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3Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 1, pp. 77-102. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210107
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 70H08, 70H12, 70H14, 70H15, 70M20
On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in the Case of Two Identical Integer or Half-Integer Frequencies
O. V. Kholostova
This paper examines the motion of a time-periodic Hamiltonian system with two degrees of freedom in a neighborhood of trivial equilibrium. It is assumed that the system depends on three parameters, one of which is small; when it has zero value, the system is autonomous. Consideration is given to a set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of the linearized equations of perturbed motion are identical and are integer or half-integer numbers (the case of multiple parametric resonance). It is assumed that the normal form of the quadratic part of the Hamiltonian does not reduce to the sum of squares, i.e., the trivial equilibrium of the system is linearly unstable. Using a number of canonical transformations, the perturbed Hamiltonian of the system is reduced to normal form in terms through degree four in perturbations and up to various degrees in a small parameter (systems of first, second and third approximations). The structure of the regions of stability and instability of trivial equilibrium is investigated, and solutions are obtained to the problems of the existence, number, as well as (linear and nonlinear) stability of the system's periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described. As an application, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.
Keywords: multiple parametric resonance, method of normal forms, stability, nonlinear oscillations, periodic motions, dynamically symmetric satellite, cylindrical precession
Received February 01, 2021 Accepted March 19, 2021
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, GSP-3, A-80, Moscow, 125993 Russia
1. Introduction
The study of the cases of multiple parametric resonances in near-autonomous, time-periodic linear Hamiltonian systems in a neighborhood of trivial equilibrium has been started relatively recently in the work of A. P. Markeev [1-3]. A systematization of all such cases was performed, an algorithm for constructing instability regions (regions of parametric resonance) was proposed, and the possibility of existence of several (from one to three) regions of instability of equilibrium in a neighborhood of the resonant point of the parameter plane of the system was shown for the first time.
A more detailed analysis of the structure of stability and instability regions in a number of cases of multiple parametric resonances was carried out in [4-7], whereas this paper deals with problems of the existence, number and stability of nonlinear resonant periodic motions of the system and problems of the existence of conditionally periodic motions [6].
This paper continues the series of studies undertaken previously. It is assumed that the system depends on three parameters, one of which is small such that, when it has zero value, the system is autonomous. An analysis is made of the set of values of the other two parameters for which, in the autonomous case, two frequencies of small oscillations of linearized equations of perturbed motion are identical and are integer or half-integer numbers. It is also assumed that the normalized quadratic part of the Hamiltonian of perturbed motion does not reduce to the sum of squares, i.e., the trivial equilibrium of the linearized system is unstable.
It should be noted that the above-mentioned case of two identical frequencies occurs rather often in autonomous systems with two degrees of freedom and has been well studied both from a theoretical point of view and in relation to specific mechanical systems [8-16]. In this paper, this case has been generalized to a near-autonomous and time-periodic system. In systems of first, second and third approximations in a small parameter, the structure of the regions of stability and instability of trivial equilibrium is studied. Solutions are found to the problems of the existence, number and (linear and nonlinear) stability of the system's periodic motions analytic in fractional or integer powers of the small parameter. For some cases, conditionally periodic motions of the system are described.
As an application to the theoretical results obtained, resonant periodic motions of a dynamically symmetric satellite modeled by a rigid body are constructed in a neighborhood of its steady rotation (cylindrical precession) on a weakly elliptic orbit and the problem of their stability is solved.
2. Formulation of the problem. Transformation of the Hamiltonian
Consider the motions of a two-degree-of-freedom Hamiltonian system 2^-periodic in time. Suppose that the Hamiltonian H(qj,pj,t; a,ft,e) of the system depends on parameters a and ft and on a small parameter e (0 < e ^ 1), and let the system be autonomous when e = 0. It is assumed that the origin of coordinates qj = 0, pj = 0 (j = 1,2) of the phase space is the equilibrium point of the system, and in its neighborhood the Hamiltonian is analytic in its arguments and is represented as the series
H = H2 + H4 + He + ..., (2.1)
H2 = H20 (qj ,pj ,a,ft) + eH21 (qj ,pj ,a,ft,t) + e2H22(qj ,Pj ,a,ft,t) + ..., H4 = H40 (qj ,pj ,a,ft) + eH41 (qj ,pj ,a,ft,t) + e2H42(qj ,pj ,a,ft,t) + ...,
which contains only terms of even degrees in qj and pj. Here, Hlk are homogeneous forms of degree l in qj and pj with coefficients constant (for k = 0) or 2^-periodic in t (for k = 1,2,...).
We assume that the value, averaged over a period (in time), of the function H21 is zero; this condition is fulfilled rather often in problems of classical and celestial mechanics.
Suppose that in the autonomous case (e = 0) for some values a = a*, ( = (* the frequencies w1 and w2 of small oscillations of linearized (in a neighborhood of trivial equilibrium) equations of motion of the system satisfy the relation 2w1 = 2w2 = n (n being a natural number), and for these frequency values the quadratic part H20 does not reduce to the sum of squares. Its normal form is (we keep the previous notation for the variables)
n
#20 = ^(Pi+P22) + ^(qiP2~Piq2). (2-2)
In this case, the trivial equilibrium of the autonomous linear system is unstable.
Next, we will explore the nonlinear oscillations of the system with the Hamiltonian (2.1) near its trivial equilibrium for the parameter values from a small neighborhood of the resonant point e = 0, a = a*, ( = (* of the three-dimensional parameter space. The structure of the regions of stability and instability of trivial equilibrium will be studied, and the problems of the existence, number and (linear and nonlinear) stability of the system's periodic motions analytic in fractional or integer powers of the small parameter will be solved. For some special cases, conditionally periodic motions of the system will be described.
We note that the double parametric resonance of this type (and that of other types) in the Hamiltonian system under consideration can arise only in the case where the autonomous part of the Hamiltonian depends on two parameters. For example, resonance detunings of two eigenfrequencies of small oscillations can serve as such parameters. By taking account of these parameters, which are chosen in the small neighborhood of the resonant point, and by adding the small parameter from the nonautonomous perturbing part, one can obtain a complete description of the bifurcation properties of the system.
Suppose the quadratic part H20 has already been reduced to the form (2.2). At the first stage, using a number of canonical transformations, we simplify (normalize) the structure of the Hamiltonian in the second- and fourth-degree terms in perturbations.
We will perform the analysis in a small neighborhood of order e of the resonant point given by the formulae
a = a* + evi, ( = (* + e^i + e2^ + e3 ¡13 + e4^4 + ..., (2.3)
where v1 and ¡k are constants. The "truncated" form of representation of the parameter a does not restrict the generality of the investigation since the coefficient v1 can, in turn, consist of terms of different powers in e. However, the adopted representation make it possible to find in the next sections in a clear and more convenient way equations describing the boundaries of the regions of parametric resonance in which the dependences ¡1 = ¡1(v1), ¡2 = ¡2(v1), ... are sequentially obtained.
We first make a near-identity canonical transformation qj,pj qj,pj (j = 1,2) (differing from an identity transformation by third-degree terms in perturbations), which normalizes the form H40 of degree four in the autonomous part of the Hamiltonian. The calculations performed, for example, using the Deprit-Hori method [10] yield (Yk = const)
K40 = Y1(q12 + q22)2 + Y2(q12 + q22)(q1 p2 - p1q2) + Y3(q'p - p1q2)2. (2.4)
Since the explicit form of the transformation is cumbersome, we do not present it here. Next, we assume that y1 = 0.
