Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 7, 299-307 Chaotic Motion in Problem of Dumbell Satellite Ayub Khan Department of Mathematics, Zakir Hussain College University of Delhi, Delhi, India Neeti Goel Research Scholar, Department of Mathematics University of Delhi, Delhi-110007, India neeti 2606@hotmail.com Abstract In our present problem we have analytically determined the chaotic parameter in the problem of a dumbell satellite. Computational studies reveal that the chaotic parameter is effective in making the regular behavior of the system chaotic. Mathematics Subject Classification: 70F15 Keywords: Chaos, Hamiltonian, Dynamical System 1 Introduction There is a great revolution in the world of celestial Mechanics due to the necessity of including the relativistic effects in the motion of celestial bodies and the discovery of new dynamical situations in the solar system, satellites etc. Investigations on chaos is the most attracting feature which is widely studied in variety of problems of Satellites. In our present paper, we consider the model of a dumbell satellite and investigate it by using the control theory of Hamiltonian systems based on [1, 2, 3]. We observe that the control parameter (as appeared in the mathematical literature obtained by the method employed is rather behaving as a chaotic parameter in our studies which has not been experienced in the earlier studies, allow us to introduce the analytically estimated control parameter as a chaotic parameter. The regular behaviour of the system under consideration at certain initial conditions is caught under chaotic situation with the inclusion of chaotic parameter which is a very interesting phenomenon in the present manuscript.
300 A. Khan and N. Goel 2 Theory of Hamiltonian Systems Let A be the Lie algebra of real functions defined on phase space. For H A, let {H} be the linear operator action on A such that {H}H = {H, H }, for any H Awhere {.,.} is the Poisson bracket. The time-evolution of a function V Afollowing the flow of H is given by dv dt = {H}V, which is formally solved as V (t =e t{h} V (0, if H is time independent, where e t{h} = n=0 t n n! {H}n. Any element V Asuch that {H}V = 0, is constant under the flow of H, i.e. t R, e t{h} V = V. Let us now fix a Hamiltonian H 0 A. The vector space Ker{H 0 } is the set of constants of motion and it is a sub-algebra of A. The operator {H 0 } is not invertible since a derivation has always a non-trivial kernel. For instance {H 0 }(H0 α = 0 for any α such that Hα 0 A. Hence we consider a pseudoinverse of {H 0 }. We define a linear operator Γ on A such that i.e. {H 0 } 2 Γ={H 0 }, (2.1 V A, {H 0, {H 0, ΓV }} = {H 0,V}. If the operator Γ exists, it is not unique in general. Any other choice Γ satisfies Rg(Γ Γ Ker({H 0 } 2. We define the non-resonant operator N and the resonant operator R as N = {H 0 }Γ R = 1 N, where the operator 1 is the identity in the algebra of linear operators acting on A. We notice that Equation (2.1 becomes {H 0 }R =0
Chaotic motion in problem of dumbell satellite 301 which means that the range Rg R of the operator R is included in Ker{H 0 }. A consequence is that any element RV is constant under the flow of H 0, i.e. e t{h0} RV = RV. We notice that when {H 0 } and Γ commute, R and N are projectors i.e. R 2 = R and N 2 = N. Moreover, in this we have RgR = Ker{H 0 }, i.e. the constant of motion are the elements RV where V A. Let us now assume that H 0 is integrable with action-angle variables (A, ϕ B T n where B is an open set of R n and T n is the n-dimensional torus, so that H 0 = H 0 (A and the Poisson bracket {H, H } between two Hamiltonians is {H, H } = H A H ϕ H ϕ H A The operator {H 0 } acts on V given by as V = k Z n V k (Ac ik ϕ {H 0 }V (A, ϕ = k iω(a.kv k (Ae ik ϕ where the frequency vector is given by ω(a = H 0 A. A possible choice of Γ is ΓV (A, ϕ = k Z n V k (A iω(a k eik ϕ ω(a k 0 We notice that this choice of Γ commutes with {H 0 }. For a given V A, RV is the resonant part of V and N V is the nonresonant part: RV = k N V = k V k (Aχ(ω(A k =0e ik ϕ (2.2 V k (Aχ(ω(A k 0e ik ϕ (2.3 where χ(α vanishes when proposition α is wrong and it is equal to 1 when α is true. From these operators defined for the integrable part H 0, we construct a control term for the perturbed Hamiltonian H 0 + V where V A, i.e. we construct f such that H 0 + V + f is canonically conjugate to H 0 + RV.
