# Chaotic Oscillation of Satellite due to Aerodynamic Torque

Authors: Rashmi Bhardwaj; Mohammad Sajid
Journal: Advances in Astronomy
 (2021)

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## Abstract

This study presents the chaotic oscillation of the satellite around the Earth due to aerodynamic torque. The orbital plane of the satellite concurs is same as the tropical plane of Earth. The half-width of riotous separatrix is assessed utilizing Chirikov’s measure. Variety of boundary techniques shows that streamlined force boundary (ɛ), unpredictability of circle (e), and mass-proportion (ω0) convert normal wavering to the disorganized one. We studied the behavior of trajectories due to change in parameters with Lyapunov exponents and time series plots. The theory is applied to Resourcesat-1, an artificial satellite of the Earth.


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## Body

## 1. Introduction

Artificial satellites are widely used in telecommunication, mass media and weather forecast, agriculture, and navigation. Satellites are widely used in agriculture and forestry for crop inventory, yield prediction, and soil/crop condition monitoring. Resourcesat-1 (also known as IRS-P6) is an advanced remote sensing satellite built by the Indian Space Research Organization (ISRO). The tenth satellite of ISRO in IRS series, Resourcesat-1, is intended to not only continue the remote sensing data services provided by IRS-1C and IRS-1D, both of which have far outlived their designed mission lives, but also vastly enhance the data quality. The major objectives of Resourcesat-1 are to provide continued remote sensing data services on operational basis for integrated land and water resources management with enhanced multispectral/spatial coverage and stereo imaging and also to develop new areas of applications to take full advantages of increased spatial and spectral resolutions. For a country like India, with populations separated by rough terrain and different languages, communications satellites provide remote populations access to education and to medical expertise that would otherwise not reach them [1].Satellite exhibits chaotic motion under the influence of different torques and, for the low-thrust tug-debris tethered system in a Keplerian orbit, experiences chaotic attitude motion. Aslanov et al. [2] introduced steady and insecure fixed answers for the in-plane movement of the framework in a roundabout circle, which rely upon the estimation of the pull’s pushed. Bhardwaj and Kaur [3] studied the satellite motion under the effect of aerodynamic torque and explained in detail about the nonresonance oscillation. Also, they discussed that under the influence of magnetic torque for different mass parameters, tumbling of satellite experiences shows the chaotic signal [4]. Bhardwaj and Sethi [5] discussed that air drag exhibits resonance criteria for nonlinear motion. Rotational nonlinear oscillation of the satellite under the influence of combined aerodynamic and magnetic torque was discussed by Bhardwaj et al. [6], and they concluded that with the change in mass parameter, the dynamics of the satellite altered. Bhardwaj and Tuli [7] discussed the nonlinear planar oscillation of a satellite under the influence of third-body torque, and it is concluded that Hyperion tumbled more chaotically with the change in the third body torque parameter. Planar oscillation of a satellite in an elliptic orbit for magnetic torque was studied by Bhardwaj and Kaur [8], and they observed that as eccentricity changed, the oscillation of satellite exhibits chaotic motion which increases with the increase in eccentricity. Bhardwaj and Bhatnagar [9–12] studied the nonlinear planar rotational oscillation of the satellite in circular orbit for magnetic torque and for third-body torque in elliptic orbit, and it is concluded that the mass parameter and torque parameter play an important role in changing the motion from regular to the chaotic one.Chegini et al. [13, 14] explored mathematically turmoil in demeanor elements of an adaptable satellite made out of an inflexible body and two indistinguishable