The further univalent canonical change of variables [2]
q[ = xi cos d tj - x2 sin(^ t), p[ = Xi cosQ tj - X2 sintj,
q'2 = xi sinQ tj + x2 cos(| tj p'2 = Xi sin tj + X2 cos(-^t
transforms the quadratic form (2.2) to
1
^20 = 2 ^i2 + 2
1
2
(2.5)
(2.6)
We note that this change of variables preserves, up to notation, the form (2.4). We now consider the nonautonomous quadratic part of the Hamiltonian. Taking (2.3) into account, we expand the form H2 from (2.1) as a series in powers of e:
H
H2 = H20 + £Ü21 + £ H22 + . ..)
dH20
(2.7)
20 =
H
20,
H21 = H2l\* +
da
vi +
dH20
dß
Vi,
Here, the symbol * means that the calculation is performed for a = a*, fi = fi*. The partial derivatives in the second and third terms in H2i are quadratic forms with constant coefficients. We write them as dH2o
da
dH20
dß
= aiq2 + a2q% + a3p1 + ap +
+ a5 qiq2 + a6qipi + a7qip2 + a8q2Pi + a9 q2P2 + awpip2, = biq"2 + b2 q2 + b3pi + bp + + b5qiq2 + b6qm + b/qip2 + bg q2pi + bgq2p2 + bi0pip2.
(2.8)
We represent the nonautonomous quadratic form H2i\* as a Fourier series
H2i\* = 2, [Dk (qj ,pj )cos kt + Ek (qj ,pj )sinkt,
k=i
(2.9)
where Dk(qj,pj) and Ek(qj,pj) are quadratic forms of variables qj, pj (j = 1,2) with constant coefficients dki and eki. In this case, the terms with dki and eki contain the same combination of variables qj, pj as the terms with al or bl in the formula (2.8).
We make a change of variables (2.5) and then, using the Deprit-Hori method, a T-periodic (in time) canonical near-identity transformation (differing from an identity transformation by terms of order e and higher), which normalizes the form H2 in terms up to the required order in e. Here, T = 4^ and T = 2n, respectively, for odd and even values of n in resonant relation. As a result, the time in the normalized part is eliminated, and the terms with constant coefficients acquire a simpler and more symmetric structure.
Keeping the notation xj,Xj (j = 1,2) for the variables, we represent the transformed quadratic form as
K2 = K20 + £K2i(Xj ,Xj ) + ... + £k K2k (Xj ,Xj ) + £k+iK2(xj ,Xj, t; £),
(2.10)
where the term K20 is defined in (2.6). The explicit form of K2\, ..., K2k for the system of the kth approximation (k = 1,2,3) will be represented in the next sections. The last term in (2.10) is T-periodic in t.
*
*
-1
Remark. In the case of two identical integer frequencies the system has third-order resonances; however, if the Hamiltonian has no third-degree terms in perturbations (which is assumed in this study), these resonances are insignificant. Conversely, if the Hamiltonian does have third-degree terms, then, after their normalization, terms with resonant harmonics remain, resulting in a more complex structure of the normalized Hamiltonians. We obtain a different problem which requires a separate study going beyond the scope of this work.
In the case of two identical half-integer frequencies there are no third-order resonances, and using an additional canonical change of variables, the third-degree terms in perturbations (if any) can be eliminated. This case does not differ from that considered in this paper.
3. A model system of first approximation
3.1. Construction of a model Hamiltonian
Simplifying the structure of the terms of order one in e using the Deprit-Hori method, we obtain the following form K21:
K21 = ^ (ci + c2 + Cq)x\ + i (ci + c2 - c0)x2 + do.T1.T2 + ^(c7 - c8)(xiX2 - x2Xi), (3.1)
Cj = djV 1 + bjß 1 (j / 0), Co = ^(e„5 + dni - dn2), d,0 = ^(d,n5 - eni + en2). (3.2)
Another canonical transformation of the form
x\ = x1 cos S — x2 sin S, X\ = X\ cos 5 — X2 sin S, (3.3)
x2 = x1 sin S + x2 cos S, X2 = X1 sin S + X2 cos S
eliminates in (3.1) the term containing the product x1x2. This yields
I<21 = ^ (ci + c2 + ad)x\ + ^ (ci + c2 - ad)x2 + ^(c7 - cg)(xiX2 - x2Xi). (3.4) Here we have introduced the notation
8 = i arctan —, a = signco, d = \Jc^ + dg.
The transformation (3.3) preserves, up to notation of variables, the forms K20 and H40. We now consider the complete nonlinear system. Let us pass to a small (of order e1/2) neighborhood of trivial equilibrium using the formulae
x
= el'2Vj, Xj = el'2Yj (j = 1,2).
The transformed complete Hamiltonian of the system can be written as
r(1) = r(10) + 0(e2), r(10) = T20 + e(r21 + r4o). (3.5)
Here, r20, r40 and r21 are, respectively, the forms Ki20, Ki40 and Ki21, in which c£j and Xj have been replaced by yj and Yj. The term 0(e2) is T-periodic in t.
We will call the approximate Hamiltonian r(10) and the corresponding autonomous system a model Hamiltonian and a model system of first approximation, respectively.
Note that, in the constructed model system, the case a = —1 reduces to the case a = 1 if one interchanges indices 1 and 2 in the variables yj, Yj and changes the sign of the coefficient in front of the last combination in the form r21 and the sign of the coefficient y2 in the form r40. However, it follows from the results obtained below that the qualitative properties of the system do not depend on the signs of the above-mentioned coefficients. Therefore, in what follows we assume that a = 1.
3.2. Stability of the trivial equilibrium of the system
In investigating the stability of the equilibrium points of the autonomous model systems under study, we consider the quadratic part of the Hamiltonian of the motion perturbed in the neighborhood of equilibrium. By virtue of the Lyapunov stability theorem, a sufficient condition for the stability of equilibrium is the sign definiteness of this quadratic part. Necessary conditions for stability are obtained in the analysis of the characteristic equation of the linearized equations of perturbed motion, which has the form
A4 + aA2 + b = 0. (3.6)
If the conditions
a> 0, b> 0, d = a2 — 4b > 0 (3.7)
hold, then the roots of equation (3.6) are purely imaginary, and the equilibrium of the model system under study is linearly stable. When the sign of at least one of inequalities (3.7) is reversed, equation (3.6) has roots with positive real parts, and this equilibrium is linearly and nonlinearly unstable.
Consider the quadratic part r20 + er21 of the Hamiltonian of the model system of first approximation. The coefficients of the corresponding characteristic equation and its discriminant are
a = 2(ci + C2)e + O(e2), b = [(ci + C2)2 — d2]e2 + O(e3), d = 4dV + O(e3). (3.8)
For sufficiently small values of e the necessary conditions for stability (3.7) of the trivial equilibrium of the system reduce to the inequality
ci + C2 > d (3.9)
under the condition d = 0. If c1 + c2 < d, then for all d ^ 0 this equilibrium is unstable.
Inequality (3.9) is for all d ^ 0 also a sufficient condition for the stability of trivial equilibrium, since in this case the quadratic part of the Hamiltonian is positive definite.
The boundary c1+c2 = d between the regions of stability and instability, with the expressions for cj from (3.2) taken into account, corresponds to a straight line in the parameter plane (a, ft), which divides the neighborhood of the resonant point into two half-planes, in one of which the trivial equilibrium of the model system is stable, and in the other, unstable.
When d = 0 and c1 + c2 = 0, the pattern of stability of the trivial equilibrium of the system in this approximation in e is not defined. In this case, it is necessary to take into account terms of the next approximations in e. This will be done in Section 4.
3.3. Nontrivial equilibria of the model system
Equating to zero the partial derivatives of the model Hamiltonian r(10) of (3.5) in the variables yj, Yj (j = 1,2), we obtain an algebraic system of equations to define nontrivial equilibrium points of the model system.
