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0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 481-512. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221101
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70H08, 70H12, 70H14, 70H15, 70M20
On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance
O. V. Kholostova
We consider the motions of a near-autonomous Hamiltonian system 2^-periodic in time, with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric resonance is assumed to occur for a certain set of system parameters in the autonomous case, for which the frequencies of small linear oscillations are equal to two and one, and the resonant point of the parameter space belongs to the region of sufficient stability conditions. Under certain restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of the system in the vicinity of the equilibrium are studied for parameter values from a small neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are obtained, which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case, for which the parameter values do not belong to the parametric resonance regions and their small neighborhoods, and both cases of secondary resonances are considered. The question of the existence of resonant periodic motions of the system is solved, and their linear stability is studied. Two- and three-frequency conditionally periodic motions are described. As an application, nonlinear resonant oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated.
Keywords: multiple parametric resonance, normalization, nonlinear oscillations, stability, periodic motions, satellite, cylindrical precession
1. Introduction
The study of near-autonomous Hamiltonian systems 2^-periodic in time, with two degrees of freedom, in the vicinity of a trivial equilibrium in cases of multiple parametric resonances was
Received June 03, 2022 Accepted October 19, 2022
This research was supported by the grant of the Russian Science Foundation (project 19-11-00116) and was carried out at the Moscow Aviation Institute (National Research University).
Olga V. Kholostova kholostova_o@mail.ru
Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, Moscow, 125993 Russia
started in [1-3]. This problem was developed in the series of articles [4-8], where for a number of cases of multiple parametric resonances the structure of instability regions (parametric resonance regions) of trivial equilibrium was studied in detail, conclusions were drawn about the existence and stability of resonant periodic motions in its vicinity, and conditionally periodic motions, if any, were described.
In this paper, we study another case of multiple parametric resonances, when for a certain set of parameters in the autonomous case the frequencies of small linear oscillations of the system in the vicinity of the trivial equilibrium are equal to two and one, and the resonant point of the parameter space belongs to the region where sufficient stability conditions for the trivial equilibrium are satisfied. It is also assumed that the Hamiltonian of perturbed motion does not contain terms of odd powers with respect to perturbations (which makes the existing third-order resonance insignificant), and the terms of even and odd powers n with respect to the small parameter in the nonautonomous quadratic part contain, respectively, only even and odd harmonics up to the number n. Such a structure of the nonautonomous part often occurs in specific problems of classical and celestial mechanics.
The boundaries of parametric resonance regions corresponding to secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies) are obtained, possible cases of the number of such regions and their relative positions are identified. It is shown that outside the parametric resonance regions and in their small neighborhoods, as well as in the region corresponding to the case of equal frequencies, the nature of the motions of the system is determined by the properties of reduced autonomous systems with one degree of freedom. In these cases, most motions of the complete nonautonomous system are two- and three-frequency conditionally periodic. In the case of zero frequency, the complete nonautonomous system exhibits motions that are time-periodic and analytic in a small parameter. The question of their existence, bifurcations and linear stability is solved.
As an application, nonlinear oscillations of a dynamically symmetric satellite (rigid body) about the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are studied in the case of a multiple parametric resonance of the type at hand. Analytical expressions are obtained for the boundaries of parametric resonance regions in the vicinity of a resonance point in a three-dimensional parameter space. The resonance periodic motions of the satellite are constructed, and a rigorous nonlinear analysis of their stability is carried out. Earlier in [2] (see also monograph [3]), the parametric resonance regions in this satellite problem were obtained in a planar section of the parameter space, in which the value of one of the parameters (the inertial parameter) is equal to its exact resonance value.
2. Problem statement
Consider the motions of a Hamiltonian system 2^-periodic in time, with two degrees of freedom, with the Hamiltonian H(qj, Pj, t; a, 3, e). Here qj and Pj (j = 1, 2) are generalized coordinates and their conjugate momenta, and a, 3 and e are parameters. The parameters a and 3 vary within a certain range; the parameter e is assumed to be small (0 < e ^ 1) and for e = 0 the system is autonomous.
Let the origin qj = Pj = 0 (j = 1, 2) of the phase space be the equilibrium position of the system, and suppose that in its vicinity the function H is analytic in its arguments and can be represented as
H = H2 + H3 + H4 + ..., (2.1)
where Hl are homogeneous forms of the Ith degree in qj and Pj.
Next, suppose that, for e = 0 and some values a = a0 and f = f0, the frequencies and w2 of small oscillations of the linearized system with the Hamiltonian H2 satisfy the equalities ш1 = 2 and ш2 = 1, that is, multiple parametric resonance occurs in the system.
We will consider the motions of the system for the parameter values from a small neighborhood of the resonance point e = 0, a = a0, f = f0 of the three-dimensional parameter space.
Note that, for the given frequencies, the system also has a third-order resonance ш1 = 2ш2. Its influence can be seen in terms H3 of the third degree (if any) in the autonomous part of the Hamiltonian of perturbed motion. If H3 = 0, then this resonance turns out to be insignificant. In this paper, which is the first stage in the study of the multiple parametric resonance at hand, we assume that H3 = 0.
We represent the forms H2 and H4 in the expansion (2.1) as series
H2 = H2o(qj, Pj, a, f) + eH2i(qj, Pj, a, f, t)+ e2H22(q3,
2 (2.2) H4 = H4o(qj, pj, a, f) + eH4i(qj, Pj, a, f,t)+ e H^qj, Pj,a,f,t) + ...,
where Hlk are forms of the Ith power in qj and Pj with coefficients constant (for k = 0) or 2^-periodic in t (for k = 1, 2, ...). The period-averaged value of the function H21 is assumed to be equal to zero ((H21) = 0). This condition is quite often satisfied in systems studied in classical and celestial mechanics.
In the resonance case considered, the autonomous parts H20 and H40 of the Hamiltonian can be transformed to the normal form (see, for example, [9])
H2o = Ql+pl + Uqi+pl), (2.3)
2
H40 = Г q + pj)2 +r2 (q? + pi) (q22 + pi) +Г3 (q22 + p22)2 . (2.4)
Here the same designations qj and Pj are left for the variables, a = 1 or —1, and Гк are constants.
For a = 1 and a = —1 in the autonomous case, the resonance point (a0, f0) lies, respectively, in the region where sufficient stability conditions and only necessary stability conditions for the trivial equilibrium of the system hold. A preliminary analysis shows that the properties of the system in these two cases have significant differences. In this paper, we will consider the case a = 1; the case a = —1 is supposed to be the subject of a separate study.
The purpose of this paper is to study the nonlinear oscillations of the Hamiltonian system at hand near the resonance point e = 0, a = a0, f = f0 for a = 1, with some restrictions on the structure of the nonautonomous part of the Hamiltonian discussed in the next section. The structure of the stability and instability regions of the trivial equilibrium is studied; the problem of the existence of periodic motions of the system is solved, and their linear and nonlinear stability is investigated). Conditionally periodic motions of the system are described. As an application, we consider the resonance oscillations of a dynamically symmetric satellite (rigid body) in the vicinity of its stationary rotation in a weakly elliptical orbit.
3. Normalization of a nonautonomous quadratic part
Consider the e-neighborhood of the resonance point given by the equalities
a = a0 + en1, ff = ff0 + e^1 + e2 ¡i2 + ..., (3.1)
where n1 and ¡¡k are constants. The quadratic part of the Hamiltonian in the terms of order e is
j — H(o) + H« Z21 — #21 + #21
#21 — h21) + H21) , (3.2)
(0) 2 2 2 2 #2l — C1 Ql + C2Q2 + C3P1 + C4P2 + C5Q1Q2 + c6Q1P1 + C7Q1P2 + C8Q2P1 + c9Q2P2 + C10P1P2,
#21) — s1(i)Q2 + «2(i)Q2 + S3(*)p? + S4 (t)p2 + «5(t)Q1 Q2 + «6(t)Q1P1 + +«7(t)Q1 P2 + Ss(t)Q2P1 + So(t)Q2P2 + S1o(t)P1 P2. Here the designations ck — akn1 + bk¡1 are introduced where ak and bk are some constants, and sk(t) are 2^-periodic functions of time.
Let us make a 2^-periodic in time canonical change of variables Qj, Pj ^ Xj, Xj (rotation transformation) of the form
q1 — x1 cos 2t + X1 sin 2t, P1 — —x1 sin 2t + X1 cos 2t,
(3.3)
q2 — x2 cos t + X2 sin t, P2 — —x2 sin t + X2 cos t.
As a result, the form (2.3) (for a — 1) in the autonomous part of the Hamiltonian is destroyed, while the form (2.4) is retained up to notation.
Next, we perform a linear canonical near-identical change of variables that differs from the identical one by 2^-periodic in t terms of order e and higher. This change of variables allows us to eliminate time in the quadratic form (3.2) transformed after the change of variables (3.3) and to simplify the structure of its autonomous part taking into account the existing resonant case. The described change of variables can be obtained, for example, using the Deprit-Hori method [9]; since its explicit form is cumbersome, it is not given here. Calculations show that the transformed quadratic form is (we leave the previous notation for the variables)
K21 = - lh) 4 + + Ih) xl + taXi + - xl + №2+
+ + ^ +(/?5" Ps)XlX'2 +(As" ^*1X2 ++ + (A, + P%)xlx2.
