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6Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, pp. 593-609. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190419
MSC 2010: 93B18, 93B52
Bifurcation Analysis of Periodic Motions Originating from Regular Precessions of a Dynamically Symmetric Satellite
E. A.Sukhov
We deal with motions of a dynamically symmetric rigid-body satellite about its center of mass in a central Newtonian gravitational field. In this case the equations of motion possess particular solutions representing the so-called regular precessions: cylindrical, conical and hy-perboloidal precession. If a regular precession is stable there exist two types of periodic motions in its neighborhood: short-periodic motions with a period close to 2n/w2 and long-periodic motions with a period close to 2n/wi where w2 and wi are the frequencies of the linearized system (W2 > Wi).
In this work we obtain analytically and numerically families of short-periodic motions arising from regular precessions of a symmetric satellite in a nonresonant case and long-periodic motions arising from hyperboloidal precession in cases of third- and fourth-order resonances. We investigate the bifurcation problem for these families of periodic motions and present the results in the form of bifurcation diagrams and Poincare maps.
Keywords: Hamiltonian mechanics, satellite dynamics, bifurcations, periodic motions, orbital stability
1. Introduction
The motion of a dynamically symmetric satellite in a circular orbit was studied in many papers [1-4, 6-16]. One of the important subjects of study in satellite dynamics is periodic motions. Since equations of motion in satellite dynamics usually cannot be integrated analytically, a relevant problem is to develop asymptotical and numerical methods for computing and studying periodic motions. The problem of numerical computation of periodic motions was
Received June 20, 2019 Accepted October 20, 2019
This work was carried out at the Moscow Aviation Institute (National Research University) within the framework of the state assignment (project No. 3.3858.2017/4.6).
Egor A. Sukhov sukhov.george@gmail.com
Moscow aviation institute (National Research University) Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993 Russia
addressed in [17] where an approach that allows a reduction of the boundary-value problem to a simpler Cauchy problem was proposed. Numerical methods based on this approach were developed in [18-22] where motions arising from the satellite's hyperboloidal precession were computed analytically and numerically and their linear orbital stability was studied.
In this paper we compute analytically and numerically families of short-periodic motions originating from regular precessions of a symmetric satellite in a nonresonant case and families of long-periodic motions originating from hyperboloidal precession in the case of third- and fourth-order resonances. We consider the bifurcation problem of these motions.
2. Formulation of the problem
We consider the motion of a satellite around its center of mass O which moves in a circular orbit in a central Newtonian gravitational field at angular velocity The satellite is considered to be a dynamically symmetric rigid body with principal moments of inertia J1, J2 and J3 (J1 = J2). To describe the satellite's motion around its center of mass, we introduce an orbital reference frame OXYZ and a mobile reference frame Oxyz. Axes OX, OY and OZ are aligned with transversal and normal vectors to the orbit and with the radius vector of the satellite's center of mass, respectively. Axes Ox, Oy and Oz are aligned with the satellite's principal axes of inertia. The relative position of these reference frames is defined by Euler's angles é, 0, p. In this case the system possesses a cyclical coordinate y and its respective impulse pv retains constant value. Following [3, 7], the equations of motion of a dynamically symmetric satellite can be written in a canonical form with the following Hamiltonian:
Pi p2 (l cos 0 , \
H = + 9 - + cosVcote ity -
2 sin2 0 2 V sin2 0 J
1 2 2 „ cos é 1 2 _ - sin #.</, + -7 cot2 0 + 7-—^ + -ó cos2 0,
2 sin 0 2
(2.1)
where p^ and pg are dimensionless impulses corresponding to variables ^ and d, 5 = 3(J3/J\ — 1)
and 7 = -j-j^ are dimensionless parameters and r0 is the projection of the satellite's absolute
angular velocity along its principal axis Oz. The independent variable is a true anomaly v = w0t. The equations of motions with Hamiltonian (2.1) possess particular solutions
n
Go = 2' = 7T, Pe0 = 0, P'lpo = (2-2)
sin do = ¡-, 00 = 0, pe0 = 0, prip0 = ¿sin 00 COS 00. (2.3)
n
do = cos 00 = —7) Pe0 = sin 00, 2%, = °> (2-4)
known as regular precessions [2, 3].
Solution (2.2) exists for all values of 7, 5 and is known as cylindrical precession. It defines a relative equilibrium of the satellite in the orbital reference frame when its principal axis Oz is aligned with the normal vector to the satellite's orbit and describes a cylinder in absolute space (Fig. 1a). Figure 1b presents a stability diagram for cylindrical precession where gray areas represent domains of instability and white areas represent domains of stability.
