = e2 [W1 + O(e2)], A2 = e2[—au2 + O(e2)], (5.9)
and the third- and fourth-degree terms H3 and H4 in the expansion (5.7) are given by the equations
H3 = e2
33/4 31/4 33/4 a2
Xix2 + irr (^i + + 6x2X1X2) + —xix\
+ e4H*3,
24a2 24
0 0.05 si s2 0.25 S
Fig. 2. Dependence of the frequency w2 on s at 0 ^ s ^ 1/4. W2 1.5 H
1 -
0.5-
0-rn-1-1-r
0.75 s3 S4 s5 1-2
s6 1.6 s7 2.0 S Fig. 3. Dependence of the frequency w2 on s at s > 3/4.
H4 = e2
6W + W2 + 9xi2)x'+ + + 24X1X2X1X2 +
+ 9x4 + 6x2x2) +
32
! o4
-tâ+xh + ^xi
+ e4H*A.
Here, HI and HI are forms of degrees three and four in x\, x2, X\, X2 with coefficients 2-n-pe-riodic in t .
6. On possible resonances
Taking into account expressions (5.9) and (5.6) for Ai, X2 and assuming that the dimen-sionless parameter s (which characterizes the stiffness of the spring connecting the pendulums) cannot be arbitrarily large, we find that at small e the resonances (5.3) inside the regions (5.2) of linear stability take only place when the integer N on the right-hand side of relation (5.3) is zero. If the order of the resonance does not exceed 4, then in the plane se at small e there exist
only seven resonant curves. In Figs. 2 and 3, they emanate from the corresponding generating points (si, 0) of the abscissa axes.
In the region 0 < s < 1/4, two resonances (see Fig. 2) are possible. For these resonances:
1 1 r- n 11^
1) wi = 2u2, si = - - - V7 ~ 0.169281; 2) ui = 3u2, s2 = - - - V3 ~ 0.211325.
2 8 2 6
And in the region s > 3/4 five resonances (Fig. 3) take place for which:
3) wi = 3w2, s3 = I - I 73 ~ 0.788675; 4) ui = 2w2, s4 = J + i V7 ~ 0.830719; 2 6 2 8
5) oji = oj2, s5 = 1; 6) uj2 = 2wi, s6 = ^ + j a/13 ~ 1.401388;
7) uj2 = 3wi, s7 = i + i v/7 ~ 1.822876.
The value of the parameter s on the ith resonant curve differs from its value si at the generating point by values of order e2.
7. A nonlinear problem inside the regions (5.2).
The nonresonant case
Let the parameter s lie inside the regions (5.2) and suppose that its value is not resonant (s = si (i = 1,2,..., 7)). Using a real near-identity 2^-periodic (in t) canonical Birkhoff transformation [7] x1,x2,X1,X2 — Vi,y2,Yi,Y2, we reduce the Hamiltonian function (5.7) to its normal form through terms of degree four in yi, y2, Yi, Y2. In the symplectic polar coordinates rj, ^j,
Vi = yffii sin Yi = V^J cos i>3 U = !> 2)
/ J DUl t/yj
the normalized Hamiltonian function can be written as
„2
H = \iTi + \2f2 + C20r2 + C11T1T2 + C02r2 + 0((v\ + f2)5/2) ,
(7.1)
(7.2)
where A1, \2 are given by equations (5.9). But finding constant coefficients cj requires rather cumbersome calculations. These calculations have been performed using methods of computer algebra, and the following expressions have been obtained for the coefficients cj :
C20
Cii
—a
8s - 3)2
1536^2 (w2 - 4w|)
+ O(e* )
CQ2 = £
(8s - 3)(1280s3 - 1376s2 + 264s - 9) 24576wf(wf^4w|)
+ O(e* )
(7.3)
In addition to considering the Lyapunov stability, we will also analyze the stability for most (in the sense of Lebesgue measure) initial conditions Uj(0), Vj(0) (j = 1,2) and the formal stability.
On the stability for most initial conditions (see [9, 34-37]). If the inequality
Cn - 4C2Q CQ2 = 0
(7.4)
is satisfied, then most trajectories uj(t), vj(t) (j = 1,2) of the system with the Hamiltonian function (4.8) which satisfy the condition u2(0) + u2(0) + v2(0) + v|(0) < i at the initial time
2
2
£
£
t = 0 remain for all t ^ 0 in a small neighborhood of the unperturbed motion Uj = Vj = 0 (j = 1,2). The set of initial conditions corresponding to the above-mentioned "majority" has a relative measure that is not smaller than 1 — O(p1/4).