(3.4)
Here we have introduced the notation
Pi = \((sa(t) ~ Si(i)) COS At + s6(t) sin At),
P2 = \((si(t) -s3(i))sin4i + s6(i)cos4i),
Pa = \((s4(t) - S2(t)) cos 21 + s9(t) sin 21),
P4 = \((s2(t) -s4(i))sin2i + s9(i)cos2i),
P5 = \((S7(t) - S8(t)) Sint + (s5(t) +s10(t))cost),
Pe = \((s5(t) - S10(t)) sin 3i + (s7(t) + s8(t)) cos31),
Pi = \((s5(t) + S10(t)) sint- (s7(t) - s8(t))cost),
Ps = ¿(M*) + s8(t)) Sin31 - (s5(t) - s10(t)) COS31), where (...) denotes the average value over the period 2^ of the function in the brackets.
(3.5)
The form K21 is obtained under the assumption that the coefficients sk (t) in the nonau-tonomous form h2 1 are arbitrary periodic functions of time, and their expansions into Fourier series can contain any harmonics. In this general case, the transformed Hamiltonian depends on a large number of parameters.
Under additional assumptions about the structure of the initial quadratic form (2.2), the form K21 in (3.4) can be simplified. In this paper, we will assume that the terms H2k in the form (2.2) contain only odd or only even harmonics up to the kth harmonic inclusive if k is an odd or even number, respectively. Such a structure of the quadratic form often occurs in Hamiltonian systems. In the described case (see Eqs. (3.5)), most of the coefficients (3k vanish, only B5 and B7 remain nonzero.
We perform another rotation transformation with respect to the first pair of variables
B n
xi = Vicos + sin X1 = —y1 sin + Y1 cos = arct.g —^ + — (1 — sign f37)
B7 2
and leave the second pair unchanged. The transformed form K21 is
K2i = 7i (Vi + Y?) + 72 {vl + if) + PMYi ~ V1Y2), (3-6)
C1 "h c3 _ c2
"2 ' 72 ~
7i = ^, 72 = ^, ß* = \ß! + ßl
Using the Deprit-Hori method, we further carry out a normalizing transformation in terms of order e2 in the quadratic part of the Hamiltonian. The transformed quadratic part K22 has the form
K22 = ¿1 {vi + Y2) + (¿2 + OLl)vl + A + «2 )Y22 +
+MviV2 + Y1Y2) + A4(V2Y1 - V1Y2) + a0V2V2, (3.7)
A = + b3)fi 2 + £11^1 + + £22^1 + Co,
A = + 64)^2 + Cii^i + Ci2^i»7i + (22^1 + Co,
A3 = C1V1 + £2^1, A4 = (1^1 + (2^1,
where £ij, (ij, , (j, aj are constants not depending on n1, and ¡i2. Without loss of generality, we assume that a1 > a2.
To eliminate the term a0v2Y2 and the group of terms with A3 in this form, we carry out the
rotation transformation in both pairs of variables by the angle —0.5arctan ^a j and then rotation transformation by the angle + O (t2) in the first pair of variables. Leaving the previous notation, we rewrite the transformed form as ^d = sj(a1 — a2)'2 + aftj
KK22 = A1 y + Y12) + A4(V2Y1 - V1Y2) +
A2 + ^(c*! +a2 + d)
v2 +
A2 + ^(c^ + a2-d)
Y22. "(3.8)
4. Stability of the trivial equilibrium of a linearized system
Consider the question of the stability of the trivial equilibrium of a linearized system with the Hamiltonian (3.6). The characteristic equation of this system has the form
A4 + aA2 + b = 0. (4.1)
Its coefficients and discriminant d — a — 4b are as follows:
a — 4 (7l2 + 72) + 201, b — (02 — 47172)2 , d — 16(71 + n)2 [(71 — 72)2 + 02] ■ (4.2) When the conditions
a> 0, b> 0, d> 0 (4.3)
are met, the roots of Eq. (4.1) are purely imaginary, and the trivial equilibrium of the linear system under study is stable. If the sign of at least one inequality in (4.3) is reversed, then Eq. (4.1) has roots with positive real parts, and the trivial equilibrium is unstable, not only in the linear, but also in the complete nonlinear system.
It follows from the expressions (4.2) that for 471 y2 — 01 and 71 — —y2 conditions (4.3) are satisfied, and the trivial equilibrium of the linear system is stable.
If 47172 — 01 (the case of zero frequency) or 71 — —y2 (the case of equal frequencies), then the nature of the stability of the trivial equilibrium in the considered system of the first approximation in e is not determined, and in the system of the next (second) approximation in e, instability regions (parametric resonance regions) arise.
The condition 47172 — 01, taking into account the expressions for 71 and y2 from (3.6) and the structure of the quantities ck introduced at the beginning of Section 3, is represented as
(b13^1 + a13n1 )(b24 ¡1 + a24 n1) — 01, (4-4)
where the notation ajk — aj + ak, bjk — bj + bk is introduced.
Equation (4.4) describes a hyperbola in the (n1 ¡1 )-parameter plane. If b13b24 < 0, there are intervals of n1 given by the conditions
2^2 2 _ 4/^*^13^24
11 < ho, ho - — 77 ~ h x2>
(b13a24 — a13 b24 )
for which there are no hyperbola points and the case of zero frequency does not take place. For n1 — ±n10, the corresponding values of ¡1 on the hyperbola are
_ b13a24 + a13b24 ^
A10 ~ 2b~b
2b13b24
If b13b24 > 0, then the hyperbola points exist for all values of n1.
In the presence of zero frequency, the bifurcation values of ¡1 as functions of n1 are given by the equalities
±/ X -r?l(&13«24 + «13^24) ± [(b13«-24 ~ «13b24)2??l + ¥*b13b24] 1/2
Ati = Ail (h) =--T7—7- (4-5)
2b13b24
following from Eq. (4.4).
The case of equal frequencies 71 — —y2 corresponds to the straight line in the parameter plane n1, ¡1 given by the equation
^ = ^o(h) =-"f^r^h- (4-6)
b13 + b24
The straight line and the hyperbola do not have common points, their relative positions are presented in Figs. 1a, 1b for the cases b13b24 < 0 and b13b24 > 0, respectively.
The stability problem of the trivial equilibrium for points of the hyperbola and the straight line requires taking into account the second-order terms in e. The corresponding systems of the second approximation will be studied in Sections 6 and 7.
5. Investigation of the general case
Let us first consider the general case when the points (n1, n1) do not belong to the hyperbola or the straight line and their small neighborhoods. We return to the complete nonlinear system, assuming that the transformations described in Section 3 are performed in it. We put Vj = e1/2Zj, Yj = e1/2 Zj and introduce a new independent variable t = et. The transformed Hamiltonian of the system takes the form
G = G(0) (zj, Zj) + eG(1) (zj, Zj, t; e), G(0) (zj, Zj) = G21 + G40, (5.1)
where G21 and G40 are the forms K21 and H40 from Eqs. (3.6) and (2.4), with changes Xj = Zj, Xj = Zj and Qj = Zj, Pj = Zj, respectively.
The main (model) part G(0) (Zj, Zj) of the resulting Hamiltonian corresponds to an autonomous system with two degrees of freedom. We will call it the model system of the first approximation. The nonautonomous part G(1) (Zj, Zj, t; e) in (5.1) has a period 2ne in t; it is analytic in all its arguments in a sufficiently small neighborhood of the origin.
In the model system with the Hamiltonian G(0) from (5.1) (obtained taking into account the formulas (3.6) and (2.4)), we introduce the symplectic polar coordinates pj, Tj (j = 1, 2) as
Zj = ^TjSmtfij, Zj = ^2r~jCOS<Pj (j = 1, 2), (5.2)
and rewrite the Hamiltonian as
G(0) = 27lr! + 272r2 + v^sin^a - + + 4T2rir2 + 4T3r|.
We perform another canonical transformation
T1 = R1 - R2, T2 = R2, = $1, P2 = $2 + $1. (5.3)
The model Hamiltonian becomes
G(0) = 2^/1R1 + AT^l + 2(72 - 7l)E2 + 2/3, s/R2 sJRl - R2 sin $2+
+ 4(r2 - 2r1)R1R2 + 4r'R2, r' = r1 - ^ + r3. (5.4)
Further, we assume that r' = 0. In the system with the Hamiltonian (5.4), the coordinate is cyclic and R1 = c = const.
5.1. Properties of the reduced system
The values of R2 must satisfy the condition 0 ^ R2 ^ c. The case c = 0 corresponds to the equilibrium position at the origin. For c > 0, we put in (5.4) R1 = c, R2 = cg (0 ^ g ^ 1) and, discarding additive constants, we obtain the Hamiltonian of the reduced system with one degree of freedom. We pass to a new independent variable r' = 4c|r'\t and introduce the notation
1
2\r'|
72 7i+2(r2-2ri)
3
2c\r' |'
c
where k > 0, and i can take values of any sign. The reduced Hamiltonian takes the form
h = ¡j,q + a1g2 + ky/g-s/l — £>sin $2, <r1 = signT'. (5.5)
In the Cartesian coordinates
u = \/2pcosci>2, v = \/2psin<I>2 (56)
this Hamiltonian can be rewritten as
ti = (v2 + v2) - (v2 + v2)2 - ^Kvy/2 - a2 - v2. (5.7)
The motions of the system with the Hamiltonian (5.7) occur inside the circle u2 + v2 =
= 2. Two semicircles with ends (±\/2,0) on the horizontal diameter are the trajectories of the system. Endpoints (± \/2,0) are unstable equilibrium positions of the system; they are connected by a separatrix inside the circle given by the equation
V2 - u2 - v2 [2i + a1 (2 + u2 + v2)] = 2kv. (5.8)
Other equilibria of the reduced system satisfy the conditions cos $2 = 0 and
2{n + 2a1p)y/py/l -p = (t.2k{2P - 1) {<j2 = sin$2). (5.9)
If p, = —<t1, Eq. (5.9) has an obvious solution g = ^ for both values of a2. Other solutions of Eq. (5.9) can be defined as the abscissas of the intersection points of the graph of the function y = = /i(i?) = + 2aip)s/p\/l — p and the two straight lines y = a2K,(2p — 1) (for a2 = ±1). Characteristic variants of the graph of the function y = /1(q) and its relative position with the straight lines are shown in Figs. 2a-2c for the case a1 = 1. For a2 = —1 the solution of Eq. (5.9) is unique, and for a2 = 1 there can be one or three solutions.