unbending boards connected to the fundamental body with springs using analytical and numerical methods. Clemson and Stefanovska [15] discussed the analysis of nonautonomous dynamics for extracting properties of interactions and the direction of couplings for chaotic, stochastic, and nonautonomous behaviour. For the chaotic class, the Lorenz system; for the stochastic class, the noise forced Duffing system; and for the nonautonomous class, the Poincare oscillator with quasiperiodic forcing discussed and gave a good review to distinguish nonautonomous dynamics from chaos or stochasticity. Doroshin [16] got altered numerical models and dynamical frameworks to give an idea of heteroclinic chaos and its local suppression in attitude dynamics for dual spin spacecraft and gyrostat satellites. Gutnik and Sarychev [17] mathematically simulated the motion of the satellite under aerodynamic torque for the control system influenced by the active dumping torques. Inarrea and Lanchares [18] examined the pitch movement elements of an awry rocket in round circle affected by a gravity inclination force and accepted that shuttle is irritated by a little streamlined drag force corresponding to the precise speed of the body about its mass community. Koupriano and Shevchenko [19] considered the issue of recognizability of clamorous systems in turn of planetary satellites utilizing Jacobian assessment approach. Kuptsov and Kuznets [20] discussed the Lyapunov analysis of strange pseudohyperbolic attractors and briefly analyzed about the angles between tangent subspaces, local volume expansion, and contraction.The phenomenon of chaos is generally related to the field of dynamical systems, and it can be characterized in the dynamics by sensitive dependence on the initial conditions. Chaos is a fascinating mathematical and physical phenomenon. The study of chaos shows that simple systems can exhibit a complex and unpredictable behaviour. The chaos in the dynamics can be identified and quantified by several techniques. A positive value of the Lyapunov exponent provides chaos in the dynamics which is discussed by Letellier [21]. Liu and Cui [22] analyzed the nonlinear model which should be adopted for the sailcraft in long duration missions, and the restricted position of the sliding mass could be selected elaborately to utilize the resultant torque by the gravitational and center-of-mass or center-of-pressure torques. Melnikov and Shevchenko [23] considered the issue of figuring the Lyapunov season of the disorganized movement region for resonances in satellite movement. Pritykin et al. [24] discussed the long-term evolution of attitude motion for defunct satellites in nearly polar orbits. Rosengren et al. [25] indicated that the sporadic and random characters of the Global Navigation Satellite Systems’ circles mirror a comparative inconsistency in the circles of numerous divine bodies in our solar framework. Rawashdeh [26] studied the attitude analysis of small satellites using model-based simulation. Efimov et al. [27] discussed about long-term attitude dynamics of space debris for sun-synchronous orbits and studied about Cassini cycles and chaotic stabilization. Chang [28] gave an idea of stability, chaos detection, and quenching chaos for the swing equation system. Wang et al. [29] developed the six‐dimensional hyperchaotic system and applied for secure communication circuit implementation. Wolf et al. [30] introduced the main calculations that permit the assessment of nonnegative Lyapunov types from an exploratory time arrangement.Apparently, none of the creators have contemplated the bedlam affected by the streamlined force in an elliptic circle. In the current examination, we contemplated the tumultuous movement of a satellite affected by a streamlined force in an elliptic circle. In this study, the condition of movement for the framework is inferred. Utilizing variety of boundaries techniques, the unrest, libration, and endless period separatrix are examined. The mathematical recreation of tumultuous movement affected by the streamlined force is examined for Earth-Resourcesat-1 satellite.