First let d = 0. Analysis of these equations shows that under the condition Y1(c1 + c2+d) < 0 the model system has a pair of equilibria given by the relations
V2 = 0, Y1 = 0, Vi = Via + O(e),
(3.10)
2 Ci + C2 + d
Vio =
471
Y2 = -e
c7 - c8 . 2 -Ö- + 722/10
Via + O(e2).
If Y1(c1 + c2 — d) < 0, then there is another pair of equilibria for which
V1 = 0, Y2 = 0, y2 = y20 + O(e),
y20 + O(e2).
(3.11)
2 ci + C2 - d
V20 =--r-,
47i
c7 - c8 . 2'
-Ö- + 722/20
In considering the corresponding characteristic equations of the linearized equations of perturbed motion, we find that in the existence region the first pair of equilibria is always unstable. When y1 < 0, the second pair is also unstable. When y1 > 0, necessary (and simultaneously sufficient) stability conditions are fulfilled in the region of existence of the second pair.
These results are illustrated in Fig. 1a, b for the cases Y1 > 0 and Y1 < 0, respectively. Here it is assumed that a1 + a2 > 0 and b1 + b2 > 0; other variations of the signs of these quantities can be considered in a similar way. The parameter plane x = a — a*, y = ft — ft* is shown for a fixed value of e.
The neighborhood of the resonant point (0,0) is divided into three regions by the parallel straight lines g- and g+ given, respectively, by the equations c1 + c2 = —d and c1 + c2 = d, or
(ai + a2)x + (bi + b2)v = Ted.
(3.12)
The straight line g+ is also the boundary of the stability region of trivial equilibrium of the system. This region is colored in Fig. 1.
2s V
0 x
2s \
\ ^g+
v-
(a) (b)
Fig. 1. The stability pattern for the first-approximation system.
The regions where the equilibria (3.10) and (3.11) exist are denoted as 1 and 2, respectively; the symbols s or u mean that the equilibria in this region are stable or unstable.
When d = 0, the model system has (under the condition 71(c1 + c2) < 0) a family of unstable equilibria for which the pairs of points (y10,y20) make up a circle
2,2 C1 + c2 fo 2/10+2/20 =--(3-13)
In this case, the solutions (3.10) and (3.11) are unstable complex singular points of the system; they correspond to four points of the above-mentioned circle.
We note that in the case d = 0 qualitative changes occur in the properties of the model system. A cyclic coordinate appears in the system, and the motion pattern is determined by the properties of the reduced system with one degree of freedom. This is discussed in detail below in Section 7.
Finally, when the conditions d = 0 and c1 + c2 = 0 are satisfied simultaneously, all equilibria of the system shrink to the origin. In this case, in a neighborhood of the straight line given by the equation c1 + c2 = 0, it is necessary to investigate the system of second approximation.
4. A model system of second approximation
4.1. Construction of a model Hamiltonian
Now let the conditions c1 + c2 = 0 and d = 0 hold simultaneously. Let us construct a model Hamiltonian of second approximation, taking into account terms of order e2 in the quadratic (in perturbations) part of the Hamiltonian.
Under the above conditions the quadratic form (3.4) becomes
I<2i = - c8)(xiX2 - x2Xi), (4.1)
where c7 and cg are the coefficients c7 and cg calculated for c1 + c2 = 0, i.e., for ¡j,1 — — = — (a1 + a2)v1/(b1 + 62) (see (3.2)).
The terms H22 from (2.7) are normalized in a similar way. The transformed form K22 reads
K22 = ^ [(61 + 62)^2 + 41} + e-o\x\ + ^ [(61 + 62)^2 + 41}" e-oH +
+ \{{b" ~ h)l*2 + 42)X*1*2 - x2X1). (4.2)
Here, d^^ (j = 1,2) are some polynomials of degree two in v1, and e0 = const. As in the first-approximation system, one can assume without loss of generality that e0 ^ 0.
Let us pass to the e-neighborhood of the origin of the phase space using the formulae
xj = eyj, Xj = eYj (j = 1,2).
The transformed Hamiltonian takes the form
r(2) = r(20) + 0(e3), r(20) = r20 + er 21 + e2 (r 22 + r40). (4.3)
Here, like the previous Hamiltonian, r21 and r22 are the forms (4.1) and (4.2) under the transformation xj = yj, Xj = Yj (j = 1,2). The term 0(e3) is T-periodic in t.
We call the approximate Hamiltonian r(20) and the corresponding system a model Hamiltonian and a model .system of second approximation, respectively.
4.2. Stability of the trivial equilibrium of the system
For the characteristic equation of the linear system with the Hamiltonian r20 + £T2i + e2r22 we obtain
a = 2
(bi + b2)ß2 + 41} + \{c7- c8)2]t~2 + 0(t~3),
b =
((&i + b2)ß2 + 41} - i (c7 - c8)2)2 - eg] e4 + 0(e5),
d = 4[(c7 - csfdh + 62)^2 + di2)) + e0] e4 + O(e5). Analyzing these expressions, we find that in the case where the quantities
1 + b2)^2 + 41} -j(cT- c8)2 and e0 (4.4)
do not vanish simultaneously, the trivial equilibrium of the linear system is stable when the condition
(bi + 62)^2 + 41} > 4(27 - c8)2 + e0 (4.5)
holds and when the following conditions are satisfied simultaneously:
(C7 - c8)2 > 2eo, (4.6)
"(c^-cg)2 < + b'2^1'2 + ^^ < ~ 28)2 ~ e°' (4'7)
In this case, inequality (4.5) is a necessary and simultaneously sufficient condition for stability, and inequalities (4.6) and (4.7) are only necessary conditions for stability of the trivial equilibrium of the model system. These conditions impose a restriction on the value of ¡i2.
If both quantities in (4.4) vanish, then the stability pattern of zero equilibrium in this approximation in e is not defined.
4.3. Nontrivial equilibria of the model system
In the model system of second approximation, as in the first-approximation system, there can exist up to two pairs of nontrivial equilibria. For the first pair, we have relations (3.10) and
2 = (C7 ~ es)2 ~ 4[(6i + 52)A*2 + 41} + eo] f48)
Vw I671 '
Y2 = ~[e{c7 - c8) + e2[(b7 - b8)ti2 + 42) + 2722/10]] Vw + 0(e3).
The second pair is described by relations (3.11), in which the expression for y20 differs from the expression for y\0 of (4.8) in the sign in front of e0; the equilibrium value of Y\ is obtained from Y2 by replacing y\0 with y20 and by removing the minus sign in front of the main part.
These equilibria exist under the condition
71 [(C7 - c8)2 - 4[(6i + 62)^2 + d{2l) ± e0]] > 0,
where the upper and lower signs refer to the first and second pairs.
The analysis shows that in the case 71 > 0 the first of the pairs of equilibria in the existence region is unstable, and for the second pair in the existence region necessary and sufficient stability conditions are fulfilled.
If Yi < 0, then the second pair of equilibria is unstable. For the first pair there is a region of fulfillment only of necessary stability conditions, which is given by conditions (4.6) and the inequality
1 3
^(c7 - c8)2 - e0 < (pi + b2)n2 + 41} < 4 (C7 - C8)2 - \/2eo |c7 - c8|.
Outside this region, this pair of equilibria is unstable.