Solution (2.3) exists for 171 ^ |5 —1| (subdomains I — III in Fig. 1d) and is known as conical precession. In this case the satellite's principal axis Oz remains orthogonal to the transversal
Fig. 1. Regular precessions of a symmetric satellite (a, c, d) and their existence and stability domains in the plane of problem parameters 7, 8 (b, e, f). Here G0 is the gravitating center, Y is aligned with the normal vector to the satellite's orbit and X is aligned with the transversal vector.
vector of the orbital reference and describes a cone in absolute space with an angle n — 26q at its tip (Fig. 1c). Figure 1d presents a stability diagram for conical precession where gray areas represent domains of instability and white areas represent domains of stability.
Solution (2.4) is known as hyperboloidal precession and defines a relative equilibrium of the satellite in the orbital reference frame when its principal axis Oz lies in a plane perpendicular to the radius vector of the satellite's center of mass and remains at a constant angle n — to the normal vector (Fig. 1d). The satellite's principal axis Oz describes a hyperboloid in absolute space. Hyperboloidal precession exists for \y| ^ 1 and is Lyapunov stable if 8 > 0 [4] (Fig. 1f).
Fig. 2. Periodic motion in the neighborhood of hyperboloidal precession. Here Go is the gravitating center, z0 is the position of the satellite's figure axis Oz in the case of hyperboloidal precession and s is the trace of the satellite's principle axis Oz on the unit sphere.
Two types of periodic motions exist in the neighborhood of a stable regular precession: short-periodic motions with period close to 2n/u2 and long-periodic motions with period close to 2n/u\, respectively. These periodic motions represent oscillations of the satellite's principle axis Oz in the neighborhood of regular precession (Fig. 2) and can be obtained as small parameter power series of the oscillation amplitude c = c(Ah) where Ah is a deviation of the energy constant h from its value for a corresponding regular precession. Here are the frequencies of the linearized system
"1,2 = \j Fj ^Fj, (2-5)
where coefficients Fj and Fj (j = r, Z,K) in (2.5) corresponding to cylindrical (Z), conical (K) and hyperboloidal (r) precession are given by the following formulae:
FF = —, F.\ =
S + 1 _r_ (S - 1)2 +472S
2 7 2 4
F? = \l2~ 7 + ^ + 1, if = ^74"73 +72 " 27<5 + <5 + i72<5 + ^2,
FK =
FK =
1
2(5 - 1)2 1
1 + 552 + 52Y2 + 3S72 - 253 - 45 1 - 4y2S5 + (Y2 + 1)2 54 + (-4 + 1072 + 6Y4) S3 +
4(6 - 1)4
+ (974 + 6 - 1072) 62 + (-4 + 272) 6
3. Analytical computation of periodic motions originating from hyperboloidal precession
In this paragraph, following [14-16, 23], we obtain families of periodic motions originating from regular precessions of a dynamically symmetric satellite in the form of small parameter power series of amplitude c.
3.1. Nonresonant case
To obtain the periodic motions originating from regular precessions of a satellite in a non-resonant case, we follow the method developed by Lyapunov [5] and introduce local coordinates
^ = ^o + 6 , 0 = Oo + 6 , p^ = p^o + ni , Ve = Ve0 + V2 , (3.1)
in the neighborhood of a regular precession and expand the Hamiltonian (2.1) into series of (i, n (i = 1, 2)
H = Hj + H3 + Hi + ..., (3.2)
where H3k (j = Z, K, r) are polynomials of degree k in variables (i,ni. We then apply a canonical transformation
6 = K1A1ixi + K2a{2x2, 6 = kiA32iyi + кl2A122y2, ni = KiAhxi + A32X2, V2 = KiAiiyi + KjAijy2, which transforms (3 . 2) to the following normal form:
H = \u)i {x\ + y\) + {xl +vl)+ Y. Krn^^A'^vT'yT +
