Rashmi Bhardwaj ∗ (New Delhi) Manjeet Kaur ∗ (New Delhi)
Aerodynamic Torque exhibits non-resonance oscillation in satellite motion
Abstract This paper deals with the non-linear oscillation of a satellite in an elliptic orbit around the Earth under the in uence of aerodynamic and gravitational torque.
It is assumed that the orbital plane coincides with the equatorial plane of the Earth.
Using BogoliubovKrylovMitropolsky (BKM) methods of nonlinear oscillations, it is observed that the amplitude of the oscillation remains constant up to the second order of approximation. Numerically time series, 2D and 3D phase spaces are plotted for Earth Moon system using Matlab. The existence of main and parametric reso- nance concludes the dierent frequency states which transit the motion from regular to an attractor that leads to chaotic state.
2010 Mathematics Subject Classi cation: 70K42.
Key words and phrases: Aerodynamic torque, Gravitational torque, Non-linear os- cillation, Non-Resonance, Resonance.
1. Introduction. Bhardwaj and co-authors [1,2,3,4, 5,6,7, 8,9,10]
studied the planar oscillation of a satellite in an elliptic orbit under the in-
uence of third body torque, magnetic torque, aerodynamic torque and ob- served that the nonlinear planar oscillation of satellite exhibits chaotic motion.
Efroimsky [11] described spin-orbit coupling for MacDonald torque and Dar- win torque. Formiga and Moraes [12] presented orbital characteristics of arti-
cial satellites in resonance and the correspondent geopotencial coe cients.
Fuse [13] investigated the in uence of Jupiter, Saturn and Uranus on the dy- namical structure of the 2:3 mean motion resonance with Neptune. Guzzo [14] detected numerically the web of three-planet resonances with respect to the variation of the semi-major axis of Saturn and Jupiter, in a model includ- ing the planets from Jupiter to Neptune. Haghighipour [15] studied stable 1:2 resonant periodic orbits in elliptic three-body systems. Ipatov [16] pre- sented the results of the numerical investigations of the evolution of orbits of trans-Neptunian bodies at the 2:3 resonance with Neptune. Ji, Kinoshita,
∗ 0000-0001-9413-3434
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Liu, Li [17] integrated the orbital solutions of the planets orbiting 55 Cancri and observed that the mean motion resonance and apsidal locking can act as two important mechanisms for stabilizing the system. Narayan, et al. [18]
discussed some non-linear oscillations of Dumbbell satellite in elliptical or- bit in the central gravitational eld of force under the combined in uence of Earth magnetic eld. Chaotic attitude motion of a satellite in Keplerian ellip- tic orbit is considered by Peng and Liu [19]. Polymilis, et al. [20] studied the geometry of the homoclinic tangle, with respect to the energy, corresponding to an unstable periodic orbit of type 1:2, on a surface of section representing a 2-D Hamiltonian system. Evaluation of aerodynamic drag for external tanks in low earth orbit is studied by Stone and Witzgall [21]. Varadi [22] investi- gated periodic orbits in the external 3:2 orbital resonance in the context of the planer, elliptic restricted three-body problem. Yokoyama, et al. [23] discussed the dynamics of some resonances of Phobos in the future. Zanardi and Real [24] discussed the classical models of gravity gradient, solar radiation, aero- dynamic and magnetic torques and analyzed the variation of magnitudes as a function of altitude relative to the Earth's surface for a spacecraft in a circular cylindrical shape. Zanardi MC, et al. [25] presented analytical attitude pre- diction of spin stabilized spacecrafts in uenced by magnetic residual torque.
Zhang, et al. [26] investigated the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam sub- jected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case.
None of the authors studied the non-resonance motion of a satellite under the in uence of aerodynamic torque in an almost elliptic orbit.