In the case a1 = —1, the graphs can be constructed similarly, and the conclusions are reversed about the number of the solutions of Eq. (5.9) for a2 = 1 and a2 = —1.
Thus, for a fixed value a1 (in each specific problem) and two variants of the value a2, Eq. (5.9) can have two or four solutions. In the (k, ¡)-parameter plane, the regions with different numbers of equilibrium points are separated by a curve given by the equation
k2 = 3\i + ct1|4/3 — 3\i + ct1|2/3 — i2 — 2a11 (k > 0).
(a)
(b) (c)
Fig. 2. Graphical solution to Eq. (5.9)
-1-
-2-1
Fig. 3. Regions with different numbers of solutions to Eq. (5.9)
(a) (b)
Fig. 4. Phase portraits of the reduced system in the general
case
Figure 3 shows the regions for the case al = 1, marked with numbers 2 and 4, with two and four equilibrium points. The corresponding boundary curve in the case a1 = —1 is located in the upper half-plane and is symmetric with respect to the abscissa axis of the curve shown in Fig. 3.
The analysis shows that, in the case of a unique solution of Eq. (5.9), the corresponding equilibrium of the reduced system is stable. In the case of three solutions to this equation, two of them, the largest and the smallest, correspond to stable equilibria of the system, and the middle solution corresponds to an unstable equilibrium.
The phase portraits of the system with the Hamiltonian (5.7) for a1 = 1 are shown in Figs. 4a and 4b for points (k, j) from regions 2 and 4 in Fig. 3, respectively. The regions with equilibrium points corresponding to a2 = 1 and a2 = —1 are separated by the separatrix (5.8). If there are three equilibria in the subdomain, there is one more separatrix passing through the unstable point (see Fig. 4b). Other motions of the system are oscillations in the vicinity of stable equilibria or oscillations corresponding to the closed trajectories covering three equilibrium points (Fig. 4b).
5.2. The motions of the nonautonomous system in the general case
The motions of the reduced system generate the motions of the model system with two degrees of freedom with the Hamiltonian (5.4), and of the complete nonautonomous system with the same Hamiltonian, to which nonautonomous part eG(1) ($j, Rj, t; e) written in the new variables is added, with period 2ne in t.
5.2.1. Two-frequency conditionally periodic motions. Each equilibrium position of the reduced system with the Hamiltonian (5.5) lying inside the boundary circle g = 1 corresponds to a one-parameter (with parameter c) family of stationary rotations
Rl=c, R.2 = cq=1-cv2, $2 = <72|, Q/1 = ^^ = 27i + 8EiC + ^^+4(E2-2E1)^
of the model system with the Hamiltonian (5.4). Here the equilibrium value of g satisfies Eq. (5.9). Let Q = 0 and, in addition, let (taking into account Eq. (5.9)),
Then, for sufficiently small e, most of the described stationary rotations generate conditionally periodic motions of the complete system with the Hamiltonian (5.4) (to which a nonautonomous part is added), with a slow frequency Q and a fast frequency e_1 [10, 11].
In the three-dimensional space of parameters n1, M1, and c, there is a surface on which, taking
into account Eq. (5.9), the condition Q = 0 is satisfied. Moreover, in the case a2 = signr1,
0)
there exists a degeneracy surface qR2 = 0. Note that, for a2 = — signE1 inequality (5.10) always holds.
Under the condition a2 = signr1, two surfaces intersect at the points that make up the curve given parametrically (with the parameter g £ (0, 1)) in the form
ni =
Mr' + 2ri + r2) + Meri - r2)]g2 - bi3(8ri - r2) + 6b24ri]^ + 4bVir
4|r1|(624a13 - 613a24)x/ë(l - g)3/2
O.. JW' +r) +
11l = fi,
[Oi3(2r' + 2ri + r2) + fl24(6ri - T2)b2 - [ai3(8ri - + 6a24r^g + 4^ri
4|r1|(624a13 - bVia2A) Jg(l - gf /2
C: = ---
161^1(1-0)3/2-
5.2.2. Three-frequency conditionally periodic motions. For the parameter values corresponding to oscillations of the reduced system with the Hamiltonian (5.5), the motions of the model system with two degrees of freedom with the Hamiltonian (5.4) are two-frequency conditionally periodic. Let us describe the motions of the complete nonautonomous system generated by them, restricting ourselves to the case k = 1, ¡i = —1, a1 = 1 corresponding to the cusp on the boundary curve in Fig. 3. The phase portrait for this case is shown in Fig. 4a.
The system under study has the first integral (energy integral)
— g + g2 + s/g-\/1 — £>sin $2 = h0 = const. (5-11)
The values h0 = \ and h0 = — § correspond to stable equilibria for which g = while $2 = = f and $2 = — §, respectively. The first of these equilibria is a complex singular point of the
system. On the interval 0 < h0 < j, the system oscillates around the first equilibrium, on the interval — | < h0 < 0 it oscillates around the second one. The value h0 = 0 corresponds to the trajectories on the boundary circle and the separatrix. Outside the segment h0 G [—f, the motions of the considered system are impossible.
Let us integrate the equations of motion of the system in the oscillation domains. To this end, we write down the differential equation for g, excluding $2 in it using the relation (5.11). We have
dg d?
= -g4 + 2g3 + 2(h0 - 1)g2 + (1 - 2h0)g - h20.
(5.12)
In the oscillation regions, the fourth-degree polynomial from the right-hand side of the equation has two real roots
gi,2
= i ± 1(1 - 4 ho)1/4 (2 - sfl^W^ (ft > ft)
1/2
and two complex conjugate roots
£>3,4 = \ ± - 4ho)1/4 (2 + s/T^)
1/2
The oscillations of the system occur on the interval ft ^ 6 ^ ft-
Integrating Eq. (5.12) under the initial condition 6(t0) = 62, we obtain [12]
Q = 2 (f?i + 62) -r2(ei- ft) cn (u, k), u = (1 - Aho)1/4(t' -t'o), k = -(2- y/l - 4 h0)
l_cn(r'-r', i)
In particular, on the separatrix (for h0 = 0) we have g =-^—
In the above relations, the standard notation for the Jacobi elliptic cosine is adopted. The frequency of the oscillations obtained is given by the formula
1/2
W(h0) =
tf(1 - 4ho )1/4
2K (k)
where K(k) is a complete elliptic integral of the first kind.
(5.13)
7
-0.5 -0.25 0 0.25
-0.6 / ho
(b)
Fig. 5. Graphs of functions oj(h0) (a) and f(h0) (b)
The graph of the function w = w(h0) is shown in Fig. 5a. The punctured point on the curve (for h0 = 0) corresponds to the separat.rix. Note that, for h0 —> \ we have oj(h0) —> 0.
In the oscillation regions of the reduced system, we introduce the action-angle variables I, w, assuming that
I = 1(h) = <b gd$2 = 77- <fvdu. (5.14)
2n
Here, the integration is carried out along a closed trajectory, covering the stable equilibrium point in the considered oscillation region (see Fig. 4a). The function h = h(I) inverse to (5.14) is the Hamiltonian of the reduced system, written in the action-angle variables.
We also introduce the action-angle variables Ij, Wj (j = 1, 2) in the model system with two degrees of freedom with the Hamiltonian (5.4) (calculated taking into account relations k = 1, H = —1, and a1 = 1). Since is a cyclic coordinate, I1 = R1. For the variable I2 we have
I
1
2vr
j) R2 2 = j 2, h0) 2,
where the integration is carried out over a complete cycle of changes in the canonically conjugate variables $2, (or 0). Note that the function q(&2, h0) is calculated using Eq. (5.11). From the above relations, we find that. h0(I) is the function of the ratio ^ of the action
variables, i.e., h0 = U ("j2)-
In the new variables, the model Hamiltonian depends only on the variables I1, I2 and is represented as
77(0) 1T T \ _ o r2 1 nT-fTTf
G {I\, I2) = 2If [71 + '2T'U(X)], 71 = f + 2IV
(5.15)
From here we find the frequencies w1 and w2 of nonlinear oscillations of the model system
w,
dL
= 4
(7i + 2T'U)Ii — T
,dU ~d\J
w2
rJG{0)
dlo
4T'L
dU_
dX'
where
A = i
dU_ d\
= w(ho).
(5.16)
Suppose that. at. the initial time u1 / 0 and the ratio ^ is not. a rational number, that, is, the frequencies w1 and w2 are incommensurable. Then the motions of the considered model system with two degrees of freedom are conditionally periodic with these frequencies.
Consider the complete system with the Hamiltonian (5.4) and an added nonautonomous part. We rewrite this Hamiltonian in the introduced action-angle variables in the form
G = G{0\lI2) +eG{1\l1, I2, wu w2, t, e).