The stability pattern in the plane of parameters x = a — a*, y = ft — ft* for a fixed value of e is shown in Fig. 2a, b for the cases y1 > 0 and y1 < 0, respectively. On the curves g±, the relations
/1) _ ^2
/1) - „ „2
h + b2)^2 + dy = ~(c7- c8)2 ± e0
are satisfied. If we assume that d2 ' = + s1v1 + so (sj = const) and k1 = (cV — c8)/v1, then the equations of these curves are
(ai + a2)x + (bi + b2)y = (^k2 - s2^x2 + es\x + t2(s0 ± e0).
go 9-
Fig. 2. The stability pattern for the second-approximation system.
On the curves g0 and g00, the quantity (b1 + b2)/j,2 + d21) takes, respectively, the values
p'2
eo
(c„ _ c8)2 and 4 ~ Cs)2 - v/^eo |c7 - c81.
The common points of the curves g0, g00 and g- correspond to the values of v1 defined by the equation (cV — C8)2 = 2e0, and the common points of the curves g00 and g+ correspond to the values of defined by the equation (cV — cs)2 = 2(1 + \/2)2eo.
The region of stability of the trivial equilibrium of the system in Fig. 2 is colored. The numbers and letters inside the regions have the same meaning as those in Fig. 1.
When e0 = 0, the equilibrium values of y10 and y20 (if any) lie on the circle
„2 , „.2 _ (C7-C8)2-4[(&1+&2)A*2 + 41)]
y2o + y2o
I671
and correspond to unstable equilibria of the model system. For the value of ¡i2 which lets the first expression in (4.4) vanish, this circle shrinks to a point.
4.4. The case e0 = 0
4-4-1- At nonzero value of the first expression in (4.4), the case e0 = 0 is an analog of the case d = 0 for the model system of first approximation (see below, Section 7). In this case, the investigation is carried out in a neighborhood of the origin of the phase space given by the relations yj = Zj, Yj = eZj (j = 1,2). This case is not considered in what follows.
4.4.2. If both quantities in (4.4) vanish, the model system of third approximation is considered which is obtained by taking account of terms of order e3 in the quadratic part of the Hamiltonian. The investigation is carried out in the e3/2-neighborhood of the origin of the phase space.
For the above-mentioned model system, the structure of the parts K22 and K23 is similar to those of the forms (4.1) and (4.2), respectively. The analysis of the system shows that, if c7 = c8, its trivial equilibrium is stable under the condition (b1 + b2)j3 + d^ > 0; when the sign of this inequality is reversed, instability takes place. There are no other equilibrium points.
When c7 = c8, the system has nontrivial equilibria. This condition leads to the equation v1 =0 and the bifurcation value j1 = 0. In this case, it is more appropriate to assume that the value of v1 is initially small (of order e) and the neighborhood of the resonant point under study has a width of order e2. This case is considered in the following section.
5. Investigation of the s2-neighborhood of the resonant point
Let us come back to the initial system and consider the e2-neighborhood of the resonant point, which is given by the relations
a = a* + e2V2, P = P* + e2j2 + e3j3 + .... (5.1)
Normalizing the quadratic part of the Hamiltonian in terms through order three in e, we obtain
r2i = 0, r22 = ±(4 + 4 + h)(y2 + y2) + l(c'7 - 4 + l2)(yiY2 - y2Yi), r23 = ^[(61 +62)^3 +ni + n0]y'i +1 [(61+62)^3 +ni -n0}y2 +^[(b7-b8)ii3 +n2](yiY2 -y2Yi).
Here the notation cj = ajv2 + bjj2 is introduced, and U and nj denote constants independent of v2 and .
In considering the second-approximation system, we find that the inequality 4 + 4 +> 0 is a necessary and sufficient condition for the stability of its trivial equilibrium. When its sign is reversed, instability takes place. Nontrivial equilibria of the system exist under the condition Yi(4 + 4 + li) < 0 and form a family of unstable singular points described by the relations
yi = y 10 + O(e), y2 = y20 + O(e), Y1 = e2Y12 + O(e3),Y2 = e2Y22 + O(e3), (5.2)
„2 , „2 c'i + 4 + h v i4 + ¿2 72(4 +4 + h)\ .. y 10 + 2/20 =--—, -2---I7Ï-)ym
F22 = _(4^±i2_72(4+^)yio_
The transition to a third-approximation system occurs under the condition ci + c'2 +11 = 0, which gives the bifurcation value
ai + a2 I2
bi + b2 bi + b2
We have
r22 = -r(C7 - c8 + h){yiY2 - U'lYl),
2
where the tilde means that the substitution j2 = j20 into the corresponding coefficient has been made.
The trivial equilibrium of the system is stable under the condition
(bi + b2)j3 + ni — U0 > 0.
The pair of nontrivial equilibria which exists under the condition Y1[(b1 + b2)j3+n1 + n0] < 0 is given by relations (3.10) and by
2 (bi + b2)j3 + Ui + U0 „ 1 2/ , , , , . , 3n
2/10 = ---^-> 2 = (c7 - 4 + ¿2)2/10 + 0(e3).
The second pair, which exists under the condition y1 [(b1 + b2)j3 + n1 — n0] < 0, is given by relations (3.11) and by
2 (bi + b2)j3 + Ui — U0 v 1 2f / / ,, A , mt 3\ i r o\
2/20 =---» = (c7 " c8 + ¿2)2/20+ 0(t)- (5.3)
In the existence region, the first pair of equilibria is always unstable, and the second pair is linearly stable for Y1 > 0 and unstable for Y1 < 0.
These results are in qualitative agreement with the results of investigation of the firstapproximation system. We have a partitioning of the plane x,y (with a fixed value of e) into three regions whose boundaries are two parallel straight lines
(ai + a2)x + (bi + b2)y + e% + e3(Ui ± U0) = 0,
which are offset from the origin by a distance of order e2; the distance between the straight lines themselves is of order e3.
The existence regions and the stability pattern of equilibria are similar to those shown in Fig. 1.
6. Periodic motions of complete systems
6.1. On periodic motions of complete systems
Let us come back to the complete transformed nonautonomous systems with the Hamiltoni-ans r(1) of (3.5), r(2) of (4.3) and with the Hamiltonian of third approximation from Section 5. In the previous sections it was shown that the corresponding model systems can have one or two pairs of isolated nontrivial equilibria. Below we exclude from consideration the families of equilibrium points and equilibria that correspond to the cases of multiple roots of characteristic equations.
It can be shown that for the model system of the kth approximation (k = 1,2,3) the roots of the characteristic equations of the linearized (in the neighborhood of the above-mentioned
equilibria) equations of perturbed motion are of order ek/2 (see also Section 6.2), whereas the frequency of small time-periodic perturbation in complete nonautonomous systems is of the order of unity.
Thus, the nonresonant case of Poincare's theory of periodic motions [18] takes place, and each nontrivial equilibrium of the model system gives rise to a unique solution, analytic in e and time-periodic with period T, of the complete transformed nonautonomous system. We recall that T = 4^ and T = 2^, respectively, for odd and even values of u in resonant relation (see the end of Section 2). In the initial variables they correspond to motions T-periodic in time and analytic in fractional (for k = 1,3) or integer (for k = 2) powers of the small parameter.
By virtue of continuity in e of the characteristic exponents of the linearized equations of perturbed motion, the periodic motions of the complete system which are generated by linearly stable and unstable equilibria of the model systems remain, respectively, linearly stable and unstable.
6.2. Nonlinear analysis of the stability of periodic motions
For periodic motions of the complete linearly stable systems, a complete nonlinear analysis of stability can be carried out. For this purpose, the Hamiltonian of the complete transformed system is normalized in terms through degree four in perturbations in a neighborhood of these motions.
Taking into account the application considered in Section 8, we describe the results of nonlinear analysis of the stability of periodic motions of systems of first and third (in the e2-neigh-borhood) approximations. The analysis for the second-approximation system is similar; however, in the general form this analysis is rather cumbersome due to a large number of parameters, and is more suitable for a concrete problem.