ni+n2+mi +m2 =3 (3 4)
+ Y^ h Xni xn2 yml ym2 +
+ / j hnin2mim2xi x2 yi y2 + . ..,
ni+n2 +mi +m2 =i
where hnin2mim2 are constant coefficients dependent upon 7 and 8. The coefficients k{, K2, Aii, A\2 (i = 1,...,4, k = Z,K, r) of the transformation (3.3) are given by the following formulae:
Afi = — (1 - 7 - 6 + wi2) to 1, Af2 = — (1 - 7 - 6 + w22),
_ 1 „ o, 1
11 Wl ' W2
Afi = A 22 = 7 — 2,
Afi = (27 — 1 — y2 — 8 + wi2), Af2 = (27 — 1 — y2 — 8 + U22),
A,zn = — (7 + S- 7^i2 - 1 + wi2), A$2 = — (7 + 5 - ^u)22 - 1 + w22), (3.5)
Wi W2
Kf = Af (Wi), i = 1,2
= -73 + (6 - 4) 72 + (6 - 2 w2 - 2 6) 7-3+ w
+ 2 8 + wl + 82 + 2 w2 — 2 8w2,
jvr 8 1 / o 0 0 0)
An =-^ (u)28 - 2W25 + U2 + U)28U)i - w27 - w2w\ ),
wiwi72 v y
jvr 8 1 /000 0)
AV2 =-o — 2W\5 + wi5W2 — W17 + — U)\U)2 ),
wiwi72 v y
A2i = A22 = 8 + 1 1
131 ~ (5 - l)2
A31 = Tx—7Y2 (wI2 - 46'2 - 6V - 3^72 + 253 + 25 - 2uji26 + UJI262) , (3.6)
All = (w22 + W2252 - 2lo225 - 5272 - 3572 + 253 + 25 - 452)
A,= ---- w2 - 2u)-20uji2 + W2Wl2^2 -
W1W172
— w2y2w12 — 3w252 — w25y2 + w253 + 3w25 + w2y2 — w2y2w12 5),
jvr 1 / Q O O 0 0
A42 =-TÎWI^ +3wi^ + wi7 +W1W2 +wiw2 5 -
w1w1y2 v
— 2w1 5w22 — w1 — w1y2w22 — w1y 2w225 — 3w152 — w15y2) ,
^(¿^W1^' i = 1'2
AK(w) = 2w2 — w4 — y254 — 2y253 + /8 — 2w2) 7252 + 74^2 + 3y45+ + /5 + 3w4 — 8w2) 5 + /1 — 2w2) y2 + /I2w2 — 3w4 — 10) 52 + 55+ + /—6 + 4w2) y25 + /—5 + 2w2) 54 + /w4 + 10 — 8w2) 53 — 1. r _ ô-ujj r _ ô-uj'j W1 W2
Ar = Ar = y a21 = a22 = y,
Ar = 5 — w2, A^ = 5 — wi
_ Az
(3.7)
^ \j Wi4 + — 2 uji2 + 5'2 ' 1 lî2"
The expressions (3.5), (3.6) and (3.7) correspond to the cases of cylindrical (Z), conical (K) and hyperboloidal (r) precession, respectively.
We then apply another canonical transformation Xi,yi ^ qi,pi (i = 1,2) which transforms (3.4) to the form
K= \ujl(q21+pj) + ^uj2(q22+p22) +
+ aio (q2 + pi)2 + an(q2 + p1)(q1 + p2) + «02 (q1 + p2)2 + K(5),
where a20, a11, a02 are constant coefficients dependent upon y and 5 and K5 are terms of order 5 and higher. The canonical system with Hamiltonian (3.8) admits a family of short-periodic motions
q2 = c sin Q2(v — Vo) , P2 = c cos^2(v — Vo), q1 = P1 = 0, (3.9)
with period T2 = 2n/Q2 where
Q2 = Wi + 4c2 aoi + O(c4).
The parameter c in (3.9) represents the small amplitude of oscillations of the satellite's axis Oz in the neighborhood of a regular precession. Returning to the initial variables 0, d, p^, pe, we obtain the following asymptotic expressions for the families of short-periodic motions originating from regular precessions:
0 =00 + cK2 A{2 sin Q2(v — Vo) + O(c2), pH =p^q + cK2 A3i2 cos Q2(v — V0) + O(c2),
(3.10)
d =d0 + ck2A322 cos Q2(v — v0) + O(c ), pe =p$q + cK2 A42 sin^2(v — V0) + O(c2),
W1 W2
wi
where coefficients k\, k32, Aj1, A32 (i = 1,..., 4, k = Z,K, r) are given by (3.5)-(3.7). In presenting further results, we will refer to the families (3.10) of short-periodic motions originating from cylindrical, conical and hyperboloidal precessions as Zs, Ks and rs, respectively.
The canonical system with Hamiltonian (3.8) also admits a family of long-periodic motions
q2 = p2 = 0, q1 = c sinQ^v — vo), p1 = c cosQ1(v — vo) (3.11)
with period T1 = (T1 > T2) where Q1 = u1 + 4c2a2o + O(c4). Upon returning to the
initial variables d, p^, pg expressions (3.11) give the following series of amplitude c:
^ =^o + c K1 A\1 sin Q1 (v — vo) + O(c2 ), m =P^c + cK1 A31 cosQ^v — vo) + O(c2 ),
(3.12)
d =d0 + ck1A321 cos Q1(v — v0) + O(c2), pg =pg0 + ck1 A41 sinQ^v — vo) + O(c2),
which represent the long-periodic motions originating from regular precessions in a nonresonant case.