(5.17)
The function G^ is analytic in all arguments, 27r-periodic in w1, w2, and 27rt-periodic in r. We introduce the quantity
A
d2G{0) d2G{0) ( d2G{oy
di2 di2
dhdh
= 16
2
2
1
2
Taking into account formula (5.16) and calculating the second derivative
d2U dw dU y ^ dw
d\2 dU d\ v 0 dh0
we find that
4rV2A y/1 - 4h0K3(k)(3 + 4h0)
/1(^0) = 2ii'(fc) [ s/T^W0 - 1 - 4/7,0) - AE(k)s/T^W0, f2(h0) = K(k) (4/7,0 + 4/7,0 V^^K - 3) - SE{k)s/l^4h0h0
When the condition A = 0 is satisfied, there is a degeneration in the model Hamilto-
nian
G(0)(Ji, h)- This condition is fulfilled for points of the surface in the space of parameters 7l, r', and /70 given by the equation
7i _ j(h x _ /2(^0)
r7"/( o)""7№)■
The graph of the function f (h0) is shown in Fig. 5b. This function increases monotonically, changing from — | (for h0 = — f) to —1 (for h0 = j)-
At A = 0, most motions of the complete system generated by conditionally periodic motions of the model system are three-frequency conditionally periodic ones with two slow (of the order of unity) frequencies u1 and w2 and one fast frequency equal to e-1. The fraction of the phase space not filled with such trajectories is of order sje [13].
6. The case of zero frequency
6.1. Model Hamiltonian
Let us now turn to the study of the case of zero frequency. Let the coefficients of the Hamiltonian (3.6) satisfy the condition 47l72 = ft"2, then the quantities ¡i1 and n1 are related by Eq. (4.5). In this case, the dependence Y±(n1) has the form
-(&13a24 - a-13ft24)i?l ± [(¿>13^24 ~ ai3b24)'2r]j + ^*b13b24] 1/2
Ab0
±f S \u 13^24 ^13^247'/1 ^ LV 24 ^13^247 H*" 13^24 J
7l (Vl) = -77-, (6-1)
y24
and the dependence y2±(v1) is obtained from (6.1) by removing the minus in front of the first parenthesis in the numerator and replacing b24 with b13 in the denominator. Here the upper and lower signs refer to the cases ¡1 = ¡'+(n1) and ¡1 = ¡~[(n1 ).
Note that the quantities y1 and y2 are of the same sign; so, we can introduce the notation 7j = | and s = sign Yj. Further, in most formulas of this section, the argument n1 and the superscript of the quantities Yj±(n1) (j = 1) 2) are omitted.
Let us substitute ¡1 = ¡±(n1 ) into the form (3.8). The coefficients Ak are rewritten as
= \b13^2 + fi(Vi), A = ¿b24fi2 + /2(î7i), A4 = /4(r?!), (6.2)
where fk(n1) are some functions of n1.
We then pass to the e-neighborhood of the origin, setting
Vj = ezj, Yj = eZj (j = 1, 2),
and introduce a new independent variable t = et. The transformed Hamiltonian is represented as
Gi = G1 )(Zj, Zj) + e^g'i )(Zj, Zj, t; e), G1 )(Zj, Zj) = g2I) + e (G22 + G4o) :
(6.3)
(6.4)
G2i = S7i (-1 + Zl) + S72 {zi + Z\) + 2v/^(z2Z1 -
G22 = Z + Z2) + A4Z Z1 — zZ )+
+
A2 + +a2 + d)
zi +
+ +a2 - d)
7 2 72 .
The Hamiltonian Gi0) (Zj, Zj) is a model one in the considered case of zero frequency. The
(2)
part G1 ) (Zj, Zj, t; e) in the complete Hamiltonian is 2-^e-periodic in t.
6.2. Stability of a trivial equilibrium
Consider a linear system with the Hamiltonian g21) + eG22 from (6.4). The coefficients of the corresponding characteristic equation (4.1) and its discriminant are
a = 4(7i + 72)2 + O(e), d = 16(7 + 72)2 + 0(e),
(6.5)
b = 4e2
7i(a! + a2) + 2 ( - sA4 s/%^~2 + A2\ ) - 7?d2
+ O(e3).
For sufficiently small e, the stability condition for the trivial equilibrium coincides with the positivity condition of the main part (of order e2) in the coefficient b. This main part vanishes if
2 [Ail2 ~ sA4 sf^f2 + A27i = —{al + a2) ± d
Y1.
(6.6)
These equations define the boundaries of the parametric resonance region in the second approximation in e.
Taking into account Eqs.(6.2) for the coefficients Ak, we find four bifurcation values of the parameter ¡2 from Eq. (6.6). For the case ¡1 = ¡i+(n1 ), the instability region corresponds to the interval for ¡2, the boundaries ¡21 (n1 ) and ¡+2(n1 ) of which are given by the formulas
sp, /4(m) — 2/1(^)7+ (m ) + ( —2/2(^1) — «1 — «2 ± d) 7+(m )
¡+1,22 =
(6.7)
b13 7+(ni) + b247+(ni)
Here the upper and lower signs in front of d correspond to ¡+2 and ¡¡+i, respectively. To obtain the boundary values ¡-i(ni) and ¡-2(ni) for ¡2 for the case ¡i = ¡-(ni), one should replace 7+(ni) with 7j~(ni) (j = 1, 2) in the above expressions. For the parametric resonance regions we obtain
min (¡j^ ¡+2) < ¡2 < max (¡j^ ¡+2) and min ¡i, ¡22) < ¡2 < max (¡2l, ¡-2). Note that, due to Eq. (6.1), the denominator of the fraction in (6.7) can be rewritten as
ai3 O24)" ni + 4p; bi3 »2^' , (6.8)
where the upper and lower signs refer to the cases ¡i = ¡+(ni) and ¡i = ¡2(ni).
brilftm) + hilt(h) = ±2 [(6i3fl24 - ai3b24)2i?i + 0!h3bM]
Note also that in the case b13b24 < 0 for the boundary points (n10, ¡10) (see Section 4), the boundaries of the parametric resonance regions are not determined by the above formulas, since the coefficient of ¡2 in the expression for b in the formula (6.5) vanishes. In this case, we put n1 = n10 + £n11, and ¡1 = ¡10 in (3.1) and obtain the model Hamiltonian in the form (6.4). Here the values of j (j = 1, 2) are calculated for n1 = n10 and ¡1 = ¡10, whereas the coefficients Ak are calculated by the formulas (6.2) for n1 = n10, with the terms b13^2 and b24¡2 being replaced with a13n2 + b13^2 and a24n2 + b24^2, respectively. The coefficient b of the corresponding characteristic equation has the form as in (6.5), and the boundary values n21 and n22 of the parametric resonance region are defined by the right-hand sides of the formulas (6.7), with b13 and b24 being replaced with a13 and a24, respectively.
6.3. Nontrivial equilibria of a model system
Nontrivial equilibria of a model system with the Hamiltonian G^ (Zj, Zj) from (6.4) can
be obtained by equating to zero the partial derivatives of the function G^ with respect to Zj and Zj (j = 1, 2). We put in the resulting equations
Zj — Zj — Zjo + £Zj i + ..., Zj — Zj — ZjQ + sZj i + ... (j — 1, 2)
(6.9)
and select systems of zero and first approximation in e. Omitting simple but cumbersome technical details, we present the main points of the study of these systems.
The zero approximation system contains two pairs of coinciding equations, from which we
find
Z10 — —s-
= z20) ^20 — s~
1o.
(6.10)
We substitute these relations into the system of the first approximation and, after some transformations, excluding combinations of variables of the first approximation in e, we get two corollary equations for z10 and z20:
no
z20
4sF(»71) (7^0 + 724)) + 27ill'2 («■\f%Ai ~ + S7i72(2A2 + «1 + «2 - d)
teFfa) (71^10 + 72-22o) + 27i723/2 (sV%ai ~ VYi^) + Hlji^i + a, + a2 + d)
F (m) — T37i2 +T27172 +Ti72.
Since the expressions in square brackets in these equations cannot vanish simultaneously, this system can only have solutions for z10 =0 or z20 = 0. Taking into account also the relations (6.10), we find two pairs of sets of solutions for the zero approximation system in e. The first pair is given by the relations
Zio — 0, z20 — —
7i (bi3T2 + b247i) (^2 - ^21)
4
F (ni)
) ^10 — ~S ^z20) ^20 — 0
and exists (considering (6.8)) when
±s {ß2 - 2 F (ni) < 0.
The second pair is given by the relations
_ n 2 _ 7i (fti372 + &247i) (»2 ~ ¿4)
^20 — U) ^10
4
F(ni)
Z'20 — s~
no,
Zio — 0
(6.11)
(6.12)
(6.13)
and exists under the condition
±s(^2 - F(m) < 0. (6.14)
Plus and minus signs in formulas (6.11)-(6.14) refer to the cases ¡1 = ¡+ and ¡1 = ¡ithe quantities ¡¡+1 and ¡+2 in these formulas are calculated by the relations (6.7) for ¡1 = ¡+ and by similar ones for ¡1 =
If we now substitute solutions (6.11) into the first approximation system, then we obtain two pairs of coinciding equations for the quantities z11, z21, Z11, and Z21. Different equations are
S\fllZll ~~ VT2^21 = 0)
-2
-2s V72Z21 + VI1Z11) 72 +
2sv/%(2r172 +t2^1)z220 (2^
-A,
z20 = 0.
To determine the above quantities, one should consider the system of the next (second) approximation in e and proceed similarly to the previous one.
The values of z11, z21, Z11, and Z21 for the second set (6.13) can be found in the same way.