6.2.1. Consider periodic motions generated by the equilibria (3.11) of the first-approximation system in the region of their linear stability, which is given by the conditions 71 > 0, C1 = c1 + c2 < d.
The frequencies of small oscillations of the linearized equations of perturbed motion are calculated from the formulae
Q =
Vide1'2 + 0(t~3/2), Q2 = fyd-C^e1'2 + 0(es/2).
After normalizing the quadratic part of the Hamiltonian of perturbed motion and normalizing the main part in the fourth-degree terms, we obtain a transformed Hamiltonian which has the following form in the new variables Zj, Zj (j = 1,2):
F = F20 + e1/2F40 + eFn + 0(e2), (6.1)
F20 = ifii (z2 + Z2) + Ul2{z2 + Z2), (6.2)
^40 = 2C'2o(zf + z2)2 + ^Cnizf + Z2)(z2 + Zf) + ~ C02 (z2 + Z2)2, (6.3)
C20 = + 0(e), Cn = , 71 + 0(e), C02 = -A ~ 71 ^ + O(e),
4d " sß^TÜ) "
F - ^
372C\ - 4c27i + 2^2d 4C271 - 72C1 - 272C0
Z1Z2 H--=----z\z2
d1/2 (d- Ci)1/2
The term O(e2) in (6.1) is T-periodic in time
\fdzl + \Jd-C\ Z2 ICild^id-Ci)1/4 '
5 Y2
A = C2n - 4C20C02 = -- „ + 0(e) (6.4)
We first consider the nonresonant case for which the quantities Q1 and Q2 are not related by the fourth-order resonant relation. Then, using another canonical change of variables, the form F41 can be eliminated.
We analyze the normalized Hamiltonian obtained above. For sufficiently small values of e the quantity
2
- -4-1 / On I /no - — _ _
4 d(d- Ci)|
is nonzero, and the periodic solutions under study are stable for most (in the sense of Lebesgue measure) initial conditions [17]. In addition, the quadratic form C20r2 + C11r1r2 + C02r2 is positive definite, and formal stability [10, 19] takes place.
Consider resonant cases. In the region of existence of the periodic solutions under study, for the values C1 = (8/9) d + O(e) and C1 = — 8d + O(e) we have fourth-order resonances Q1 = 3Q2 and 3Q1 = Q2, respectively. In these cases, the form F41 cannot be completely eliminated: there remain resonant terms in it.
We continue the normalization of the Hamiltonian in the terms of degrees six, eight etc. in perturbations, taking into account the existence of resonances of orders eight, twelve etc. Since the eigenfrequencies Qj (of order e1/2) are incommensurable with the frequency of external perturbation (of the order of unity), time t can be eliminated at each step of normalization. As a result, we have an autonomous Hamiltonian which is a formal first integral of the system, whose main part, of order el/2, is positive definite (and the resonant terms are of order e and higher). In a sufficiently small neighborhood of the origin, this formal integral is also positive definite. Therefore, in resonant cases the periodic motions of interest are formally stable.
6.2.2. For the third-approximation system from Section 5, we consider periodic motions generated by the equilibria (3.11) and (5.3) in the linear stability region y1 > 0, B = (b1 + b2)j3 + n1 < no.
The frequencies of small oscillations of the linearized equations of perturbed motion are calculated from the formulae
fil = \/2no£S/2 + 0(t~5/2), Q2 = V'2(n0 - B) e3/2 + 0(e5/2), and the normalized Hamiltonian of perturbed motion has the form
F = F20 + e3/2F4o + e2Fa + 0(e3), in which the parts F20 and F40 are calculated from the formulae (6.2) and (6.3) and
C2o = 7—+0(e), C11 = -r=== + 0(e), C02 = + 0(e),
4 no y/no (n0 - B) 4 (n0 - B)
_ \/2 (c7 - c8 + l2) 71(^1^2 y/»o ~ ZiZ2 s/np - B)(y/n^zl + s/np - B Z\)
\B\4/A(no-Bf/±
Analyzing the expressions for Cj and the quantity A, which was introduced in (6.4), we conclude that the periodic solutions under study are stable for most initial conditions and are formally stable.
In the region of existence of periodic motions, for the values B = (8/9)n0 + O(e) and B = —8n0 + O(e) we have fourth-order resonances Q1 = 3Q2 and 3Q1 = Q2, respectively. Following the same reasoning as above, we conclude that in these resonant cases the periodic motions are formally stable.
7. The first-approximation system: the case d = 0
7.1. Further transformations of the model Hamiltonian
Let us return to the analysis of the first-approximation system, assuming that the quantity d introduced in Section 3.1 vanishes. In this case, the form r21 reads
r2i = -2Ci(y{ + y22) + -2C2(yiY2 -y2Y1), C\ = c\ + c2, C2 = c7-c8. (7.1)
In what follows, we will assume that C1 = 0.
In the complete system with the Hamiltonian r(1) of (3.5), we perform some additional canonical transformations. We first make the change of variables
Vj = zj, Yj = e1/2Z3 (j = 1,2).
As a result, this Hamiltonian becomes
5 = e1/2
\(Z2 + zf ) + ±cl(z? + ,1) + 71(*? + 4)*
+ O(e).
Here, as above, the term O(e) is T-periodic in t.
Next, we make the transformation Zj, Zj — Uj, Uj (j = 1,2)
(7.2)
which is similar to the transformation inverse to (2.5). This gives the Hamiltonian
n
S='~ (UlU2 - u2lh) + e1'2 [-(U'i + Hi) + -Ciiui + ui) + 7i(ui + utf j + O(e).
We now transform to polar coordinates by making the canonical transformation u1 ,u2,U1 ,U2 ^ r,d,R,J by the formulae
u1 = r cos 0, u2 = r sin 0,
J J (7.3)
U\ = R cos 9--sin 9, U2 = R sin 9 -\— cos 9.
1 "-2 I tt2\ I ^ („.2 , „ 2\ , !„,2 , „2x2
The transformed Hamiltonian becomes n
S = ^J + eV2
+ O(e).
2 r2 2
Finally, we set
r = |71|"1/2 Q, R = |71|"1/2 P, 0 = v, J = I71I-11. (7.4) The complete transformed Hamiltonian takes the form
F = F' + O(e), (7.5)
F'= F0 + e1'2 FU F0 = ^I, F, = \ (p2 + + ± Cl62 + a^4. (7.6)
Here the notation a1 = sign71 is introduced and the term O(e) is T-periodic in time.
We note that the Hamiltonian F' of (7.6) coincides, up to notation, with the corresponding approximate Hamiltonian describing the motions of the autonomous Hamiltonian system with two degrees of freedom in the case of identical frequencies under consideration. A detailed analysis of the nonlinear oscillations of such an autonomous system in the case ai = 1 is carried out in [15].
We describe the properties, necessary for further analysis, of the approximate system with the Hamiltonian (7.6) for both cases ai = 1 and ai = —1. In this system, the coordinate p is cyclic and there is a first integral I = c = const. Without loss of generality we assume that c ^ 0.
7.2. A reduced system with one degree of freedom
Consider a reduced system with one degree of freedom with the Hamiltonian Fi in which the substitution I = c has been made. The system has the first integral
1 / c2 \ 1
2 (p2 + J2 ) + 2 Ciq2 + <t^ = h = const- (7J)
The phase portraits of the reduced system for c> 0 are shown in Fig. 3 for the cases ai = 1 and all Ci = 0 (a); ai = —1 and Ci < 3(2c)2/3 (b); ax = —1 and Ci > 3(2c)2/3 (c). At the equilibrium points of the system we have P = 0, and the value q = q0 is the root of the equation
4aig6 + Ciq4 — c2 = 0.