2. Equation of Motion. Let us consider a rigid satellite S moving in an elliptic orbit around the Earth E such that the orbital plane of the satellite coincides with the equatorial plane of the Earth. The satellite is assumed to be a tri-axial body with principal moments of inertia A < B < C at its centre of mass and C is the moment of inertia about the spin axis which is perpendicular to the orbital plane. Let ~r be the radius vector of the centre of mass of the satellite, ν be the true anomaly, θ be the angle that the long axis of the satellite make with a xed line EF lying in the orbital plane and η2 be the angle between the radius vector and the long axis as shown in Figure1.
Figure 1: Motion of Satellite S in an elliptic orbit around the Earth
The equation of motion for the system has been obtained by Bhardwaj and Kaur (2014) ([10]) as
d2η
dν2 + n2η = −e cos νd2η
dν2 + 2e sin νdη
dν + 4e sin ν + n2(η − sin η) + (A∗ν2sin ν + B∗ν sin ν + C∗sin ν + D∗ν + E∗),
(1)
where
n2 = 3(B − A)
C ,
= ρSCdl2
CΩ2 = parameter due to aerodynamic torque, A∗= a2(1 − e)
Ω2l , B∗= ωa(2e − 1) cos i
Ω +2V1a(1 − 2e) Ωl
,
C∗ =
ωV1(2e − 1) cos i + V12(1 − 2e)
l +ωae
2Ω sin i
,
D∗ = ωa(2e − 1) Ω sin i
, E∗ = ωV1(2e − 1) sin i are all constants.
In the equation (1), the non-linearity (η − sin η) is su ciently weak and therefore it can be taken of the order of e, so by taking n2 = ceand = e1, we get
d2η
dν2 + n2η = ef (ν, η, η0, η00) (2)
where
f (ν, η, η0, η00) = − cos νd2η
dν2 + 2 sin νdη
dν + 4 sin ν + c(η − sin η) + 1(A∗ν2sin ν + B∗ν sin ν + C∗sin ν + D∗ν + E∗) As e is very small, so the solution of the equation can be obtained by BKM method.
For e = 0 the generating solution of the zeroth order is given by η = a cos ψ, ψ = nν + ψ∗
where amplitude a and phase ψ∗are constants, which can be determined from initial conditions.
The solution of equation (2) is obtained in the form
η = a cos ψ + eu1(a, ν, ψ) + e2u2(a, ν, ψ) + ... (3) where
da
dν = eA1(a) + e2A2(a) + ... (4) dψ
dν = n + eB1(a) + e2B2(a) + ... (5) Substituting the values of η, dη
dν and d2η
dν2 in the equation (2), and com- paring the coe cients of like powers of e and e2, we get
n2∂2u1
∂ψ2 + 2n∂2u1
∂ν∂ψ +∂2u1
∂ν2 − 2nA1sin ψ − 2naB1cos ψ + n2u1 (6)
= 4 sin ν + 1 A∗ν2sin ν + B∗ν sin ν + C∗sin ν + D∗ν + E∗ + c [a cos ψ − sin(a cos ψ)] − na [cos(ν − ψ) − cos(ν + ψ)]
+n2a
2 [cos(ν + ψ) + cos(ν − ψ)]
and
n2∂2u2
∂ψ2 + 2n∂2u2
∂ν∂ψ +∂2u2
∂ν2 −
aB12− A1dA1 da
cos ψ (7)
−
2A1B1+ aA1
dB1 da
sin ψ + 2A1
∂2u1
∂a∂ν + 2nB1
∂2u1
∂ψ2 + 2nA1
∂2u1
∂a∂ψ+ 2B1
∂2u1
∂ν∂ψ − 2nA2sin ψ − 2naB2cos ψ + n2u2
= 2 sin ν
A1(a) cos ψ − aB1(a) sin ψ +∂u1
∂ν + n∂u1
∂ψ
− cos ν
n2∂2u1
∂ψ2 + 2n ∂2u1
∂ν∂ψ +∂2u1
∂ν2
− 2nA1(a) sin ψ − 2naB1(a) cos ψ
+ c(u1− u1cos(a cos ψ)) Using Fourier expansion
sin(a cos ψ) = 2
∞
X
k=0
(−1)kJ(2k+1)(a) cos(2k + 1)ψ
cos(a cos ψ) = J0(a) + 2
∞
X
k=1
(−1)kJ2k(a) cos(2k)ψ
where Jk(a) is Bessel's function of order k in the equation (6) and equating the coe cients of like powers of sin ψ and cos ψ, so that u1(a, ν, ψ) do not contain the resonant terms, we get
A1(a) = 0 and B1(a) = c
2an[2J1(a) − a] .