6.4. Stability of nontrivial equilibria
In the Hamiltonian G^, we introduce perturbations of variables with respect to the found equilibrium values zj = , Zj = Zj (j = 1, 2) and compose the corresponding characteristic equations of the form (4.1). Calculations show that for both pairs of equilibria the coefficient a and the discriminant d of these equations are represented as (6.5), and the coefficient b for the first and second pairs of the equilibria is
b = 16d71 (b1372 + b2471) (¡2 - ¡±1) e2 + O (e3)
and b = 16d71 (b1372 + b2471) (¡2 - ¡±2) e2 + O ,
respectively. Here the upper and lower signs refer to the cases ¡1 = ¡+ and ¡1 = ¡1-.
Taking into account the existence conditions (6.12) and (6.14), as well as the relation (6.8), we find from here that in the case ¡1 = ¡+(n1) for sufficiently small e the linear stability condition b > 0 yields the system of inequalities
F(m) < 0, s (¡2 - ¡+1) > 0 and F(m) > 0, s (¡2 - ¡+2) < 0 (6.15)
for the first and second pairs of equilibria, respectively. In the case ¡1 = ¡1 (n1), the first and second pairs of equilibria are linearly stable, respectively, at
F(m) < 0, s (¡2 - ¡1) < 0 and F(m) > 0, s (¡2 - ¡-2) > 0. (6.16)
When the signs of both inequalities are simultaneously reversed in the systems (6.15) or (6.16), the corresponding pair of equilibria becomes unstable, and when the sign of only one of the inequalities is reversed, the pair does not exist.
Thus, a change in the nature of the stability of nontrivial equilibria of the system occurs at the boundaries of the parametric resonance regions, and also when passing through the values of n1 satisfying the equation F(n1) = 0. Given the expressions for 7j and Eq. (4.5) this equation can be rewritten as
± (0-13624 - bi3bM) (r1624 - T3b213) 'ft sj(«13624 - &13&24)2^i + M^bl3b24 =
= (a13b24 - b13b24)2 (rb24 + ^3) n2 + 2&2b13b24 (r3b23 + r2b13b24 + r^). (6.17) Here the upper and lower signs correspond to the upper and lower signs in (4.5).
We square both sides of this equation and, after transforming, we obtain the corollary equation
r^13&24 - MnA4 + P*2(ai3- 6-3624)2 (r^^ + 4^3613^ + r3r262^
+ P4 ^3b23 + ^613624 + ^4) =0,
which is biquadratic in n1. Its discriminant is KK (r2 — 4r1 r2), where KK > 0.
For r2 < 4r1r2 the biquadratic equation has no real roots, i.e., the equation F(n1) = 0 has no solutions, and F(n1) has a constant sign coinciding with the sign of r3 (or r1). For r2 > 4r1 r2 the equation quadratic in n2 has two real roots. We select positive roots, one root when r1r3 < 0 and two roots when the following conditions are met:
r^3 > 0, r^2624 + 4Tr3&13624 + ^2^3 < 0.
Thus, a biquadratic equation can have 0, 1, or 2 pairs of solutions that differ in sign.
Note, however, that when n1 is replaced by —n1, the sign of the left-hand side of Eq. (6.17) is reversed, while its right-hand side is preserved. Therefore, out of two values n1 and —n1 of each pair of solutions, one satisfies Eq. (6.17) for the bifurcation value ¡j+ , and the other one satisfies the equation for n-. When passing through the described boundary values of n1, the nature of the stability of nontrivial equilibria of the system changes (or these equilibria disappear).
6.5. Periodic solutions of the complete system in the case of zero frequency
For the found equilibrium positions of the model system with the Hamiltonian G-0 from (6.4), two roots of the corresponding characteristic equations (4.1) have the order of unity, and the other two roots have the order of e. In the complete transformed nonautonomous system with the Hamiltonian G1 from (6.4), the frequency of a small t-periodic perturbation has order e-1.
Therefore, the nonresonant case of the periodic Poincare motion theory [14] takes place, and each nontrivial equilibrium position of the model system generates a unique solution, analytic in e and periodic in t (with period 2ne), of the complete transformed nonautonomous system. These solutions correspond to the motions of the initial system that are analytic in e and 2n-periodic in time t.
Since the characteristic exponents of the linearized equations of perturbed motion are continuous in e, the periodic motions of the complete system which are born from the linearly stable and unstable equilibria of the model system remain linearly stable and unstable, respectively.
7. The case of equal frequencies
7.1. Stability of a trivial equilibrium
Let us now consider the case of equal frequencies in the first approximation system for which 7l = —Y2 and the bifurcation value of fi1 is given by (4.6). We substitute fi1 = n10(n1) from (4.6) into the forms (3.6) and (3.8). We then move to the e-neighborhood (6.3) of the origin and introduce a new independent variable t = et.
The Hamiltonian of the system is transformed to the form
G2 — g2 ^(zj, Zj) + ^2 ^(zj, Zj, T5 £), g2 ^(zj, Zj) — G72i) + £ (G22 + G4o), (7.1)
13^24 ~ ^13a2 2(&24+&13)
^2? = 71 (*? + ^i2 - z22 - Zl) + ßiz.Z, - Zlz2), 71 =
The form G^ coincides in structure with the form G^ from (6.4); the coefficients Ak are calculated by the formulas from (3.7) for ¡i = ¡ii0(ni).
The coefficients and the discriminant of the characteristic equation of the linear system with the Hamiltonian r^ + er^ are as follows:
a = 2 (47l2 + 02) + O(e), b = (47l2 + 02)2 + O(e), d = 4e2 [4 (02 + 47l2) (Ai + A2)2+ +4 0 + 472) (ai + a2)(Ai + A2) + 47?(ai + a2)2 + 02 ((ai + a2)2 - d2)] + O (e3).
The expression in the square brackets in the main part of d is a square trinomial with respect to Ai + A2, with the discriminant 1602 (0* + 472) d2 and the roots
2 2V^i+47i
Taking into account the expressions for Ai and A2 from the formulas (3.7) for ¡i = ^i0(ni), we have
A1 + A2 = ~(b13 + b.M)n2 + s^l + Sq .
Here si and s0 are constants independent of ni and ¡2.
Equating these expressions, we find the boundary values for a parametric resonance region as
2
¡ 2 =
b1s + b2
ai + a2 2 , 0*d
- SiV2 - so ±
13 24
This region is given by the inequality
min ¡+) <¡2 < max ¡+).
7.2. Reduced Hamiltonians
In the system with the model Hamiltonian G2° from (7.1), there are no nontrivial equilibria. To study the motions of this system, we carry out a series of canonical transformations that take into account the structure of the form g21 .
For y1 =0 (that is, for n1 = 0) we put in the model Hamiltonian from (7.1)
z1 = n1z1, z2 = n1z2, Z1 = n2z2 + n3Z1, Z2 = n2z1 + n3Z2, (7.3)
1 ] ' v^pyTT "
The form G^
is transformed to
G{£ = (zi2 + Zi2 - Z22 - 42), s = signQ = yfirf + ff. (7.4)
Then, taking into account the structure of this form, we normalize the model Hamiltonian in the terms of the second and fourth orders with respect to perturbations and of the first order
in e. We pass to the symplectic polar coordinates pj, Tj and rewrite the complete transformed Hamiltonian in the form
HF = Q(t! - T2) + e liTi + l2T2 + t^t/2 (¿3 + m2T1 + mAT2) sin((^1 + ^)+
+mlT\ + (m-3 + m3i sin2(^i + TiT2 + m^T2] + O (e2), (7.5)
f2 A + 8Ai 7l2 + 2A4 7i&
li =
2s7i\/4 ll + ß2*
, _ P2A + 47l2(2A2 + a2 + al)- 2At7l& _ ß,A
h — , - , i 3 — ,
2S7i\/47? + Ä2
7i
_ 128ri74 + 16/j272(r2 + 2r1) + &4f _ fßl + 87l2 (r2 + )
ml — -n 9 /. 9 ,—-) m2 — ~SP*
872 (472 + Ä2) ß2 (ß2 + 872) r + 64r274
rry~i — _____
472 (472 + ßl) fß2 + 872 (2r3 + T2)
m31 = ß
272^472 + ß2
2 ß2r + 872 (r3 + r2 +
m4 = —sß;
27?V/47Ï + Â2 A = Ai + A2 +
m*
272 (472 + ß2) ß4f + 16ß272(2r3 + r2) + 128r374 872 (472 + ß2)
a1 + a2 — d
2 '
r = 3r, +r2 + 3r
3-
The term O (e2) in (7.5) has a period 2ne in t. Assuming further
= $1, V2 = $2 — $1, ^ = + R2, r2 = R2
where R1 + R2 ^ 0, R2 ^ 0, we get the Hamiltonian
HF = sQR1 +1 l1R1 + m^! + \JRl + E2\fR2[¿3 + m2i?i + (m2 + m4)E2] sin $2+
+[li + l2 + (2mi + m3 + m3i sin2 $2)Ei]R2 + (mi + m3 + m5 + m3i sin2 $2)R2] + O (e2).
(7.6)
In the main (model) part of this Hamiltonian the coordinate is cyclic. In this model Hamiltonian we have Ri = c = const and c + R2 ^ 0, R2 ^ 0. The constant c can take values of any sign or be equal to zero.