The solutions of the reduced system (except for solutions corresponding to the asymptotic motion in Fig. 3c) are written in terms of Jacobi elliptic functions. We restrict our attention to the closed trajectories shown in Fig. 3a, c, which correspond to oscillations of the system in a neighborhood of stable equilibrium points. Let ui, u2, u3 (ui > u2 > u3) be the real roots of the equation
—2aiu3 — Ciu2 + 2hu — c2 = 0. (7.8)
The dependence Q(t) for the oscillations in Fig. 3a is described by the relation
e2(t) = u1 - (Ui - u2)sn2(V'2(Ul - u3)t, h), kj = (7.9)
ui — u3
Fig. 3. Phase portraits of the reduced system for c > 0. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2021, 17(1), 77-102
where 0 < u2 ^ Q2 ^ u\ and it is assumed that Q2 (0) = u\. For the oscillations in Fig. 3c we
have
ß2(t) = u3 + (u2 - t/,3)sn2( v/2(i/,i - u3)t, k2), k22 =
u2 - «3 u1 — u3'
(7.10)
where 0 <u3 ^ q2 ^ u2 and q2(0) = u3.
The frequencies of the oscillations described above are calculated from the formulae
Wl,2
7T y/2(t/,l ~U3)
K(kl:2) '
(7.11)
where K(fc1>2) is the complete elliptic integral of the first kind.
After finding the dependence g(t) the corresponding dependence P(t) is defined using the integral (7.7).
For c = 0 the phase portraits of the reduced system have the form shown in Figs. 4a-4d for the cases a1 = 1, C1 > 0 (a), a1 = 1, C1 < 0 (b), a1 = -1, C1 > 0 (c), a1 = -1, C1 < 0 (d). In these figures, it is formally assumed that the variable r can take values of any sign. The phase trajectories are symmetric relative to the ordinate axis, and the symmetric parts of the trajectories from the right and left half-planes correspond to the same motion of the system.
The solutions corresponding to oscillations in the case c = 0 are described by the same formulae as in the general case c > 0. For a1 = 1 and h > 0, for trajectories corresponding to oscillations in a neighborhood of trivial equilibrium (Fig. 4a) and the closed trajectories enclosing the separatrix in Fig. 4b, we use the formula (7.9), where u1 = gf, u2 = 0, u3 = q2.
(a)
(b)
\ 1 P p ^^f
1 _
BSi y \A '
/ 1 \ f A
(c) (d)
Fig. 4. Phase portraits of the reduced system for c = 0.
For a1 = 1 and —C2/16 < h < 0, for oscillations in the neighborhood of stable equilibria in Fig. 4b we take the same formula, in which we assume u1 = q2, u2 = q2, u3 = 0. Here the notation g'f 2 = (~Ci ± \/C2 + 16/1/4 (g\ > g2) is introduced.
In the case a1 = —1 and 0 < h < C2/16, for oscillations in the neighborhood of trivial equilibrium in Fig. 4c we use the formula (7.10), where u1 = q2, u2 = q2, u3 = 0, assuming that e\:2 = (Ci ± sjcf -16/?./4 (Ql > 02).
The frequencies of the above oscillations are calculated from the formulae (7.11) for the chosen values of uj.
7.3. Verification of the nondegeneracy condition in the regions of oscillations of the reduced system
In the ranges of parameters corresponding to oscillations of the reduced system with one degree of freedom, we introduce action-angle variables I,w1 [17], assuming that
h = h(h,c) = l-£pdQ.
Here integration is performed along the closed trajectories under consideration. The quantity P = P(Q,h,c) in the integrand is defined from relation (7.7). We have
= ± (7.12)
2ft J Q
The upper and lower bounds of integration correspond to the boundaries of the variation interval of q for the trajectory under study.
The function h = h(I1, c) inverse to (7.12) is the Hamiltonian of the reduced system, which is written in the action-angle variables. The frequency of the corresponding oscillation is equal to w = dh/dl1.
For further analysis we need to verify the nondegeneracy condition
d2h
H/0 (7,3)
of the Hamiltonian h = h(I1,c). Taking into account that
d2h duj duj 'dlf=dh =UJdh.
and using the formulae (7.8)-(7.11) and some relations from the theory of Jacobi elliptic integrals, we find that on the trajectories corresponding to oscillations in the neighborhood of stable equilibria in Fig. 3a, c the left-hand sides of (7.13) are, respectively, equal to
d'2h _ 7r2[2fc|A'(fci)(m - u2)(3uiu3 + h) + (E(ki) - fc|/i'(fci))(9c2 + Cih)] dlf ~ 'IK^hKui-ii^iiia-us)2
and
d2h = TT2[2k2K(k2)(u2 - u3)(-h + 3U1U3) - (E(h2) - k2K(k2))(-9c2 + Cih)] dl2 ~ 2/i 3 (k2)(Ul- u2)2(u2 - u3)2
(7.14)
(7.15)
We note that the first of these formulae was obtained previously in [15], but there it contained some inaccuracies; in (7.14) these inaccuracies were removed.
Numerical calculations have shown that in the case a1 = 1 the numerator in the formula (7.14) vanishes on some two-dimensional surface in the space of parameters c, C1, h when C1 < 0 and, thus, on this surface, the nondegeneracy condition is violated. In the other part of the parameter space in the case a1 = 1, and for all admissible parameter values in the case a1 = -1, the nondegeneracy condition is fulfilled.
In the case c = 0, for the three oscillation patterns described at the end of Section 7.2, we have, respectively,
d2h _ 7r'2[—'2CiE(ki) + K(ki)(C\ + y/C'{ + 16/?,)] ~ 4K3(ki)h '
d2h _ 2tt2(Ci - sJC2 + 16/?,)[Ci£(fci) - A'(fci)(Ci + \/C2 + 16/?)] dlf ~ K3(fci)(C2 + 16/?) (C\ + sJC2 + lQh)
d2h _ 2tt2(Ci + y/C'{ - 16h)[C\E(k2) ~ VP2 - 16h,K(k2)} K3(k2)(C2 - 16/?)(Ci - s/Cl^lQh) '
where k1 and k2 are calculated by the formulae from (7.9) and (7.10) for the values of Uj described at the end of this section. It is easy to verify that for all three cases the nondegeneracy condition is satisfied.
7.4. On the motions of the complete system with the Hamiltonian (7.5), (7.6)
7.4-1- On the motions of the model system. Let us come back to the model system with two degrees of freedom with the Hamiltonian F' of (7.6). Let g(t), R(t) be the solutions of the reduced system as described in Section 7.2.
The trivial equilibrium of the reduced system in Fig. 4 (the case c = 0) corresponds to the trivial equilibrium of the model system (and the initial nonautonomous system). The nontrivial equilibria in Fig. 4b correspond to the equilibrium point g = -C1/4, P = 0, I = 0 of the system with the Hamiltonian F'. Note that in the model system with the Hamiltonian r(10) of (3.5) (with d = 0) this equilibrium corresponds to the family of equilibria (3.13).
The trajectories corresponding to oscillations, as shown in Fig. 4, correspond to periodic solutions (with frequency el/2Uj) of the model system
g = g(e1/21), R = R(e1/21), I = 0.
In the case c > 0, the equilibria of the reduced system in Fig. 3a, c correspond to a one-parameter family (with parameter c) of solutions of the model system
n c
g = g0, P = 0, I = c, ^ = n0t, i10 = -+el/2—. (7.16)
2 go
The trajectories corresponding to oscillations in the neighborhood of stable equilibria, as shown in Fig. 3a, c, correspond to a one-parameter family of two-frequency conditionally periodic solutions
g = g{el'2t), R = R{el'2t), I = c, O = | + e1'2 ^ (7.17)
with frequencies el/2Uj (j = 1 or 2, see (7.11)) and Q.