Substituting the values of A1(a) and B1(a) in the equation (6) and solving, we get
u1(a, ν, ψ) = 4 sin ν
n2− 1− 1 2n + 1
na +n2a 2
cos(ν + ψ)
+ 1
2n − 1
n2a 2 − na
cos(ν − ψ) + A∗1
1 n2− 1
ν2sin ν − 4ν cos ν
n2− 1 −2(n2+ 3) (n2− 1)2 sin ν
+ B∗1 1 n2− 1
ν sin ν − 2 cos ν n2− 1
+ C∗1
sin ν n2− 1+ ν
n2D∗1+ 1 n2E∗1
+ c 2n2
∞
X
k=0
(−1)kJ2k+1(a)cos(2k + 1)ψ k(k + 1)
(8)
Thus, we have dA1
da = 0 dB1
da = c
2an[2J10(a) − 1] − c
4a2n[2J1(a) − a]
Putting the values of A1(a), B1(a),∂u1
∂ψ,∂2u1
∂ψ2 ,∂2u1
∂ν∂ψ, ∂u1
∂ν ,∂2u1
∂ν2 , ∂2u1
∂a∂ψ, dA1 da , dB1
da and u1 in the equation (7), and then equating the coe cients of cos ψ and sin ψ to zero, to avoid the resonant term, we obtain
n2∂2u2
∂ψ2 + 2n∂2u2
∂ν∂ψ +∂2u2
∂ν2 − aB12cos ψ − 2nA2sin ψ − 2naB2cos ψ (9) + n2u2+ c
a(2J1(a) − a)h(n + 1)(2na + n2a)
2n(2n + 1) cos(ν + ψ)
+(1 − n)(n2a − 2na)
2(2n − 1) cos(ν − ψ)
− c 2n2
∞
X
k=0
(−1)kJ2k+1(a)(2k + 1)2cos(2k + 1)ψ k(k + 1)
i
= c
a(2J1(a) − a)(− sin ν sin ψ + n cos ν cos ψ) +6 sin 2ν n2− 1
+
n2− 1 2(2n − 1)
n2a 2 − na
− n2− 1 2(2n + 1)
na +n2a 2
cos ψ
−n2+ 4n + 3 2(2n + 1)
na +n2a 2
cos(2ν + ψ) −−n2+ 4n − 3 2(2n − 1)
n2a 2 − na
cos(2ν − ψ)
+ A∗1
sin ν cos ν n2− 1
−2 + 3ν2− 16
n2− 1 −6(n2+ 3)
(n2− 1)2 − 4 n2
n2− 1ν cos ν
+8ν sin2ν n2− 1
n2+ 1 n2− 1
+ B∗1
2 sin ν n2− 1
3
2ν + tan ν +2 tan ν
n2− 1 − n2 n2− 1cot ν
+ C∗1 3 sin 2ν
2(n2− 1)+ 2D∗12 sin ν n2
− c nsin ν
∞
X
k=1
(−1)kJ2k+1(a)(2k + 1)sin(2k + 1)ψ k(k + 1)
+c 2cos ν
∞
X
k=1
(−1)kJ2k+1(a)(2k + 1)2cos(2k + 1)ψ k(k + 1)
+ ch4 sin ν n2− 1 − 1
2n + 1
na +n2a 2
cos(ν + ψ) + 1 2n − 1
n2a 2 − na
cos(ν − ψ)
+ A∗1 1 n2− 1
ν2sin ν −4ν cos ν
n2− 1 −2(n2+ 3) (n2− 1)2sin ν
+ B∗1
1 n2− 1
ν sin ν − 2 cos ν n2− 1
+ C∗1
sin ν n2− 1 + ν
n2D∗1+ 1 n2E∗1
+ c 2n2
∞
X
k=0
(−1)kJ2k+1(a)cos(2k + 1)ψ k(k + 1)
i
×
"
1 − (
J0(a) + 2
∞
X
k=1
(−1)kJ2k(a) cos(2k)ψ )#
Equating the coe cient of sin ψ and cos ψ, we get A2(a) = 0 and
B2(a) = − c2
8a2n3(2J1(a) − a)2+3n(n2− 1) 4(4n2− 1) + c2
2n3a
∞
X
k=1
J2k+1(a)J2k+10 (a) k(k + 1) ,
where J2k+10 (a) = 12[J2k(a) − J2k+2(a)]. Thus in the rst approximation, the solution is obtained as η = a cos ψ
da
dν = 0 (since A1(a) = 0) ⇒ a =constant (10)
dψ
dν = n + eB1(a) = n + n