3.2. Third-order resonance
To analytically obtain the families of periodic motions originating from hyperboloidal precession in the neighborhood of third-order resonance (w2 = 2u1 ), we apply a canonical transformation ^, d, p^, pg ^ £2, n1, n2 which transforms the Hamiltonian (2.1) into the following normal form:
K1 = + VÏ) + + 4) + AMvl - ¿1) - 2r?i6m] + O4, (3.13)
where A is a resonance coefficient. In the case of precise resonance w2 = 2w1 this coefficient is given by the expression
3wf *
Following [10, 16], we introduce new time t = u1v, apply a scaling canonical transformation = eaxi, ni = £ayi, (i = 1,2) with valence n = 1/e2a2 and then canonical transformations Xi = s/2FiS'm4>i, yi = s/WiCoscpi, (i = 1,2) and <pi = Q1, (j)2 = Q2 + 2Qi, n = Pi~2P2, r2 = P2 which transform (3.13) to the form [10, 15]
■-11
K11 = Pi + ¡iP2 + e(Pi - 2P2 )VP2 sin Q2 + O(e2), (3.14)
where ¡1 = u2/u1 — 2 is a small parameter, e is a scaling coefficient and P1 ^ 2P2. The system with Hamiltonian (3.14) possesses a first integral P1 = C using which and omitting the terms O(e2) we arrive at a truncated system
-hi
K111 = 1P2 + e(C — 2P2)VP2 sin Q2, (3.15)
which describes long-periodic motions in the neighborhood of hyperboloidal precession in the case of the third-order resonance. The system (3.15) possesses equilibria
n* 71 p* vV+ 6 e2C , . Q-2 = g > 2 =-^-, (3-16)
and
Qi = -§, J? = ->'+ vV + fe2C (3 17)
In the initial system with Hamiltonian (2.1) the equilibria (3.16) and (3.17) correspond to families of long-periodic motions, which we will refer to as r and r2, respectively.
Depending upon the values of the problem's parameters 7, 5 and h there may exist one or both of these families in the neighborhood of the third-order resonance curve
Yo = ±--Above the surface given by the following series of h
55
which emanates from the resonant curve 70, there exist both r1 and r2. Below the surface (3.18) there exist one family r1 in the case y < Y0 and one family r2 in the case 7 > y0. Due to their size we do not present the expressions for the families r1 and r2 in the initial variables in this work.
3.3. Fourth-order resonance
To obtain the long-periodic motions in the case of precise fourth-order resonance (w2 = 3w1), we apply a canonical transformation ^, 6, p^, pe — q1, q2, p1, p2 that transforms the Hamiltonian (2.1) into normal form [11, 23, 24]
K1 = +p\) + \u2{ql + Pi) + jCao(9? + pj)2 + ]cn(qj +pj)(q22 + p22) +
2 2b 4 4 (3.19)
+ ^co 2{ql +plf --[qf{qiq2 +3pip2) ~ p'lipm + 3<?i<?2)] + 06,
where c11, c20, c02 are constant coefficients dependent upon the frequencies (2.5). In the case of precise fourth-order resonance they take on the following form:
cn =7^-13 (678w? + " 20u)? + !).
560w1
1
Oa 1
1
C20 - 319wi + 60w? - 3)>
320w1
cm =---—r (906w? - 983wf - 220w? + 11).
6720w®V 1 1 1 '
In (3.19) b is a resonant coefficient given by the following expression:
1 (1 - 15wf + 56uf) v/-3 + 30 u)j - 27 uf
b=
120
We introduce new time r = uj\u, apply a scaling canonical transformation qi = s/euji/bxi, Pi = sj£U)\/byi, (i = 1,2) with valence n = b/(ewi), a canonical transformation Xi = s/2risin <pi, yi = s/WiCos(f)i, (i = 1,2) and omit the terms of order e2 and higher, which brings us to a truncated canonical system with a Hamiltonian [11, 23]
K11 = r1 + 3r2 + £
C20r2 + Cnrir2 + c02r| + rf y^cos <t>\ - 302 (3.20)
and a first integral 3r1 + r2 = const where e is a scaling coefficient. We apply a canonical transformation ^ = 01, © = 02 — 301, J = 3r2 + r1, R = r2, which transforms a system with the Hamiltonian (3.20) to the following form:
d.0 1 y/J - 3R(J - 12R) cos 8 T „ d,R r-, T ^m3/2. „ — = e-------h a J + ¡jlR, — = e VR(J - 12 R)' sin 8 (3.21)
dR
, = -:—1=-:--h a j + ij,k, —
d,T 2 s/R d,T
with a first integral J = J0 = const where a = c11 — 6c20 and i = co2 + 9c2o — 3c11 are constant coefficients. The system (3.21) possesses equilibria
n
e = -1 - sign[(12.T* - 1)(<7 + 2nx*)],R = x* Jo, (3.22)
where x* is a real-value root of a cubic equation
(432 + 16i2 )x3 + (16ia — 216)x2 + (4a2 + 27)x — 1 = 0. (3.23)
For the values of the problem's parameters 7 and 5 considered in this work, Eq. (3.23) possesses three real-valued roots x* which upon substituting into (3.22) and returning to the initial variables d, p^, pg give three families of long-periodic motions that originate from the hyper-boloidal precession in the case of fourth-order resonance. We refer to these families as r2, r3 and r4. Numerical analysis carried out in [23] shows that the family r2 obtained here coincides with the family r2 which was obtained in the previous subsection.