7-4-2. On the motions of the complete system. Let us establish correspondence between the motions of the model system and the motions of the complete system with the Hamilto-nian (7.5), (7.6). We restrict us to the case c> 0.
It obviously follows from relations (7.5) and (7.6) that the condition
d2F
dI2
= 0
holds. Therefore, according to [17, 22, 23], most steady rotations (7.16) of the model system give rise to two-frequency periodic motions of the complete system with frequencies Q0 from (7.16) and 2n/T.
For the parameter values corresponding to oscillations of the reduced system, we rewrite the Hamiltonian of the model system and then the Hamiltonian (7.5), (7.6) of the complete system in the variables I1, I, w1, p. We obtain
n -__-
${I1,I,w1,<p,t) = -I + sl/2h(h, I) + (h, /, Wl, <p, t), (7.18)
where the last term is 2n-periodic in the angle variables wi and p and T-periodic in t.
In the Hamiltonian (7.18) the case of eigendegeneracy [17] takes place since the main part (of the order of unity) of the Hamiltonian depends only on one action variable.
Consider the nondegeneracy conditions [17]
d$ , d$ , d2$ , ,
The first of these inequalities obviously holds by virtue of (7.18). The second condition holds since the frequencies w1>2, which are calculated by the formulae (7.11), are nonzero. Finally, the third condition is equivalent to the condition for nondegeneracy of the reduced Hamiltonian which was verified in Section 7.3 and holds for all admissible values in the three-dimensional parameter space, except for points of some two-dimensional surface in the case a1 = 1.
This implies [17] that most two-frequency motions (7.17) of the model system in the complete system give rise to three-frequency conditionally periodic motions with frequencies Q + O(e), ewj + O(e) and 2n/T. The portion of the phase space which is not filled with such trajectories is exponentially small and is defined by the expression O(-5'e-1/2), where 5' = const > 0.
8. Resonant motions of a symmetric satellite in a neighborhood of cylindrical precession
As an application, we consider the problem of the motion of a dynamically symmetric satellite (rigid body) in a central Newtonian gravitational field in an elliptic orbit. The orientation of the principal central axes of inertia of the satellite in the orbital coordinate system OXYZ (whose axes OX, OY, OZ are directed, respectively, along the transversal to the orbit, along the binormal to the orbit's plane and along the radius vector of the center of mass relative to the attracting center) will be specified by Euler angles ^, 6, p.
The motion of the symmetry axis of the satellite is described by a reduced system with two degrees of freedom with the Hamiltonian [20]
4 , P2e +n , a/i(l-e2)3/%cos0
H = —- no-2/3 + 777^—i--To ~ 'P'<t> COt 0 COS V ~
2(1 + e cos v )2 sin2 9 2(1 + e cos v )2 v (1 + e cos v )2 sin2 9
. , aft(1 - e2)3/2 cos é a2fi2(1 - e2)3 cot2 9 . 2n
- p0 sin 0 + -r^-- + ^ ,- To— + - a - 1 1 + ecos v) cos2 9.
sin 9 2(1 + e cos v )2 2
Here, p^ and pe are dimensionless momenta conjugate to the angles ^ and 6, and v is the true anomaly of the orbit of the center of mass, which is taken to be an independent variable.
The Hamiltonian depends on three parameters: the eccentricity of the orbit e, which we assume to be small (0 < e ^ 1) in what follows, the inertial parameter a = C/A, where A and C are the equatorial and axial moments of inertia of the satellite (0 ^ a ^ 2), and / = tq/wq, where t is the projection of the angular velocity vector of the body in the orbital coordinate system onto the axis of dynamical symmetry (r = tq = const), and w0 is the average motion of the satellite's center of mass along the orbit.
There exists [21] a partial solution to the reduced system
n
■Ip = ir, 0=2> = P0 = 0,
which corresponds to the steady rotation of the satellite with angular velocity tq about the axis of dynamical symmetry perpendicular to the plane of the orbit (cylindrical precession).
We will investigate the motions of the satellite's axis in the neighborhood of this partial solution. Let us introduce perturbations by the formulae
0 = 7T + <?2, 0 = - + qi, P4,=P2, Pe=Pi
and represent the Hamiltonian of perturbed motion in the form (2.1), in which e = e and t = v.
The equation for the frequencies of small linear oscillations of the system in the case of a circular (e = 0) orbit of the center of mass of the satellite has the form [21]
w4 - (3a - 1 + a2/2 - 2a/)w2 + (a/ - 1)(a/ + 3a - 4) = 0. (8.1)
From this equation we find two points from the admissible part of the parameter plane (a, /) at which the resonant cases considered in this paper take place:
Point 1: a = 0.829862, /i = 1.082050, uj1=uj2 = -; (8.2)
3
Point 2: a = 0.826677, (3 = -1.215694, u)i = w2 = -.
Points 1 and 2 lie on the boundary where only necessary conditions for stability of the cylindrical precession of the symmetric satellite in the circular orbit are satisfied. For both points the normalized quadratic part of the Hamiltonian does not reduce to the sum of squares.
We note that there is also a point a = 1, / = 2 at which equation (8.1) has a double root w\ = w2 = 1 and the above-mentioned quadratic form reduces to the sum of squares. This case was previously examined in detail in [6].
Based on the results from the previous sections, we find resonant periodic motions of the symmetry axis of the satellite near its cylindrical precession for the parameter values from small neighborhoods of resonant points 1 and 2 in the three-dimensional parameter space.
8.1. Point 1
Consider the e-neighborhood of point 1 of the form (2.3) for e = e and resonant values of a* and from (8.2). Making the univalent canonical linear change of variables
qi = 0.352645p1 - 0.839103q2, Q2 = -1.313363q'l + 0.551959p'2, pi = 0.893811qi - 1.567384p2, p2 = -1.001397pi + 0.571053q2,
we reduce the autonomous quadratic part of the Hamiltonian to the form
H20 = l(p'l+lA) + -2(q'1p'2-p'1q'2).
The subsequent nonlinear change of variables transforms H40 to the form (2.4), where
71 = 0.0887855, 72 = -0.299151, 73 = 0.50129344.
Next, we make a transformation similar to (2.5) for n = 1 and normalize the nonautonomous quadratic part of the Hamiltonian in the terms of order e. As a result, we obtain the form (3.1), where
C1 + C2 = C1 = 0.550601^1 + 1.774066^1, Co = -0.292965, do = 0, c7 - eg = C2 = -0.768972^ - 1.270572^1.
We set up a corresponding model system of first approximation. Exploring its properties, we find that the trivial equilibrium of the linear system is stable under the condition
0.550601^1 + 1.774066^1 - 0.292965 > 0
and is unstable when the sign of the inequality is reversed.
Nontrivial equilibria of the nonlinear model system are described by the relations
V2 = 0, Y1 =0, y1 = y1o + eyn + O(e2), Y2 = eY21 + O(e2), y20 = -1.550367^1 - 4.995368^1 +0.824924, yn = (2.165044^2 + 9.090629^1 f 1 + 5.775834v? - 1.617792f 1 - 5.738694^1 + 0.879997)y10 Y21 = -(1.109882^1 - 0.171493f 1 - 0.246777)yw
and
y1 = 0, Y2 = 0, y2 = y20 + ey21 + O(e2), Y1 = eYn + O(e2),
1
y20 = -1.550367f 1 - 4.995368^1 - 0.824924,
y21 = (2.165044f1 + 9.090629^1 f 1 + 5.775834v? + 1.617792f 1 + 5.738694^1 + 0.879997)y. Y11 = (1.109882^1 - 0.171493f 1 + 0.246777)y20.