2a(2J1(a) − a) (since n2= ce) (11)
and in the second approximation, the solution is obtained as
η = a cos ψ (12)
+ eh4 sin ν
n2− 1− 1 2n + 1
na + n2a 2
cos(ν + ψ) + 1 2n − 1
n2a 2 − na
cos(ν − ψ) + A∗1 1
n2− 1
ν2sin ν − 4ν cos ν
n2− 1 −2(n2+ 3) (n2− 1)2 sin ν
+ B∗1 1 n2− 1
ν sin ν −2 cos ν n2− 1
+ C∗1 sin ν n2− 1+ ν
n2D∗1+ 1 n2E∗1 + c
2n2
X(−1)kJ2k+1(a)cos(2k + 1)ψ k(k + 1)
i
da
dν = 0 (∵ A2(a) = 0) ⇒ a =constant (13) and
dψ
dν = n + n
2a(2J1(a) − a) − n
8a2(2J1(a) − a)2+ e23n(n2− 1) 4(4n2− 1) + n
2a
∞
X
k=1
J2k+1(a)J2k+10 (a) k(k + 1)
(14)
From the equations (10) and (13), it is observed that the amplitude of the oscillation remains constant up to the second order of approximation and the main resonance occurs at n = ±1 and parametric resonance at n = ±1/2.
3. Results and discussions. The 2D and 3D phase spaces are ploted for Earth Moon system. For natural satellite Moon, it is assumed that
Semi major axis = a = 3.8 × 105 Km, Eccentricity = e = 0.0549,
Inclination = i = 5.14500,
Angular velocity = ω = 2.425 × 10−6 rad/sec.
The eect of mass parameter and aerodynamic torque parameter is studied on the non linear oscillation of a satellite in an elliptic orbit. The phase spaces for dierent values of mass parameter, aerodynamic toque parameter and constants are plotted as described in tables and gures.
Table 1 gives the behavior for Earth-Moon system at xed values of A∗, B∗, C∗, D∗, E∗, e and for the variable parameters , c with variation in the values of n.
Table 2 gives the behavior for Earth-Moon system at xed values of A∗, B∗, C∗, D∗, E∗, e and for the variable parameters n, c with variation in the values of .
Table 3 gives the behavior for Earth-Moon system at xed values of A∗, C∗, D∗, E∗, e , c, n with variation in the values of B∗.
Table 4 gives the behavior for Earth-Moon system at xed values of A∗, B∗, D∗, E∗, e , c, n with variation in the values of C∗.
Table 5 gives the behavior for Earth-Moon system at xed values of A∗, B∗, C∗, E∗, e , c, n with variation in the values of D∗.
Table 6 gives the behavior for Earth-Moon system at xed values of A∗, B∗, C∗, D∗, e , c, n with variation in the values of E∗.