4. Bifurcation analysis of periodic motions originating from regular precessions
Analytical representations of the families Zs, Ks, rs, r (i = 1,..., 4) obtained in the previous section are only valid for small values of the amplitude c and small deviations Ah of the energy constant h from its value on a corresponding regular precession. To obtain the existence domains of the said families for nonsmall values of c, a numerical method has to be used. In Refs. [14-16, 23], using a method described in [18, 22], the families rs, (i = 1,..., 4) were continued numerically to the borders of their existence domains. Linear orbital stability of these periodic motions was studied in [15, 22, 23]. In this section we construct existence domains of the families of short-periodic motions Zs, Ks, rs and consider the bifurcation problem of the families Zs, Ks, rs and r (i = 1,...,4).
4.1. Existence domains of periodic motions
Using the method described in [18, 22] the families Zs, Ks, rs of short-periodic motions arising from regular precessions of a symmetric satellite were numerically continued to the borders of their existence domains in the problem's three-dimensional parameter space. In Fig. 3 we present existence domains of the aforementioned families for fixed values of 5 = 0.5, 5 = 1.0, 5 = 2.8 and assuming 7 ^ 0.
The family Zs of short-periodic motions originating from cylindrical precession exists in a domain between the curves SZ and SZ. In Fig. 3 this domain is colored gray. The family Zs arises from the cylindrical precession on the curve SZ and terminates on curve SZ. In the case 5 ^ 1.0 the existence domain takes on the form shown in Figs. 3a, 3b. The form of the aforementioned domain in the case 5 > 1.0 is shown in Fig. 3c. Numerical computation has
(a) Families Ks, Za, Ts, 5 = 0.5 (b) Families ZS,TS, 5 = 1.0
(c) Families ZS:TS: 5 = 2.8 (d) Families Ks, 5 = 2.8
Fig. 3. Existence domains for the families of short-periodic motions Zs (gray areas), Ks (horizontally-hatched areas), rs (cross-hatched areas) arising from cylindrical, conical and hyperboloidal precession of a symmetric satellite.
shown that in this case the family Zs cannot be continued using the method described in [18, 22] into two subdomains enclosed by the curves .
The horizontally-hatched areas in Fig. 3 correspond to the existence domain of the family Ks of short-periodic motions arising from conical precession. For 6 < 1.0 (Fig. 3a) the aforementioned domain is enclosed by curves SK, SK and SK. Here the curve SK corresponds to conical precession. The family Ks terminates on the border SK. On the curve a bifurcation occurs and the family Ks coincides with the family Zs of short-periodic motions arising from cylindrical precession. With increasing 6 point Po 1 of the curve SK approaches point Po2 and at the value 6 = 1.0 the existence domain becomes degenerate. For 6 > 1.0 the existence domain of the family Ks takes on the form shown in Fig. 3d.
The cross-hatched areas in Fig. 3 correspond to the existence domain of the family rs
of
short-periodic motions arising from hyperboloidal precession. The existence domain of the family rs is enclosed by curves Sq , Sr and S2 where the curve Sq corresponds to hyperboloidal precession. The family rs arises from hyperboloidal precession on the curve Sq and terminates on the border S2 . The curve Sr corresponds to the bifurcation values of the problem's param-
uj1=2uj2 0ji=3uj2 uji—4:uj2 1.0 7
Fig. 4. Existence domains for the families of periodic motions rs and r (i = 1,...,4) emanating from hyperboloidal precession. Gray corresponds to the existence domain of the family rs of short-periodic motions.
eters. On Sr the family rs coincides with the family Zs of short-periodic motions arising from cylindrical precession.
Figure 4 shows existence domains for the families of long-periodic motions r (i = 1,..., 4) arising from hyperboloidal precession for fixed value of 5 = 1.0. This pattern of the existence domains remains valid for 0.115 < 5 < 3.0. Numerical computation shows that these domains are symmetric with respect to the axis h, so further results are presented assuming 7 > 0. In this figure the solid lines divide the parameter space into subdomains in which there may exist different numbers of long-periodic motions. On the borders of these subdomains, a bifurcation of families rs, r (i = 1,... , 4) occurs.
4.2. Bifurcation analysis of short-periodic motions
It has been noted above that a bifurcation of short-periodic motions originating from regular precessions may occur on the borders of their existence domains.