In the case c > 0 we put R2 = cg (g ^ 0), and in the case c < 0 we make the change = = — $i, Ri = —Ri and then put Ri = |c|, R2 = \c\g (g ^ 1). We introduce a new independent variable t = e\c\t for c = 0 and t = et for c = 0. Discarding additive constants, we obtain the Hamiltonians of the reduced systems with one degree of freedom. For c = 0, c > 0 and c < 0 they have the form
h0 = (li + l2 + l3 sin $2)R2 + [mi + m3 + m5 + (m2 + m4) sin $2 + m3i sin2 $2]R2,
h± =
^ + ± (2m 1 + m3 + m31 sin2 $2)
\c|
± m,2 + g(m,2 + m4)
+ Vq± 1VQ
Here the upper and lower signs refer to the cases c > 0 and c < 0.
0 + (7-7)
sin $2 + (m1 + m3 + m5 + m31 sin2 $2) g2-
Let us separately consider the case y1 =0 (n1 = 0). First, in the model Hamiltonian from (7.1) we put
zi = z2 + z2 = —z[ — Z2, Zx = -(—z'x + Z2), Z2 = ~(z2 — Z[)
and get as a result
Then we carry out normalization in terms of order e and subsequent transformations, as in the case y1 = 0. The transformed Hamiltonian has the form (7.6). The corresponding reduced Hamiltonians with one degree of freedom have the structure as in (7.7) where one should put
h = + A2) + A4 + 7-ai + ^a2 + l2 = j(A1 + A2) - A4 + 7-ai + ^a2 +
3fA A ^ 5 1 3T 59 41
h = 2 ^ 1 + + 4«! + 4«2 + 2d' mi = m5 = 32 (ri + r3) + —r2,
15- 9- 91 9
m2 = m4 = -g-r, m31 = -I\ m3 = — + T3) + — r2.
7.3. Phase portraits of the reduced systems
7.3.1. The reduced system with the Hamiltonian h0 from (7.7) has an equilibrium R2 = 0, its stability was studied in Section 7.1. Other equilibrium positions are described by the system of equations
cos $2 [(m2 + m4 + 2m31 sin $2)R2 + l3] = 0,
l1 + l2 + l3 sin $2 + 2 (m1 + 2m3 + m5 + (m2 + m4) sin $2 + m31 sin2 $2) R2 = 0.
A simple analysis, taking into account the above expressions for lk and mk, shows that in both cases 71 = 0 and y1 = 0 this system has solutions only for cos $2 = 0. In the case y1 = 0, the equilibrium values of R2 are given by the formulas
VW+~M (V47i2 + № + <T2S0*) [V47'f +02MAl + A2) + + a2] + <r2spj
R0 =--
8|7i l^i(7i)
where a2 = sin$2 and Fi (7i ) = r0l + 8(ri + r2 + r3)72. The corresponding equilibria exist under the condition
Fi(YI)
A + a2 + Ch +"2 +
< 0
2 2^472 + /i2_
and are linearly stable in the existence region if the inequality a2sF1(7l) < 0 holds.
Thus, the number of nontrivial equilibria of a model system can be two, one, or zero. The boundaries of the regions of their existence and stability coincide with the boundaries of the parametric resonance region (see the formulas (7.2)). The boundaries can also be two symmetric values of 7l, defined by the condition F1(7l) = 0, they exist if r(r1 + r2 + r3) < 0. For these values of 7l, there are no nontrivial equilibria in the model system. Similarly, in the case 7l =0 we have
(5 + 3a2)(A1 + A2) + 4(a1 + <r2) + a2(l - <r2)
lío -----•
(17 + 15a2)r
The corresponding equilibria are linearly stable in the existence region R2 > 0 if r(3 + 5a2) < 0. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 481-512_
2
7.3.2. The reduced systems with the Hamiltonans h+ and h- have equilibria for which cos $2 = 0, and the equilibrium values of g satisfy the equation (a2 = sin $2)
2-\/g ± l s/g[li +1-2 + c(2nii + m3 + m31) ± 2cg(m1 + m3 + m5 + m31)] =
= — a2 [g(2l3 + 5m2c + 3cm4) ± 4cg2(m2 + m4) ± (l3 + m2c)], (7.8)
where the upper and lower signs correspond to the system with the Hamiltonian h+ and h-.
Squaring both sides of Eq. (7.8), we obtain a corollary equation containing a polynomial of the fourth degree in g.
The leading coefficient of this polynomial is given by the expression
16c2 ((mi + m3 + m5 + m3i)2 — (m2 + m4)2)
and reduced to the form 14 400c2 (28n2 + 45)2 n4 for 7i =0 and to the form 64r2c2 for 7i = 0. Moreover, in the case c> 0 the constant term of the polynomial equal to —(l3 + m2c)2 is negative. In the case c < 0, the value of the polynomial at the boundary point g = 1 equal to —(l3 — cm4) is also negative.
It follows from the above that on the interval g > 0 or g > 1 (for c > 0 or c < 0) the described polynomial can have one or three real roots. Each real root of the polynomial from these intervals is also the root of the initial equation (7.8) for one of the values of a2 (1 or —1). Thus, in the systems with the Hamiltonians h+ or h-, there are one or three equilibria for which cos $2 = 0. This conclusion applies to both cases 7i = 0 and 7i = 0.
There are no other nontrivial equilibria in the system with the Hamiltonian h+. In the system with the Hamiltonian h- there are two more unstable equilibria g = 1, $2 = 0 or n lying on the boundary circle g = 1 of the existence region of the motions.
A more detailed analysis of the existence and stability of the equilibria of the considered reduced systems in general form is difficult due to the large number of parameters.
Variants of the phase portraits of the reduced systems with the Hamiltonians h0 (in the plane of Cartesian variables u = \J2R2 cos <J>2, v = \J2R2 sin<J>2), h+ and h_ (in the plane of u = ^cos $2, v = y^sin <J>2) are shown in Figs. 6a-6c, 6d-6f and 6g—6i, respectively. The chosen values of the parameters correspond to the systems given below, in Section 8.3 where the satellite problem for ni =0 is considered. Regions inside the boundary circle u2 + v2 = 2 in Figs. 6g-6i where motion is impossible are shaded.
7.4. On the motions of the complete system
The motions of the reduced systems (positions of equilibrium, oscillations, and rotations) generate the motions of a complete nonautonomous system with two degrees of freedom with the Hamiltonian (7.6).
Consider the case c > 0. Each equilibrium point (a2-g*) of the reduced system with the Hamiltonian h+ corresponds to a one-parameter family of stationary rotations of the model system with two degrees of freedom with the Hamiltonian (7.6) (without the last term) of the form
Rl=C, R'2 = CQ*1 = =S\j4^+ß2 +
yffh<T2(h + 3m2c + 3m2 c6* + C6*m 4)
+ £
2' dT
I1 + 2m-1c(1 + gj + (m-3 + m^ )cg* +
Zy/T+Q*
,v
\ H1
V
(g) 00 (i) Fig. 6. Phase portraits of the reduced systems in the case of equal frequencies
A degeneracy condition similar to that given in Section 5.2.1 has the form
V^2H3 + (3(x + Q*)m2 - ß*m4)C1
4 (C(1+ ft)3/2)
+ 2m1 = 0.
In the absence of degeneracy, for sufficiently small values of e, most of these stationary rotations generate conditionally periodic motions with frequencies Q and e_1 in the complete nonau-tonomous system.
For the reduced system with the Hamiltonian h+, in the parameter regions corresponding to oscillations in the vicinity of stable equilibria or rotations (the closed trajectories in Figs. 6e, 6f, covering all singular points) we introduce the action-angle variables I, w, as in Section 5.2.2. We
write the reduced Hamiltonian h+ as a function of the variable I, i.e., h+ = h+(I). We also introduce action-angle variables Ij, Wj (j = 1, 2) in the model system with the Hamiltonian (7.6)
(without the last term), setting R1 = /1, I2 = ^/(ft^). Hence, we find that. h+ = h+ (i2)-The model Hamiltonian is rewritten as (a = f- j
HF(IU I2) = H^ilJ+eH^il,, I2), H{F](h) = ssJkfiVfth + ^ (hh + mjl), I2) = lfh+( A).
The frequencies of the nonlinear oscillations of the system with this Hamiltonian are 9HP r~7, ~T x dHp r dh,
= wf = sv4-'i+«+°<*>. ■* = -§£=
The complete Hamiltonian (7.6) in the variables Ij and Wj is written in the form
HFI, I2, Wi, w2, r; e) = HP (Ij) + eH^ (I1, I2) + £2hP (Il9 I2, wl7 w2, r; e). (7.9)
Here the last term is analytic in all arguments, 2^-periodic in w1 and w2, and 2^e-periodic in r.
In the system with the Hamiltonian (7.9) there is an eigendegeneracy case, since the main (of the order of unity) part of the Hamiltonian depends on only one variable I1 [13]. If the conditions
dH{F0\ H' 92HF" , , ,
-wr^0' ("0)
hold, then most motions of the complete system are conditionally periodic, with two slow frequencies u1 ~ 1 and w2 ~ e and one fast frequency equal to e-1. The fraction of phase space
not. filled with these motions is defined by the value O (exp where d1 = const. > 0 [13].
The first condition in (7.10) is obviously met. The verification of the second and the third ones requires a study of the oscillation frequency for the reduced system and a study of the nondegeneracy condition for the reduced Hamiltonian h+; this study is not carried out in this paper.
Similar conclusions are drawn in the case c < 0.