302 A. Khan and N. Goel If H 0 is resonant and RV = 0, the controlled Hamiltonian H = H 0 + V + f is conjugate to H 0. In the case RV = 0, the series (6 which gives the expansion of the control term f, can be written as f(v = f s, (2.4 s=2 where f s is of order ε s and given by the recursion formula f s = 1 s {ΓV,f s 1} (2.5 where f 1 = V. 3 Application to the problem of a Satellite The equation of planer oscillation of a dumbell satellite in the central gravitational field of the Earth under the influence of the solar radiation pressure together with the effects of the Earth s shadow and some phenomenological factor [4] is given by (1 + e cos vψ 2e sin vψ + 3 sin ψ cos ψ + ρ 3 k cos ε sin(ψ + α =2esin v + E sin nv (3.1 where ψ is the angular deviation of the line joining the satellites with stable position of equilibrium, e is eccentricity of the orbit of the centre of mass, ρ is variable radius of circular orbit, ε is the Inclination of the osculating plane of the orbit of the centre of mass of the system with the plane of ecliptic, α is the angular separation of the solar position vector projected on the orbital plane, v is the true [ anomaly] of the centre of mass of the system in elliptical orbit and k = ρ3 B 1 πμ m 1 B 2 m 2 δ r sin θ, where μ is the product of gravitational constant and mass of the Earth, B i(i=1,2 are the absolute values of the forces due to direct solar radiation pressure exerted on masses of satellites m 1 and m 2 respectively, δ r is the Earth s shadow function, θ is the angle between the axis of cylinder and line joining the Earth s centre and the end point of the orbit of the centre of mass, E is the phenomenological parameter characterizing the periodic term, n is the frequency of the external periodic force. Substituting 1 ρ = 1+ecos v, 2ψ = q, kcos ε = ek 1, E = E 1 e, (3.2 the equation (3.1 reduces to (1 + e cos v d2 q dq 2e sin v dv2 dv + 3 sin q +2k 1e(1 + e cos v 3 sin 2 + α =4esin v +2E 1 e sin nv. (3.3
Chaotic motion in problem of dumbell satellite 303 The Hamiltonian for the above equation is H = 2p + 1 2 p2 3 cos q { +e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α (3.4 In order to apply the theory developed by Vittot [5], we need to put the Hamiltonian in an autonomous form. We consider v as an additional angle whose conjugate action is E. Then in the autonomous form Hamiltonian can be perceived as H(p,q,E,v= 2p + 1 2 p2 3 cos q + E { + e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α (3.5 where the actions are A =(p, E and the angles are φ =(q, v. The unperturbed Hamiltonian to be used for constructing the operator Γ is H 0 = 2p + p2 2 3 cos q + E (3.6 The action of {H 0 } and Γ on { V = e p 2 cos v 2E ( 1 q } n p cos nv 3 cos q cos v 4k 1 cos 2 + α,v A are expressed as: [ {H 0 }V = e 3(p 2 sin q cos v +2(p 2k 1 sin 2 + α + p 2 sin v ] +2E 1 p sin nv (3.7 [ 3 sin q cos v ΓV = e + 2k 1 sin + α 2 + sin v + 2E ] 1 sin nv (3.8 p 2 p 2 p 2 p for p 0, 2. The term f is given by f = 1 2 {ΓV,V } = 1 { ΓV 2 p V q ΓV q } V. p