However, the trajectories starting in an addition to the set of "most" initial conditions which has a relative measure of order /j,1/4 can, generally speaking, go arbitrarily far away from the initial position (this phenomenon is known as "Arnold diffusion"). But it was shown in [38-40] that for almost any system with a Hamiltonian function of the form (4.8) there exist positive numbers ^ and 52 such that the difference of the quantity u2(t) + u2(t) + v2(t) + v|(t) from its small initial value does not exceed a quantity of order for all values of t satisfying the inequality
0 <t < exp(^-^2).
This implies that, if there does exist "Arnold diffusion", it occurs exponentially slowly.
Based on the formulae (7.3) and (5.6), the left-hand side of inequality (7.4) can be rewritten as
c2i — 4c20C02 = e4 [d + O(e2)], (7.5)
where the parameter d does not depend on a and has the same expression in both regions 0 < s < 1/4 and s > 3/4:
(8s - 3)(81920s5 - 169984s4 + 110336s3 - 22944s2 + 360s + 243)
9437184^2 (w2 — 4w|)2 ' ( }
When e is small, the inequality d = 0 is [9, 37] a sufficient stability condition for most initial conditions.
1. Analysis shows that in the region 0 < s < 1/4 the function d(s) defined by equation (7.6) is positive for all values of s (recall that the resonant values si (i = 1,2,..., 7) are not considered in this section). Figure 4 shows a graph of the function d(s) in the interval 1/8 <s< 1/4. The curve d(s) has two vertical asymptotes: s = s1 (resonance u1 = 2w2) and s = 1/4 (the boundary of the region); the point s = 0.216774 is a point of local minimum of the function d(s), at this point d = 2.644152. We also note that d(0) = 1/768 ~ 0.001302 (Fig. 4 does not show the point with coordinates 0 and d(0)).
Thus, if s belongs to the interval 0 < s < 1/4 (such that s = s1,s = s2), then for sufficiently small e the periodic motion of the pendulums is stable for most initial conditions.
2. The behavior of the function d(s) with s > 3/4 is more complex (see Figs. 5 and 6). The graph of the function d(s) has two vertical asymptotes: s = 3/4 (the boundary of the region) and s = s4 (resonance u1 = 2w2). In the interval 3/4 < s < s4 the function d(s) is positive and has the minimal value d = 0.441034 at the point s = 0.770815 (Fig. 5); when s4 < s < 1, the function decreases monotonically from infinitely large values to the value d(1) = 115/186624 ~ 0.000616. When 1 < s < s* = 1.004417, the function d(s) is positive, and when s > s*, it is negative
(Fig. 6); when s = s*, the function d(s) vanishes. The point s = 1.279761 is a minimum point at which d = —0.006942. As s increases, the graph of the function d(s) approaches from below the horizontal asymptote d = —5/1152 ~ —0.004340.
It follows from the analysis of the function d(s) that in the region s > 3/4 outside the resonant points si (i = 3,..., 7) and the point s*, at which d = 0, stability takes place at small e for most initial conditions.
On the formal stability (see [41-48]). The concept of formal stability was introduced by J. Moser for systems possessing a sign-definite formal integral which is a (possibly divergent)
Fig. 4. Graph of the function d(s) for 1/8 <s< 1/4. Fig. 5. Graph of the function d(s) for 3/4 <s< 1.
power series. This concept is very important in investigating the stability on a finite (but very large) time interval. The presence of formal stability implies that the Lyapunov instability (if there is any in the problem of interest) does not occur if terms up to an arbitrarily high (but finite) order in the coordinates and momenta of the perturbed motion are taken into account in the expansion of the Hamiltonian function. In the presence of formal stability, even if there exist trajectories going far away from the unperturbed motion, the motion along them is extremely slow. The corresponding estimates were obtained by J. Moser [42, 43], C.L.Siegel [45], J. Glimm [46], N. Nekhoroshev [39, 40] and others (see [9, 48]). Formal stability is quite sufficient to solve the question of stability in most physical and technical problems described by Hamiltonian differential equations.
1. In the region 0 < s < 1/4 the quantities Xj (j = 1,2) from (5.9) at small e have the same signs (they are positive). Therefore [41], in this region, for all s (including the resonant values si and s2) formal stability takes place if e is sufficiently small.
2. In contrast, in the region s > 3/4 the quantities (5.9) with small e have different signs. Here, to solve the question of formal stability, we use the results of [47]. We assume that there are no resonances up to order four (i.e., s = si, i = 3,..., 7).
Consider the expression
D = C2oo\ + ciiaia2 + Co2&1 (oj = \Xj|, j = 1, 2). (7.7)
If the value of (7.7) does not vanish for o2 = ro1, where r is a rational number, then the unperturbed motion is formally stable [47].