The time series and 2D phase plots are plotted for the values of n = 0.1, c = 0.182149, = 0.00000001, A? = 3.1E + 16, B? = −8.12389, C? =
−5.4E + 11, D? = −1.3E + 12, E? = 0 and e = 0.0549 which is shown in Fig.2and 3D phase plots are plotted for the values of n = 0.1, c = 0.182149,
= 0.00000001and 0.005, A? = 3.1E + 16, B?= −8.12389, C?= −5.4E + 11, D? = −1.3E + 12, E? = 0 and e = 0.0549 which is shown in Fig.3.
Fig. 4 gives time series and 2D phase plots for the values of n = 0.5, c = 4.553734, = 0.0001, A∗ = 3.1E +16, B∗ = −8.12389, C∗ = −5.4E +11, D∗ = −1.3E + 12, E∗ = 0 and e = 0.0549 and g.5gives 3D phase plots for the values of n = 0.5, c = 4.553734, = 0.0001 and 0.005, A∗ = 3.1E + 16, B∗ = −8.12389, C∗ = −5.4E +11, D∗ = −1.3E +12, E∗ = 0 and e = 0.0549.
Fig. 6gives time series and 2D phase plots for the values of n = 0.8, c = 11.657559, = 0.000005, A∗ = 3.1E + 16, B∗ = −8.12389, C∗ = −5.4E + 11, D∗ = −1.3E + 12, E∗ = 0 and e = 0.0549 and Fig.7gives 3D phase plots for the values of n = 0.8, c = 11.657559, = 0.000005 and 0.001, A∗ = 3.1E +16, B∗ = −8.12389, C∗ = −5.4E +11, D∗ = −1.3E +12, E∗ = 0 and e = 0.0549.
From tables and gures, it is observed that the phase space plots almost remain spiral for dierent values of constants, mass parameter, eccentricity and aerodynamic torque parameter.
4. Tables and gures
Table 1: For xed values of A? = 3.1E+16, B?= −8.12389, C? = −5.4E+11, D? = −1.3E + 12, E? = 0, e = 0.0549 variable c, and variation in n the graphical behavior are spiral.
Sr. No. n c
1. 0.182149 1 × 10−8 2. 0.1 0.182149 0.0001
3. 0.182149 0.005
4. 4.553734 1 × 10−8 5. 0.5 4.553734 0.0001
6. 4.553734 0.005
7. 14.754098 1 × 10−8 8. 0.9 14.754098 0.3
9. 14.754098 0.6
Table 2: For xed values of A∗ = 3.1E +16, B∗ = −8.12389, C∗ = −5.4E +11, D∗ = −1.3E + 12, E∗ = 0, e = 0.0549 variable c, n and variation in the graphical behavior are spiral.
Sr. No. c n
1. 0.182149 0.1
2. 0.000005 1.639344 0.3
3. 6.557377 0.6
4. 11.657559 0.8
5. 0.182149 0.1
6. 0.0005 1.639344 0.3
7. 6.557377 0.6
8. 11.657559 0.8
9. 0.182149 0.1
10. 0.001 1.639344 0.3
11. 6.557377 0.6
12. 11.657559 0.8
Table 3: For xed values of A∗ = 3.1E + 16, C∗ = −5.4E + 11, D∗= −1.3E + 12, E∗ = 0, e = 0.0549 variable c, , n and variation in B∗ the graphical behavior are spiral.
Sr. No. c n B∗
1. -8.12389
2. 0.0001 11.657559 0.8 552289.6
3. 920481.4
Table 4: For xed values of A∗= 3.1E +16, B∗= −8.12389, D∗ = −1.3E +12, E∗ = 0, e = 0.0549 variable c, , n and variation in C∗ the graphical behavior are spiral.
Sr. No. c n C∗
1. -5.4 E+11
2. 0.0001 11.657559 0.8 -5.5E+11
3. -8.7E+10
Table 5: For xed values of A∗ = 3.1E+16, B∗r = −8.12389, C∗= −5.4E+11, E∗ = 0, e = 0.0549 variable c, , n and variation in D∗ the graphical behavior are spiral.