304 A. Khan and N. Goel The explicit expression of f for p = 1 is given by f = 1 [9 2 e2 sin 2 q cos 2 v +12k 1 sin 2 + α sin q cos v + 6 sin v cos v sin q +6E 1 sin nv cos v sin q +4k1 2 sin2 2 + α +4k 1 sin v sin 2 + α +4E 1 k 1 sin nv sin 2 + α + 6 cos 2 v cos q +2k 1 cos 2 + α cos v + 6E 1 cos v cos q cos nv n + 2E ( 1k 1 q ] cos n 2 + α cos nv. (3.9 By introducing a central parameter β in the expression of f we get f = 1 [9 2 βe2 sin 2 q cos 2 +12k 1 sin 2 + α sin q cos v + 6 sin v cos v sin q +6E 1 sin nv cos v sin q +4k1 2 sin2 2 + α +4k 1 sin v sin 2 + α +4E 1 k 1 sin nv sin 2 + α + 6 cos 2 v cos q +2k 1 cos 2 + α cos v + 6E 1 n cos v cos q cos nv + 2E 1k 1 cos n 2 + α cos nv ]. (3.10 4 Results and Discussion Figure 1(a,b and Figure 2(a,b depict the Poincare surface of section and Poincare map of the Hamiltonian given by (3.4 without and with the inclusion of (3.9 respectively for e =0.2, k 1 =0.01, α =0.001, E 1 =0.09, n =0.009, β = 0.6. Figure 3(a,b and Figure 4(a,b depict the Poincare surface of section and Poincare Map of (3.4 without and with with inclusion of (3.9 respectively for e =0.07, k 1 =0.01, α =0.001, E 1 =0.09, n =0.009, β =1.50. From these figures we observe that for e =0.2 and e =0.07 the system exhibits regular behavior without the expression of f while with the addition of the term f the system becomes chaotic for a particular value of the parameter β.
Chaotic motion in problem of dumbell satellite 305 Figure 1 (a Poincare surface of section for e =0.2 without the control term. (b Poincare Map for e =0.2 without the control term. Figure 2 (a Poincare surface of section for e =0.2 with the control term. (b Poincare Map for e =0.2 with the control term. Figure 3 (a Poincare surface of section for e = 0.07 without the control term. (b Poincare Map for e =0.07 without the control term.
306 A. Khan and N. Goel Figure 4 (a Poincare surface of section for e =0.07 with the control term. (b Poincare Map for e =0.07 with the control term. 5 Conclusion We observed that the system started to behave in chaotic manner for a particular value of the parameter β which asserts that the term obtained analytically is effective to make the system under consideration chaotic which is completely opposite to the research studies that are made in this direction so far now; and gives the new dimension to the analytical studies of the chaotic phenomenon. References [1] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M., 2004, Control of Chaos in Hamiltonian Systems, Celestial Mechanics and Dynamics Astronomy, 90:3 12. [2] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M. Figarella, C. and Ghendrih, Ph., 2004, Controlling chaotic transport in a Hamiltoman model of interest to magetized plasmas, J. Phys. A. Math. Gen., 37, 3589. [3] Ciraolo, G., Chandre, C., Lima, R., Vittot, M., Pettini, M. Figarella, C. and Ghendrih, Ph., 2004b, Contol of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas. Phy. Rev. E 69(4, 056213. [4] Sharma, S. and Narayan, A., 2001, Non-Linear Oscillation of interconnected satellites system under the combined influence of the solar radiation pressure and dissipative force of general nature, Bull. Astr. Soc., India, 29, 115 126. [5] Vittot, M., 2004, Perturbation Theory and Control in Classical or quantum Mechanics by an inversion formula, J. Phys. A. Math. Gen., 37:6337 6357.
Chaotic motion in problem of dumbell satellite 307 Received: January, 2010