Calculations show that, when o2 = ro1, the expression for D with an error of order e4 can be represented as
jj _ -2_243P_
" 16384w2(3-64w2)Q' 1 ;
where P and Q are polynomials of the form
P = 16r12 + 184r10 + 577r8 - 1508r6 + 290r4 + 108r2 - 15, (7.9)
Q = 8192s6 - 24576s5 + 38016s4 - 41984s3 + 28416s2 - 8928s + 783.
But it is easy to see that the polynomial P with integer-valued coefficients has no rational roots r = l/m, where l and m are coprime integers. Indeed, otherwise (see, e.g., [49]) l and m must be the denominators of the numbers 15 and 16, respectively, and a simple enumeration of l and m shows that the numbers r = l/m are not the roots of the polynomial P(r).
Thus, for all s > 3/4 (except perhaps for the values s = si (i = 3,..., 7) which have not been considered so far) the motion under study is formally stable.
8. A nonlinear problem in the region s > 3/4. Resonant cases
Resonance X1+2X2 = 0 (s = s4+0(e2)). Using the Birkhoff transformation, the Hamiltonian function (5.7) can be reduced to the following normal form:
H = Xir'i + A2r2 + xVnr2 sin (01 + 2ip2) + 0((r 1 + r2)2),
(8.1)
where
X =
(1 +
+ O(eA ).
2 123/4
For small e the parameter % is nonzero. Therefore [11], instability takes place.
Resonance 2X1 + X2 = 0 (s = s6 + 0(e2)). Since the variables x2, X2 appear in the term H3 of expansion (5.7) in powers whose sum is equal to two, the form H3 contains no resonant terms and can be completely eliminated using the Birkhoff transformation. The Hamiltonian function normalized through terms of degree four in y1, y2, Y1, Y2 will be given by an equation of the form (7.2) in which
C20 =
C02 =
C11 = £
2051 + 328 y/l3 17280
53 + ■
2160
+ O(£2 )
+ O(£2)
(8.2)
For the left-hand side of inequality (7.4) we obtain the expression (7.5), in which
d = -
80227 + 13064 y/T3 18662400
~ -0.006823.
(8.3)
It follows from (7.4), (8.2) and (8.3) that for sufficiently small e the form, quadratic (in r1, r2), in equation (7.2) will be sign-definite. Therefore [46], the motion under study is formally stable. It will also be stable for most initial conditions, since inequality (7.4) is obviously satisfied for small e.
Resonances X1+3A2 = 0 (s = s3+O(e2)) and 3A1+A2 = 0 (s = s7+O(e2)). These four-order resonances are investigated similarly to the above-mentioned third-order resonance 2A1 + A2 = 0. Here one can also completely eliminate the third-degree terms H3 in (5.7) and, similarly to the third-order resonance, the presence of commensurabilities A1 + 3A2 = 0 and 3A1 + A2 does not lead to a change in the structure of normalized terms of degree four in y1, y2, Y1, Y2. They are given by Eq. (7.2).
For the resonance A1 + 3A2 = 0 we have
C20
Cii
19 + ;
40
+ O£ )
C02
413 +
480
+ O£ )
d =
17843 + 9664 v/3
57600
0.600374,
and for the resonance 3Ai + A2 = 0
C02
C20 = £
3347 + 5
1
30240
+ O£)
Cii = £
113 +
7560
+ O(£2 )
d = -
1001437 + 154048 y/7 228614400
0.006163.
In both resonant cases, both formal stability and stability for most initial conditions take place.
2
2
£
£
2
£
2
£
9. Conclusion
The above analysis shows that the resonant periodic motion of the pendulums with small values of the amplitude of horizontal oscillations of their suspension points is Lyapunov unstable if the dimensionless parameter s, which characterizes the stiffness of the spring connecting the pendulums, lies in the interval 1/4 < s < 3/4.
In the interval 0 < s < 1/4 the motion is stable for most (in the sense of Lebesgue measure) initial conditions and is also formally stable.
If s > 3/4, formal stability takes place for all s except for one value, s = s4 = 1/2 + + V7/8 ~ 1.401388, which corresponds to the third-order resonance; for this value of s the motion under study is Lyapunov unstable. If s > 3/4 and s = s4, s = s* = 1.004417, then the motion is stable for most initial conditions.
The stability at the boundary points of the intervals (5.2) and at the point s = 1, which lies inside the second interval, has not been investigated. We note that, when s = 0, we arrive at the problem of the stability of resonant periodic oscillations of one mathematical pendulum under horizontal oscillations of its suspension point; it is well known (see the monograph [50, Chapter 4]) that these oscillations are Lyapunov stable. The problem of the stability of the above-mentioned periodic motions of coupled pendulums in the cases s = 1/4, 3/4 and 1 are a subject for a separate study.
Conflict of Interest
The authors declare that they have no conflict of interest.
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