Figure 5 shows existence domains and a diagram of periods Tk , Tz , Tr of the families Zs, Ks, rs for fixed values of 5 = 0.5 and h = 0.35. In Fig. 5b the solid lines represent intervals of linear orbital stability and the dashed lines represent intervals of orbital instability of the aforementioned families of periodic motions. The method of investigating the linear orbital stability of the said periodic motions is described in Refs. [19, 22].
For 7 < 0.16 there exist only one family of linear orbitally stable short-periodic motions - rs. At the value 7 = 0.16 a family of linear orbitally stable short-periodic motions Zs arises from cylindrical precession and at the value 7 = 0.42 a family of orbitally unstable short-periodic motions Ks arises from conical precession. With increasing 7 the periods Tk and Tz of the families Ks and Zs converge and coincide at the point B1 (7 = 0.78). At this point a bifurcation occurs: the family Ks coincides with Zs and periodic motions of the family Zs become orbitally unstable. With further growth of parameter 7 the period Tp of the family rs converges with the period Tz of the family Zs and these two families coincide at the point B2 (r = 1.11). To the right of the point B2 only one family of short-periodic motions remains — the family Zs of short-periodic motions arising from cylindrical precession. Periodic motions of the family Zs are linear orbitally stable for 7 > 1.11. The family Zs coincides with cylindrical precession at point P3 (7 = 1.64). This bifurcation pattern remains valid for 0 < 5 < 1.0.
Fig. 5. Existence domains and a bifurcation diagram of the families Zs, Ks, rs of short-periodic motions emanating from cylindrical, conical and hyperboloidal precession for fixed values of S = 0.5, h = 0.35. Here TZ, TK, Tr are periods of the periodic motions belonging to the families Zs, Ks, rs, respectively.
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Q
(b)
Fig. 6. Poincare maps computed in the neighborhood of bifurcation points B\ (a) and B2 (b) for fixed values of 7 = 0.77 and 7 = 1.09, respectively.
To verify the results described above, Poincaré maps were constructed in the neighborhood of the bifurcation points B\ and B2. Figure 6a shows a map constructed near the point B\ for the values 7 = 0.77, ô = 0.5, h = 0.35. On this Poincare map a point marked Zs corresponds to a linear orbitally stable short-periodic motion of the family Zs and the point marked Ks corresponds to an orbitally unstable short-periodic motion of the family Ks. With 7 approaching the bifurcation value the points corresponding to the families Ks and Zs converge and coincide forming a point corresponding to an orbitally unstable periodic motion of the family Zs (Fig. 6b).
Figure 6b shows a Poincare map constructed near the point B2 for the values 7 = 1.09, ô = 0.5, h = 0.35. On this map the point marked Zs corresponds to an orbitally unstable short-periodic motion of the family Zs and the point marked rs corresponds to a linear orbitally stable short-periodic motion of the family rs. With 7 approaching the bifurcation value at point B2 the points corresponding to the families Zs and rs converge and coincide forming a point corresponding to a linear orbitally stable periodic motion of the family Zs (Fig. 7).
Fig. 7. Poincare map computed in the neighborhood of the bifurcation point B2 for fixed value of 7 = 1.15.
In Fig. 6 and Fig. 7 there also exist points marked Zi (i = 1,..., 5), K1, Г3)4 corresponding to long-periodic motions arising from regular precessions. The families Г3 4 of long-periodic motions arising from hyperboloidal precession have been obtained analytically and numerically in [22]. Numerical computation shows that the families Zi (i = 1,...,5) and Ki arise from cylindrical and conical precessions, respectively.
4.3. Bifurcation analysis of long-periodic motions originating from hyperboloidal precession
Figures 8 and 10 show bifurcation diagrams of periodic motions originating from the hyperboloidal precession of a satellite for S =1.0 and different values of the energy constant h. In these diagrams the periods Ts, T (i = 1,..., 4) of the families rs, Г (i = 1,... , 4) are mapped against the parameter 7 for fixed values of h. The curves NTs represent period Ts of the family rs multiplied N times. The solid lines represent intervals of linear orbital stability, while gray lines represent intervals of orbital instability.
7
Fig. 8. Bifurcation diagrams for families of periodic motions originating from the hyperboloidal precession of a symmetric satellite obtained for fixed values of h = 0.001, S = 1.0 and h = 0.1, S = 1.0.