Conclusions. Let us sum up the results of the study of nonlinear oscillations of the system at the resonance 2:1:1 considered. For sufficiently small values of e, in a small neighborhood of the resonance point of the three-dimensional parameter space, there are instability regions of the trivial equilibrium of the system (parametric resonance regions). They are adjacent, see Fig. 1, to two branches of the hyperbola (the case of zero frequency) and to the straight line (the case of equal frequencies), obtained by considering the first approximation system; their width is of order e2. For the case related to Fig. 1b, the relative positions of the parametric resonance regions in the sections e = e = const and n1 = const look like in Figs. 7a and 7b, respectively, for the satellite problem studied below in Section 8. Here e is eccentricity of the satellite's center of mass (0 < e ^ 1). For the case corresponding to Fig. 1a, the conclusions are similar; in the sections n1 = const for n2 > n2o or n2 < n2o there are three or one parametric resonance region, respectively. Outside the parametric resonance regions and their small (of order e2) neighborhoods, as well as in the case of equal frequencies, most motions of the complete
(a) (b)
Fig. 7. Stability picture in the parameter plane a, ¡3 (a) and e, ¡3 (b)
nonautonomous system are two- and three-frequency conditionally periodic, of different orders
In the case of zero frequency, the initial nonautonomous system exhibits motions analytic in e and 2^-periodic in time. Inside the parametric resonance region (in the neighborhood of the hyperbola) there is one pair of linearly stable periodic motions. Outside this region, in a small neighborhood (of order e2), on one side there are two pairs of periodic motions (one linearly stable and one unstable), on the other side there are no periodic motions.
8. Resonance oscillations of a dynamically symmetric satellite
Let us consider the motions of a dynamically symmetric satellite (rigid body) in a central Newtonian gravitational field on an elliptical orbit of small eccentricity e (0 < e ^ 1). The orientation of the satellite in the orbital coordinate system will be given by three Euler angles. The angle of proper rotation is a cyclic coordinate; the reduced system with two degrees of freedom describes the motions of the satellite's symmetry axis and can be represented in Hamiltonian form with the Hamiltonian [15].
We will investigate these motions in the vicinity of a stationary rotation for which the satellite's dynamic symmetry axis is perpendicular to the orbit plane and the satellite itself rotates around this axis with a constant angular velocity r0 (cylindrical precession) [16]. The Hamiltonian of perturbed motion depends on a small parameter e and two dimensionless parameters a = ^ (0 < a ^ 2) and /3 = ^ where C and A are axial and equatorial moments of inertia of the satellite, and u0 is the average motion of the center of mass along the orbit. The true anomaly v is taken as an independent variable.
The frequency equation for small linear oscillations of the system in the case of a circular (e = 0) orbit of the satellite's center of mass has the form [16]
u4 — (3a — 1 + a232 — 2a3) u2 + (a@ — 1)(a3 + 3a — 4) = 0. (8.1)
For the values a = 2, / = 1 from the region of sufficient stability conditions for cylindrical precession, this equation has a solution w1 = 2, w2 = 1, and the resonance of the form considered in this paper takes place.
8.1. General case. Model system of the first approximation
Let us put a = 2 + en1, 3 = 1 + e^1 + e2^2 + ... In the autonomous (e = 0) part of the Hamiltonian we make the linear [2] and then nonlinear normalizing changes of variables that bring H20 and H40 to the form (2.3) (for a = 1) and (2.4) where
r2 = i, r, = -I
1 128' 2 16' 3 32
Then we perform a rotation transformation similar to (3.3), and normalize the quadratic part of the Hamiltonian in terms of order e. We get
K2i = + 2Vl) {xj + X2) + ±(2^ + Vi) {4 + X2) + ^(x2X, - XlX2).
The coefficients and the discriminant of the characteristic equation of a linear system with this Hamiltonian are
5 5 9 1 2
« = + + '-^l + 4, b = — + 10^7?! + 4rfi - 9) ,
9
d = — |>i - Vif + 18] i^+ Vi)2-Hence, we find that the case of zero frequency takes place at
(h = (4(il 1) = ± ^ \Jili + 4,
and the case of equal frequencies, at ¡1 = ¡10(n1) = —n1.
After replacing (5.2) and (5.3), we obtain a model Hamiltonian (see (5.4)) of the form
G(0) = ^(Ah+2r?1)E1+^atl-r?1)E2+^^(E1-E2)JR2sin$2-^E2+^iJR2-^Ri- (8-2)
We put R1 = c = const > 0, R2 = eg (0 ^ g ^ 1) in it, introduce the notation
16,
(i = 7r((h-Vi)+ 2, K =
9cvr 1 3c
and a new independent variable r = ^ct. As a result, we get a reduced Hamiltonian of the form (5.5) for a1 = —1.
According to the results of Sections 5.1 and 5.3, most motions of the complete system with the Hamiltonian (8.2) (with an added nonautonomous part) generated by the equilibrium positions or oscillatory motions of the reduced system are two- and three-frequency conditionally periodic. Appropriate criteria are given in Section 5.1.
8.2. Model systems of the second approximation
8.2.1. The case of zero frequency. In the case ¡l = ¡+(ni), we normalize in the terms of order e2 in the quadratic part of the Hamiltonian of perturbed motion and compose the model Hamiltonian from (6.4), in which
= ^ (»71 + ^+4) (-1 + Zf) + | (-7,1 + \fi+4) (z\ + Zl) + ^■(z2Z1 - Zlz2),
022 - 2 + 25g~ 256V ^ 640 J +
Z1 31 2 27 rz-- 117\ 2
+ " ^++4"
(1 31 27 /- 93 \ 3 \/2 / /- \
2^2 - 32 ^ + 32 »71 + 4"64j^ + l28 (21V ^ + 4 " 19i?0 ^ " ^
An analysis of the coefficient b in the corresponding characteristic equation makes it possible to determine the region of parametric resonance given by the inequality ¡1+ < ¡2 < ¡+2, where
+ _ 3r?! (MOr?2 + 299) - 19y/rfi + 4 (3 + 20r?2) ^21 —
160 y/rji + 4
¡22 =
377! (I40r?f + 319) - 071 +4 (380r?f - 3)
160V»7Ï + 4
In particular, for r,1 = 0 we have — < ,i2 < j|q, which matches the result from [2].
Similarly, in the case ¡l = (nl), we find the interval ¡-l < ¡2 < ¡-2, where ¡-l and ¡-2 are obtained from ¡+l and ¡+2 by changing the sign in front of the radicals in the numerator and the denominator.
Nontrivial equilibria of the model system for ¡l = ¡+(nl) are given by the equalities
- -0 Z -0 ~2 (¿¿2 - z .
— u> ^20 — u) ^"20 / , - \ ) ¿10 — o I Vl — V ll T 4 I ^20)
h [Zrii - 5??101 + 4 + 12J 2 v /
(8.3)
- -0 Z -0 ~2 (/*,-/*+)
^20 — u> zio — u> ^10 —--7—-, , -¿20 — —p I + V + 4
h [ni + h Vri + 4 + 12j 4 V y
(8.4)
Solutions (8.3) exist under the condition n1 (¡2 — ¡22) < 0, they are linearly stable in the existence region for n1 > 0, ¡1 < ¡+2 and unstable for n1 < 0, ¡2 > ¡22. Solutions (8.4) exist for n1 (¡2 — ¡+0 < 0, they are linearly stable for n1 < 0, ¡2 > ¡+1 and unstable for n1 > 0, ¡2 < ¡+1.
Nontrivial equilibria of the model system for ¡1 = ¡2-(n1) are obtained from (8.3) and (8.4) by replacing by ¡-j (j = 1, 2) and sign change before the radicals. Taking into account the existence region, the first of these pairs is linearly stable for n1 < 0, ¡2 < ¡¡-2 and unstable for n1 > 0, ¡2 > ¡-2; the second pair is linearly stable for n1 > 0, ¡2 > ¡1— and unstable for m < 0, ¡2 < ¡21.
Note that the function (6.15) in this satellite problem has the form
F{r]l) = (~2i?1 +
and vanishes at the only point = 0; the existence and stability regions of the studied equilibria change when passing through it. For the value = 0 in the model system under consideration there are no nontrivial equilibria.
Periodic motions of the satellite's symmetry axis in the initial nonautonomous system generated by the described nontrivial equilibria of the model system will be obtained in Section 8.3, where a rigorous nonlinear analysis of their stability will also be carried out.
8.2.2. The case of equal frequencies. Calculations show that in the case of equal frequencies the model Hamiltonian has the form as in (7.1) where
G21 — (zf + Z\ — z\ — Z2) H —{z2Zi — zlZ2),
72 2 72\ '1 — z2 — Z2)
+ (^'2 - §4 - If) - ^hfez, - HZ2).
The main part of the discriminant of the characteristic equation of the corresponding linearized system is
d0 = e
^ (277? + 9) - (277? + 9) (185/7? - 63) ii2+
1369 6 5809 4 5607 2 4131
+7777^1 + ™ m - T^T^i
2304 2560 3200 12 800 From the condition d0 < 0 we obtain the parametric resonance region in the form
37 2 7 3 37 2 7 3
- 7TT ~ , „ < Lh < nil - +
72 40 4y/2Ïj[+9 72 40 4^/ЩТя
In particular, for r]1 = 0 we have — < ц2 < ^ (see also [2]).
After performing the transformations described in Section 7.2.1, we obtain a Hamiltonian of the form (7.6) for e = e and the corresponding reduced Hamiltonians of the forms (7.7). In particular, in the case = 0 the coefficients lk and mk of the model Hamiltonian are
15 3 Ï6At2 + Ï28'
9 87 269 75
8th ~ 320 ' ml = 4096' m-2 = 1024'
45 75 269
1024' m4 ~ 1024' m5 ~ 4096'
- 128 >
19
mo =--—, m 01
3 204g) 31
As follows from the results of Section 7.2.3, most motions of a complete nonautonomous system with the Hamiltonian (7.6) are two- and three-frequency conditionally periodic, with frequencies of different orders in e.