-1
20 ,
These equilibria exist if the right-hand sides of the expressions with squares are positive. The equilibria from the first pair are linearly stable and those from the second pair are unstable in the existence regions. The stability pattern is illustrated in Fig. 1a.
The equilibria described above correspond to solutions, 4n-periodic in v and analytic in e, of the complete nonautonomous transformed system. In the initial system these solutions are 4n-periodic in v and analytic in e1/2. Note that periodic solutions generated by pairs of equilibria arise from each other by shifting the independent variable v by 2n and hence physically correspond to the same periodic motion of the symmetry axis of the satellite.
We present the explicit form of the periodic solution to the initial system, which corresponds to the first pair of equilibria:
91 = e1/2
-(0.172292^1 - 0.413139i/i + 0.495985) sin ^ +
3
+ (0.104100^1 + 0.335416i/i + 0.174681) sin - v
y10 +
+ e3/2 [(0.0420513^1 - 1.921111vi - 1.059478) yio +
+ (0.104375^1 + 1.304577vi - 0.643195) yn - (8.3)
- (0.554885^ii + 2.501462i/i + 0.159989) Y21] sin ^ +
+ [(-0.262580^1 - 0.846047vi - 0.389228) yio + + (0.312301^1 + 1.006250vi + 0.063901) yn +
3 1
+ (—0.392473^1 - 1.264569i/i - 0.370296) Y2i] sin - i/J + 0(e5/2);
92 = e1/2
—(0.269671^ii - 0.646644z/i + 1.522315) cos ^ +
3
+ (0.0185910^ii + 0.0599013i/i - .208294) cos - v
+ e3/2
yio +
[(0.0679485^1 + 2.876580^1 + 0.390343) yio +
+ (0.163368^1 + 2.041920^1 - 1.752728) yn + (8.4)
- (0.812287^1 + 3.970059i/i - 1.400310) Y21] cos ^ +
+ [-(0.178975^1 +0.576669^1 - 0.347254) y1o + + (0.0557731^1 + 0.179704^1 - 0.228078) yn +
3 i
+ (0.321776^1 + 1.036781i/i + 0.281380) Y2i] cos - i/J + 0(e5/2).
The periodic solution corresponding to the second pair of equilibria is in its structure the same, and the sine and cosine functions change places; y10 is replaced with y20, y11 with y21, and Y21 with Y11; also, the signs of some coefficients in terms with ¡1 and v1 and the values of some terms that do not contain these quantities are reversed.
According to the results of Section 6.2, in nonresonant cases the periodic solution (8.3), (8.4) is stable for most initial conditions and is formally stable; for the cases of fourth-order resonance it is formally stable. The second periodic solution described above is unstable.
If we neglect terms of order e3/2 and higher in the formulae (8.3), (8.4) and in the expressions for the second solution, we obtain equations of closed curves in the plane of q1, q2, with a typical size of order e1/2. These curves are symmetric relative to both coordinate axes and intersect each axis at two or four points. The form of the curves depends on the chosen values of ¡1, v1 from the region of existence of the solution in a neighborhood of the resonant point. A number of typical variants of these curves are shown in Figs. 5a-5f.
8.2. Point 2
Consider the e2-neighborhood of the resonant point P3, which is defined in (5.1) for e = e. We first make the following linear normalizing change of variables:
q1 = -0.018664^1 + 0.972395q2, q2 = 1.030098q1 - 0.019772p2, P1 = 0.428495q1 + 1.020164p'2, P2 = 0.963018p1 + 0.404492q2,
which reduces the form H20 to
13
H2o = T2(pl2i+Pll) + T2(QliPl2-pliQl2),
r >92 9 ^^^ 9i •92 9i ( '92 \ 9i
0 0 0
(d) (e) (f)
Fig. 5. Periodic motions in a neighborhood of point Pi.
and a normalization of the form H40. This yields the expression (2.4) with the coefficients
71 = 0.060828 , 72 = 0.126671, 73 = 0.168391.
Next, we transform the nonautonomous quadratic part in terms through order three in e. This gives r21 = 0 and
T22 = (-0.206325^2 + 1.012581^2 - 0.0444830)(y2 + y2) +
+ (-0.414942^2 + 0.637429^2 - 0.00205811)(yi Y2 - Y1y2), T23 = (-0.206325 № + 0.000880917)y2 + (-0.206325 ^3 - 0.000880917)y| -- 0.414942 ^3 (yiYî - Y№).
Let us introduce the notation
H20 = -0.215597 + 4.907708 ^2, №0 = -0.000426957.
It follows from the above relations that the conditions for stability of the trivial equilibrium of the system are given by the inequalities
1^2 < 1^20, 1^3 < 1^30.
When ¡2 = ¡20, a model system of third approximation is considered for which the form r22 becomes
T22 = (0.0874022 - 1.398986 V2)(yiY> - Y^)-
This system has isolated equilibrium points: for ¡3 > ¡30, it has a linearly stable pair of equilibria given by
yi = 0, Y2 = 0, y2 = y 20 + ey2i + O(e2), Yi = e2Yi2 + O(e3 ), y20 = 0.00724101 + 1.695959^3, Yi2 = (0.0874022 - 1.398986^2)y20, y2i = (4.676218v| + 2.100087V2 - 0.0327068e)y;TC)i,
and for ¡3 > — ¡30, it has an unstable pair of equilibria defined by the conditions
y2 = 0, Yi =0, yi = yi0 + eyn + O(e2), Y> = e2Y>2 + O(e3 ), y20 = -0.00724101 + 1.695959^3, Y22 = (1.398986v2 - 0.0874022)yi0,
yii = (4.676218^1 + 2.100087^2 - 0.0327068)y
-1 10 .
The periodic solution to the initial complete system, which is generated by the first pair of equilibria, can be represented as
91 = e3/2
+ e
5/2
v 3 5 -
0.00888998 sin - + 0.972395 sin - v - 0.00615846 sin - v 2 2 2 .
(0.104079yio + 0.00888998yn + 0.001613573^1 ) sin ^ +
yio +
3
+ (0.972395yn + 0.01866443-21 ) sin -v-
5 1
- (0.852579yio + 0.00615846yn + 0.00222669Y2i) sin - i/J +0(e7/2);
92 = e3/2
+ e
5/2
v 3 5
-0.00555937 cos - + 1.030098 cos -v + 0.00845298 cos - v 2 2 2 .
(0.143855yio - 0.00555937J/11 - 0.01773813-21 ) cos ^ +
yio +
3
+ (1.030098J/H - 0.01977203-21) cos - v +
5 1
+ (—0.911498yio + 0.00845298yn + 0.002518633^1 ) cos - i/J + 0(e7/2).
The periodic solution generated by the second pair of equilibria is obtained if (as for point 1) in the above expressions we interchange the sine and cosine functions, replace yi0 with y20, yii with y2i, and Y2i with Yii. In addition, in the expression for qi we need to reverse the signs in front of the terms with Yii, and in the expression for q2, the signs of all terms except for terms with Yii.
The first of these periodic solutions is stable for most initial conditions and is formally stable in the absence of fourth-order resonances and is formally stable in resonant cases; the second periodic solution is unstable.
Let us neglect terms of order e5/2 and higher in the solutions found above, and let us neglect in the remaining part terms with harmonics cos(v/2), sin(v/2), cos(5v/2), sin(5v/2) in view of the smallness of the coefficients. As a result, we obtain equations of an ellipse, close to a circle, with semiaxes of order e3/2.
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