Figures 8 show a diagram for h = 0.001 which represents relatively small deviations of the energy constant from its value on hyperboloidal precession and corresponds to the results presented in the analytical studies [10, 13]. In this case, for small values of 7 there exist a family of short-periodic motions rs and a family of long-periodic motions ri. At point Bi a family of long-periodic motions r2 branches off from the family rs, so between the points Bi and B2 there exist two families of long-periodic motions which arise from hyperboloidal precession. At point B2 the period Ti of the family r1 becomes twice as large as the period Ts of the family rs and the families r and rs coincide. The family r2 can be numerically continued to the point B4 where it coincides with rs. At this point the period T2 becomes equal to 3Ts. At point B3 two families arise: r3 and r4. The family r3 can be continued to the point B4 where it coincides with rs. Upon approaching point B3 the period T3 gets closer to 3Ts . The family r4 coincides with rs at point B5 with its period T4 becoming equal to 4Ts. The family rs can be continued along the 7 axis to the value 7 ~ 1.01 where it coincides with a family of periodic motions that arises from cylindrical precession.
Figure 9 shows Poincare maps computed for fixed values of h = 0.001 and 5 = 1.0 in the neighborhood of the bifurcation point Bi. The left map shows families rs and r before the bifurcation. The right map shows the family r2 branching off from rs with rs becoming orbitally unstable.
Fig. 9. Poincare maps computed for fixed values of h bifurcation point Bi shown in Fig. 3a.
0.001 and 5 = 1.0 in the neighborhood of the
Figure 10a shows a diagram for h = 1.0. In this case, for small values of 7 there exist a family rs and a family rs. At point BI families r| and r2 appear. The family r| exists for h > 0.05 and has been obtained numerically using Poincare maps. With increasing 7 the period Ti becomes closer to 2Ts and at point B2 the families ri and rs coincide. r2 coincides with rs at point B3 with its period T2 becoming equal to 2Ts. Approaching the point Bi, period T^ of the family gets closer to 2Ts and at point Bi the family coincides with rs.
At point B3 two families of long-periodic motions arise: r3 and r4. The family r3 can be continued to the point B4 where it coincides with rs and its period T3 becomes equal to 3Ts. The family r4 coincides with rs at point B5 where its period T4 equals Ts. The family rs can be continued along the 7 axis to the value 7 ~ 1.04 where it coincides with a family of periodic motions that arises from cylindrical precession.
/ 4 Ts
T-2 \l3 3 Ts
B{
2 Ts
~ ■------- Ts
T4 4 Ts
....... T2f TA 3 Ts
B{ T^Bi
2 Ts
~ ________ Ts
(a) 7 (b)
7
1
T4
Tif
BI Bi
" T{
--Ts
(c)
7
Fig. 10. Bifurcation diagrams for families of periodic motions originating from the hyperboloidal precession of a symmetric satellite obtained for fixed values of h = 0.205, S =1.0 and h = 0.3, S =1.0.
The diagram shown in Fig. 10b represents the case of h = 0.205. In the interval 0.204 < h < 0.206 there exists a value h* for which the curves T* and T2 form a sharp angle at the point B* which coincides with the curve T1. For h > h* the family r* takes the place of r and coincides with r at the point B2, while the family r coincides with r at the point Bi. The families r2, r3 and r4 retain similar behavior as in the previous case. A similar change in the bifurcation diagram occurs at the value h & 0.25 where the curves T2, T3 and T4 converge at the point B3 and the families r2 and r3 switch places. Figure 10c shows a diagram computed for h = 0.3 which demonstrates the aforementioned change.
Conclusion
In this work we have constructed families of periodic motions emanating from the regular precession of a dynamically symmetric satellite for all admissible values of the problem parameters. A bifurcation problem for the said families of periodic motions has been addressed. The results have been presented in the form of bifurcation diagrams and Poincare maps.
References
[1] Duboshin, G. N., On Rotational Movement Of Artificial Celestial Bodies, Bull. ITA AN SSSR, 1960, vol.7, no. 7, pp. 511-520 (Russian).
[2] Kondurar, V. T., Particular Solutions of the General Problem of the Translational-Rotational Motion of a Spheroid Attracted by a Sphere, Sov. Astron., 1960, vol. 3, no. 5, pp. 863-875; see also: Astron. Zh., 1959, vol.36, no. 5, pp. 890-901.
[3] Beletsky, V. V., Motion of a Satellite about Its Center of Mass in the Gravitational Field, Moscow: MGU, 1975 (Russian).
[4] Chernous'ko, F. L., On the Stability of Regular Precession of a Satellite, J. Appl. Math. Mech., 1964, vol.28, no. 1, pp. 181-184; see also: Prikl. Mat. Mekh, 1964, vol.28, no. 1, pp. 155-157.
[5] Lyapunov, A.M., The General Problem of the Stability of Motion, London: Fracis & Taylor, 1992.
[6] Sokol'skiy, A. G. and Khovanskiy, S.A., Periodic Motions Close to Hyperboloidal Precession of a Symmetric Satellite in Circular Orbit, Kosmicheskie Issledovaniya, 1979, vol. 17, no. 2, pp. 208217 (Russian).
[7] Markeev, A. P., Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass, Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).