Sr. No. c n D∗
1. -1.3E+12
2. 0.0001 11.657559 0.8 -7.7E+12
3. -1E+13
Table 6: For xed values of A∗ = 3.1E +16, B∗ = −8.12389, C∗ = −5.4E +11, D∗ = −1.3E + 12, E∗ = 0, e = 0.0549 variable c, n, and variation in E∗ the graphical behavior are spiral.
Sr. No. c n E∗
1. 0
2. 0.0001 11.657559 0.8 -61.2204
3. -5.05119
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
x 1014
−2
−1.5
−1
−0.5 0 0.5 1 1.5
2x 1013
η
dη/dν
0 200 400 600 800 1000
−0.5 0 0.5 1 1.5 2 2.5 3 3.5
4x 1014
ν
η,dη/dν
Figure 2: 2D and time series Plot of n = 0.1, c = 0.182149, = 0.00000001, A? = 3.1E + 16, B?= −8.12389, C? = −5.4E + 11, D? = −1.3E + 12, E? = 0 and e = 0.0549.
−1 0 1 2
3 4
x 1014
−2
−1 0 1 2
x 1013 0 200 400 600 800 1000
dη/dν η
ν
−5 0 5 10
15 20
x 1019
−1
−0.5 0 0.5 1
x 1019 0 200 400 600 800 1000
dη/dν η
ν
Figure 3: 3D Plots of n = 0.1, c = 0.182149, = 0.00000001 and 0.005, A? = 3.1E + 16, B?= −8.12389, C? = −5.4E + 11, D? = −1.3E + 12, E? = 0 and e = 0.0549.
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 1018
−2
−1.5
−1
−0.5 0 0.5 1 1.5
2x 1017
η
dη/dν
0 200 400 600 800 1000
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 1018
ν
η,dη/dν
Figure 4: 2D, 3D and time series. Plot of: n = 0.8, c = 11.657559, = 0.000005, A∗= 3.1E + 16, B∗ = −8.12389, C∗= −5.4E + 11, D∗ = −1.3E + 12, E∗ = 0 and e = 0.0549.
−2 0
2 4
6
x 1018
−2
−1 0 1 2
x 1017 0 200 400 600 800 1000
dη/dν η
ν
−1 0
1 2
3
x 1020
−1
−0.5 0 0.5 1
x 1019 0 200 400 600 800 1000
dη/dν η
ν
Figure 5: 2D, 3D and time series. Plot of: n = 0.8, c = 11.657559, = 0.000005, A∗= 3.1E + 16, B∗ = −8.12389, C∗= −5.4E + 11, D∗ = −1.3E + 12, E∗ = 0 and e = 0.0549.
−5 0 5 10 15 20 x 1016
−8
−6
−4
−2 0 2 4 6 8 10x 1015
η
dη/dν
0 200 400 600 800 1000
−5 0 5 10 15 20x 1016
ν
η,dη/dν
Figure 6: 2D, 3D and time series. Plot of: n = 0.1, c = 0.182149,
= 0.00000001, A∗ = 3.1E + 16, B∗ = −8.12389, C∗ = −5.4E + 11, D∗= −1.3E + 12, E∗= 0 and e = 0.0549.
−5 0 5
10 15 20
x 1016
−1
−0.5 0 0.5 1
x 1016 0 200 400 600 800 1000
dη/dν η
ν
−1 0 1
2 3 4
x 1019
−2
−1 0 1 2
x 1018 0 200 400 600 800 1000
dη/dν η
ν
Figure 7: 2D, 3D and time series. Plot of: n = 0.8, c = 11.657559, = 0.000005 and 0.001, A∗ = 3.1E + 16, B∗ = −8.12389, C∗ = −5.4E + 11, D∗= −1.3E + 12, E∗= 0 and e = 0.0549.
5. Conclusion. Using BKM method, it is observed that the amplitude of the oscillation remains constant up to the second order of approximation.
The main resonance occurs at n = ±1 and parametric resonance at n = ±1/2.
Using time series, 2D and 3D phase plots it is concluded that mass parameter, eccentricity, aerodynamic torque parameter, variable constant almost have the same eect and motion almost remains constant only with change in the size of oscillations.