The parametric resonance regions obtained, as well as the existence and stability regions for nontrivial equilibria in the vicinity of the resonance point, are shown in Fig. 7a in the (а, в)-plane (for a fixed value of e) and in Fig. 7b in the (e, e)-plane (in the section a = const).
The parametric resonance regions are shaded; in the unshaded regions, the trivial equilibrium of the system is linearly stable. The numbers 1 and 2 denote the regions where the first and second pairs of nontrivial equilibria exist, the letters s and u denote linearly stable and unstable equilibria. The problem in question corresponds to the case represented in Fig. 1a.
8.3. Resonant periodic satellite motions
8.3.1. Construction of periodic motions. Let us return to the case of zero frequency. Nontrivial equilibria (8.3) and (8.4) generate periodic solutions of the complete nonautonomous system which correspond to motions of the satellite's symmetry axis that are analytic in e and 2^-periodic in v.
Note that, when considering the system of the second approximation, only the main parts (of zero order in e) of the generating equilibrium positions can be determined. For a more complete understanding of the form of the studied periodic motions, the terms of the third approximation in e were additionally found in the model Hamiltonian. This made it possible to find the corrections of order e in the expansions of the equilibrium values of the variables, see the formulas (6.9) for e = e.
Omitting details, we present the results obtained for the case ¡¡1 — The additional terms z21 and Z11 for the first pair of the generating equilibria values are 1
z21
20
38
400 (/*2 " /4)
300
Vi sjvi + 4 (21 + 1077?) - (5r?? + 18) (2T?? + l)
z20 +
+
— 800^2 (13n2 + 12) + 376 860^2 + 98 600^? + 161100+
+207?!\fq[+ 4 (520^i2 - 9153 - 4930T??) z$q + 12 800^ - 69 600^7?! \Jrfi + 4+ +160^2 (4257?? + 177) - 194 2007?f - 558 555T?? -2922 + 225^\fij[+4 (899 + 872T??)
12
Zn = TT7 + \Jrft + 4J + (^-3 - rji + 7?! ^rfi + 4 )
+ (~8/J2 + 237?? + 45 - 217?! ^7?? + z20
The values of z11 and Z21 for the second pair have a similar form and are not given here. Moreover, z11 = Z21 = 0 for the first pair and z21 = Z11 = 0 for the second one.
In the initial variables q1 and q2 (perturbations of the nutation and precession angles relative to their values for the satellite's cylindrical precession), the resonance periodic motions for the first pair of generating equilibria are determined by the expressions
q^v) = e—Z10 sin2z/ + e
1 \/2 -(2^+rh)z.20 + —Z10
sin v—
— (27?! + ill)Zw - —Zn - —z20
sin 2v--——^Zw sin 3is \ +0 (e3),
20
q2(v) = ez20 cos v + e2 —
"(2Ati +h)Zw + z2o
4(2^1 + ni)z20 - Z21 + —Z10
cos v
cos 2is H--Zw cos 3is + z2n + O (e3),
16 10 20
3
and for the second pair, by the expressions
Qi (v) = e—zw cos 2v + e2
\/2 \/2 1 — (2r?i + Ati)-io - —zn +
3(2^1 - m)z2o + —-10
cos V —
cos 2z/ + - Z20--cos I + o (e3
q2(v) = eZ2Q sin v + e2 -
— r]i)Z20 — Z21 + ~zio
sin v+
+
— + - Z20
sin 2v--zw sin 3v } + 0 (e3).
16
Here the terms O(e3) are 2^-periodic in v. Figures 8a, 8b shows the characteristic forms, in the (ql, q2)-plane, of the described motions born from the first and second pair of equilibria, respectively; the terms O (e3) are discarded when plotting.
<Q2 Ql <h \\ ft
u
(a) (b)
Fig. 8. Geometric interpretation of periodic motions
Since the generating equilibria that make up each pair differ in the signs of the quantities z20 (or zl0), it follows from the above expressions that the periodic motions of the satellite that make up the pair are obtained from one another by changing to opposite signs of ql and q2. Due to the symmetry of the first and second solutions about the Oq2 axis and the Oql axis, the periodic solutions that make up the first and second pairs are obtained from one another by symmetrical reflection about the Oql and Oq2 axes, respectively.
8.3.2. Nonlinear stability analysis. In the regions of linear stability of the found satellite's periodic motions, a nonlinear stability analysis is carried out. For this purpose, the Hamil-tonian of perturbed motion is normalized in terms up to the fourth power inclusive with respect to perturbations. First, we make a linear normalizing change of the form
zl = nl3Zl + nl4Z2, z2 = n2lzl + n22z2, Zl = n3lzl + n32z2, Z2 = n43 Zl + n44 Z2,
yfhy/% _ _ yfh yfh
«13 — 7 ; \ ) n14 — ~n32 — / 1 ! n22 — n44 — ,-i-)
(7i+72)%! V7I + 72 V7i + 72
n
Yi
n
43
n
21
(7i+72)%i' "31 yfh RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 481 512
^ 7i = ¿(^1 + \A?i +4)' 72 = I (~Vi + \jrft +
Here for the first and second pair of solutions we have, respectively,
n.
21
7i
1/2
(7i + 72 )1/2
7i
Vn2 +4(^22 - ^2)
1/4
and n21 =
71
1/2
(71 + 72)1/2
772^1+4(^2-^21)
1/4
In the above expressions, terms of order e and higher are discarded.
As a result, the quadratic part of the perturbed Hamiltonian is reduced to the form
-(ÎÎ! + 0(e)) (42 + Z'2) + - (eQ2 + O (e2)) (z'22 + Z!22), where for both pairs of solutions Q1 = 2(71 + 72), and
43 200
Q22 =
U =
2240 sjrjl + 4 (3 + 2rjf) - 640m (7r/j + 25) fa - + 4 (15 120i?i + 4800î?i + +5200n4 + 13 113n2 - 2178) + 5200n5 + 23 503n? + 4800n4 + 25 200n2 + 25 920 + 13 236m
for the first pair of solutions and
q2 =
sjrft + 4(3/^ + 4-^ (3^+4-2^ (m + sjrft + (fi2-fi21),
160+4 (rfi - 42) + 160ï7! (rft + 16) fa + /r?2 + 4 (5646^ + 5r?f + 2160 m-
10 800
U2 =
-240n? - 2898) - 25 920 + - 240^4 - 7920^2 - 17832m - 4744n?
for the second pair. Corrections O(e) and O (e2) to frequencies are constants.
Next, using the Deprit-Hori method, we find a nearly identical change of variables that cancels the terms of the third degree in the Hamiltonian and normalizes the terms of the fourth degree with respect to perturbations. In symplectic polar coordinates pj, rj (j = 1, 2) we have
H = + O(e)K + (e^2 + O (e2)) r2+
+ e [(C20 + O(e))r2 + (cn + O(e))r1 r2 + (co2 + O(e))r2] + O (e2),
where the quantities Cj + O(e) are constants, and the last term O (e2) is the set of terms whose degree in Vj is at least with coefficients 2-/r-periodic in v. The explicit, form of the normalizing change and the quantities Cj are not given due to their cumbersomeness. If, for sufficiently small values of e, the quantity
A = C?1 — 4C2Q C02
(8-5)
is nonzero, then the periodic motions under study are stable for most (in the sense of Lebesgue measure) initial conditions [13]. If the quadratic form c20r2 + c11 r1r2 + c02r| is sign-definite for r1 ^ 0, r2 ^ 0, then there is a formal stability [9, 17].
2- 1M2
-1 ^ \ -0.5 0 Vi
-2
-4
-6
(b)
Fig. 9. Nonlinear stability analysis of the first (a) and second (b) periodic motions
The results of numerical calculations for the first and second periodic motions are presented in Fig. 9 in the (n1, ¡2)-plane. For the first periodic motion in the linear stability region shown in Fig. 9a, there are two regions where formal stability conditions are met. The first of the regions is shaded, its boundary is the curve on which c20 = 0 (thin line), and inside the region all three coefficients Cj are negative. The second region (not shown in the figure) is very narrow; it adjoins the upper boundary ¡2 = ¡22(n1) (thick line) of the linear stability region; inside it, all three coefficients c^ are positive. Note that inside this narrow region there is a curve on which A = 0; for points outside this curve in the linear stability region, the periodic motion considered is stable for most initial conditions.
Figure 9b, which refers to the second periodic motion, shows the boundary curve ¡ 2 = = ¡21(n1) (thick line) of the linear stability region and the curve c02 = 0 (thin line), which is the lower boundary of the region where formal stability conditions are satisfied. This region is shaded; three coefficients Cj are negative in it. For all points of the linear stability region, the condition A > 0 is satisfied, i.e., the motion considered is stable for most initial conditions.
Remark. Let us compare the results obtained here with the results of the study of the same satellite problem at a 1:1:1 resonance (when a =1, 3 = 2) [6]. In the problem from [6], the parametric resonance regions appear only when the system of the third approximation in eccentricity e is considered. In the general case, in the case of equal frequencies (for the systems of the second and third approximations in e), as well as in the system of the second approximation for the zero frequency case, most of the motions of the system are conditionally periodic, of different orders in e. In the system of the third approximation for the zero frequency case, there are periodic solutions, their number and stability results are similar to those obtained in this paper. Thus, both systems demonstrate to a large extent a qualitative similarity in the nature of motions, but also a number of differences.
Conflict of interest
The author declares that she has no conflicts of interest.
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