[8] Markeev, A. P. and Bardin, B. S., On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit, Celest. Mech. Dynam. Astronom., 2003, vol.85, no. 1, pp. 51-66.
[9] Bardin, B. S., On Orbital Stability of Planar Motions of Symmetric Satellites in Cases of First and Second Order Resonances, in Proc. of the 6th Conference on Celestial Mechanics (Señorío de Bértiz, 2003), J.Palacian, P. Yanguas (Eds.), Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, vol. 25, Zaragoza: Real Acad. Ci. Exact. Fís.-Quím. Nat., 2004, pp. 59-70.
[10] Markeyev, A. P., Non-Linear Oscillations of a Hamiltonian System with 2 : 1 Resonance, J. Appl. Math. Mech, 1999, vol.63, no. 5, pp. 715-726; see also: Prikl. Mat. Mekh., 1999, vol.63, no. 5, pp. 757-769.
[11] Bardin, B. S. and Chekin, A. M., Non-Linear Oscillations of a Hamiltonian System in the Case of 3 : 1 Resonance, J. Appl. Math. Mech., 2009, vol. 73, no. 3, pp. 249-258; see also: Prikl. Mat. Mekh, 2009, vol. 73, no. 3, pp. 353-367.
[12] Bardin, B.S., On Nonlinear Motions of Hamiltonian System in Case of Fourth Order Resonance, Regul. Chaotic Dyn, 2007, vol. 12, no. 1, pp. 86-100.
[13] Bardin, B. S., On the Orbital Stability of Periodic Motions of a Hamiltonian System with Two Degrees of Freedom in the case of 3 : 1 Resonance, J. Appl. Math. Mech., 2007, vol.71, no. 6, pp. 880-891; see also: Prikl. Mat. Mekh., 2007, vol.71, no. 6, pp. 976-988.
[14] Sukhov, E. A. and Bardin, B. S., Numerical Analysis of Periodic Motions of a Dynamically Symmetric Satellite Originated from Its Hyperboloidal Precession, Inzhenern. Zhurnal: Nauka i Innovatsii,
2016, no. 5(53), 10pp. (Russian).
[15] Sukhov, E. A. and Bardin, B. S., On Periodic Motions Originating from Hyperboloidal Precession on a Symmetric Satellite, in Abstracts of thee 13th All-Russian Conf. on Problems of Dynamics, Physics of Plasma Particles and Optoelectronics (Moscow, RUDN, 2017).
[16] Sukhov, E. A. and Bardin, B. S., Numerical and Analytical Plotting of Periodic Motion and Investigating Motion Stability in the Case of a Symmetric Satellite, Inzhenern. Zhurnal: Nauka i Innovatsii,
2017, no. 11(71), 3pp. (Russian).
[17] Deprit, A. and Henrard, J., Natural Families of Periodic Orbits, Astron. J., 1967, vol.72, no. 2, pp.158-172.
[18] Karimov, S. R. and Sokolskiy, A. G., Method of Numerical Continuation of Natural Families of Periodic Motions of Hamiltonian Systems, Preprint No. 9, Moscow: Institute of Theoretical Astronomy of the Academy of Sciences of the USSR, 1990.
[19] Sokolskiy, A. G. and Khovanskiy, S.A., On Numerical Continuation of Periodic Motions of a La-grangian System with Two Degrees of Freedom, Kosmicheskie Issledovaniya, 1983, vol.21, no. 6, pp. 851-860 (Russian).
[20] Lara, M., Deprit, A., and Elipe, A., Numerical Continuation of Families of Frozen Orbits in the Zonal Problem of Artificial Satellite Theory, Celestial Mech. Dynam. Astronom., 1995, vol. 62, no. 2, pp.167-181.
[21] Lara, M. and Pelaez, J., On the Numerical Continuation of Periodic Orbits: An Intrinsic, 3-Di-mentional, Differential, Predictor-Corrector Algorithm, Astron. Astrophys., 2002, vol. 389, no. 2, pp. 692-701.
[22] Sukhov, E. A. and Bardin, B. S., On the Algorithm for Numerical Computation of Periodic Motions of a Hamiltonian System with Two Degrees of Freedom, in Abstracts of the 14th All-Russian Conf. on Problems of Dynamics, Physics of Plasma Particles and Optoelectronics (Moscow, RUDN, 2018).
[23] Sukhov, E. A., Analytical and Numerical Computation and Study of Long-Periodic Motions Originating from Hyperboloidal Precession of a Symmetric Satellite, in Proc. ofhte 8th Polyahov Readings (St. Petersburg, St. Petersburg State University, 2018).
[24] Schmidt, D. S., Periodic Solutions near a Resonant Equilibrium of a Hamiltonian System, Celestial Mech., 1974, vol. 9, pp. 81-103.