6. Acknowledgements. Authors are thankful to Guru Gobind Singh Indraprastha University, Delhi (India) for providing research facilities.
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Nierezonansowe oscylacje aerodynamicznego momentu obrotowego dla satelity okoªoziemskiego
Rashmi Bhardwaj, Manjeet Kaur
Streszczenie Artykuª po±wi¦cony jest nieliniowej oscylacji satelity w eliptycznej orbicie wokóª Ziemi pod wpªywem grawitacji i aerodynamicznego momentu ob- rotowego. Przyjmuje si¦, »e pªaszczyzna orbity pokrywa si¦ z pªaszczyzn¡ równi- kow¡ Ziemi. Po zastosowaniu metody nieliniowych oscylacji BogoliubovaKrylova
Mitropolsky'ego (BKM) obserwujemy, »e amplituda oscylacji jest staªa przy aprok- symacji rz¦du drugiego. Ilustracje szeregów czasowych w przestrzeni fazowej 2- i 3- wymiarowej wykonano z wykorzystaniem procedur zaimplementowanych w MatLa- bie. Istnienie gªównej skªadowej rezonansu i skªadowych parametrycznych wyja±nia chaotyczny charakter cz¦stotliwo±ci.
2010 Klasy kacja tematyczna AMS (2010): 70K42.
Sªowa kluczowe: moment aerodynamiczny• moment grawitacyjny• oscylacje nie- liniowe• oscylacje swobodne• rezonans.
Dr. Rashmi Bhardwaj is a professor of Mathematics at the Uni- versity School of Basicand Applied Sciences, Guru Gobind Singh Indraprastha (GGSIP) University. She received Ph.D. at the fac- ulty of Mathematics, University of Delhi in 1997. Her specializa- tion are Celestial Mechanics and Space Dynamics, chaotic mo- tion of satellites in particular. She became Assistant Professor at Acharya Narendra Dev College, University of Delhi in 1996 and worked up to 1999. Then she joined GGSIP University in 1999.
Visiting Associate of Inter University Centre for Astronomy and Astrophysics IUCAA). Awardee of Young Scientist Award from (All India Council for Technical Education (AICTE). She was awarded Best Re- searcher Award of GGS Indraprastha University (GGSIPU) thrice. She has published more than 80 research papers, 2 books and edited Proceedings of 1 international con- ference. She has supervised 10 Ph.D. scholars and 7 are presently working. She is Managing Editor of research Journal published by the Society (www.siam-india.in).
Visited, attended and delivered invited Lectures in International National confer- ences. Subject Reviewer for many International/ National Journals. Life member of several International National recognized Organizations in the sphere of Mathemat- ical Sciences. She has established Nonlinear Dynamics Research Lab & Mathematics Lab in GGSIPU. Presently working on complexity dynamics and nonlinear analysis of real world problems specially in the eld of environment, weather forecasting,
nancial modeling. Government of India, Ministry of Earth Sciences, India Meteo- rological Department (IMD) awarded 27th Biennial MAUSAM Award. References to her research papers are found in MathSciNet under ID: 353148.
Ms. Manjeet Kaur is pursuing Ph.D. (Mathematics) in Univer- sity school of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, New Delhi, India. She has completed her M. Phil Mathematics from Kurukshetra University in 2003.
She has completed her M.Sc. Mathematics from Guru Jambhesh- war University Hisar in 2002.
Rashmi Bhardwaj
University School of Basic and Applied Sciences Non-Linear Dynamics Research Lab
Guru Gobind Singh Indraprastha University, New Delhi, Ind E-mail: rashmib22@gmail.com, rashmib@ipu.ac.in
Manjeet Kaur
University School of Basic and Applied Sciences Non-Linear Dynamics Research Lab
Guru Gobind Singh Indraprastha University, New Delhi, Ind E-mail: manumail05@yahoo.com
Communicated by: Wojciech Mitkowski
(Received: 15th of May 2016; revised: 22nd of September 2016)