## 2. Mathematical Model

Let an inflexible satellite S revolve in elliptic circle around Earth E with the end goal that orbital plane concurs with central plane of Earth. S is thought a trihub body with head snapshots of inactivityA<B<C at its focal point of mass, and C is the snapshot of idleness about turn hub which is opposite to the orbital plane. Let r⟶ be the sweep vector of focal point of mass of S, ν be the true anomaly, θ be the point that the long hub of S makes with fixed line EF lying in the orbital plane, and η/2 be the point between the span vector and long pivot as shown in Figure 1.Figure 1
S revolving around Earth E.Equation of motion for the system, see details as given in [3], is obtained as(1)d2ηdν2+n2η=−ecosνd2ηdν2+2esinνdηdν+4esinν+n2η−sinη+εA∗ν2sinν+B∗νsinν+C∗sinν+D∗ν+E∗,where n2=3B−A/C=mass parameter;ε=ρSCdl2/CΩ2= aerodynamic torque parameter; A∗=a21−e/Ω2l = constant; B∗=ωa2e−1/Ωcosi+2V1a1−2e/Ωl  = constant; C∗=ωV12e−1cosi+V121−2e/l+ωae/2Ωsini = constant; D∗=ωa2e−1/Ωsini = constant; E∗=ωV12e−1sini = constant; and(2)d2θdt2=μr3−2esinν−esinνdηdν+121+ecosνd2ηdν2.From equations (1) and (2), we get(3)d2θdt2=−μ2r3n2sinδ−ε1A∗ν2sinν+B∗νsinν+C∗sinν+D∗ν+E∗.Takingn2=ω02=3B−A/C;θ=ν+δ/2⇒δ=2θ−ν, equation (3) becomes(4)d2θdt2=−μ2r3ω02sin2θ−ν−ε1A∗sinν3+B∗sinν2+C∗sinν+D∗ν+E∗.In condition (4), if units are picked to the point that orbital time of S is 2π and its semisignificant pivot is 1, at that point dimensionless, time is equivalent to mean longitude or genuine inconsistency which is 2π intermittent and μ = 1. As r and ν are 2π occasional as expected, utilizing Fourier-like Poisson series (Wisdom et al. [31]), equation (4) becomes(5)d2θdt2+ω022∑Hm2,esin2θ−mt−ε2A∗sinν3+B∗sinν2+C∗sinν+D∗ν+E∗=0,Hm/2,e corresponds to e2m/2−1 and is given by Cayley [32] and Goldreich and Peale [33]. At the point, if e is little, Hm/2,e≅−e/2. The half whole number m/2 is signified by the image p. Resonances happen at whatever point one of the contentions of the sine or cosine capacities is almost fixed, for example, at whatever point dθ/dt−p≪1/2. In such cases, it is to rework the condition of movement as far as the gradually changing reverberation variable vp=θ−pt⇒d2vp/dt2=d2θ/dt2⟹2vp=2θ−mt. Equation (5) can be written as(6)d2vpdt2+ω022Hp,esin2vp−ε2A∗sinν3+B∗sinν2+C∗sinν+D∗ν+E∗=0.This is pendulum perturbed byε/2A∗sinν3+B∗sinν2+C∗sinν+D∗ν+E∗. When ɛ ≠ 0, condition (6) speaks to the condition of movement of upset pendulum given by(7)d2xp/dt2+f′xp=mpg′xp,t,where xp=2vp;f′xp=k1p2sinxp;k1p2=ω02Hp,e;mp=ε; and g′xp,t=A∗sint3+B∗sint2+C∗sint+D∗t+E∗. The unperturbed piece of condition (7) is d2xp/dt2+f′xp=0⇒dxp/dt2=2k1p2cosxp+c1p. The integration constant is defined as c1p. If c1p+2k1p2≥0, then motion is said to be real. Three kinds of motions are defined based on the conditions c1p>2k1p2,c1p<2k1p2, and c1p=2k1p2.


### 2.1. Category-I

We considerc1p>2k1p2. Forc1p>2k1p2, the value of dxp/dt never vanishes; it is either certain or negative, and the pendulum is seeming well and good or the other. For this situation, the unperturbed arrangement is(8)xp=lp+c1psinlp+oc1p2,lp=npt+ε1,c1p=k1p2np2;1np=12π∫02πdxpc1p+2k1p2cosxp1/2,where c1p and ε1 are the discretionary constants, and lp is a contention. Intermittent segment of this arrangement can be viewed as swaying about the mean condition of movement which is unrest with a period 2π/np. Half plentifulness of wavering is clearly not exactly π, and it diminishes as np increments. Here, we may see that dxp/dt≠0, and the movement is supposed to be of type I, for example, upheaval. Brown and Shook [34] proposed the theory of variation of parameters for the perturbed pendulum which gives(9)dc1pdt=mkp∂x∂lg′,dlpdt=n−mkp∂x∂c1g′,kp=∂∂c1pnp∂x∂l∂x∂l−np∂2x∂l2∂x∂c1p,since c1p=k1p2/np2. Therefore,(10)∂np∂c1p=−np2c1p,∂xp∂lp=1+c1pcoslp,∂2xp∂lp2=−c1psinlp,∂xp∂c1p=sinlp,∂2xp∂c1p∂lp=coslp.Putting the above values and writingkp=k1p, equation (9) can be written as(11)k1p=−np2c1p−npc1pcos2lp2+npc1p≅−np2c1p.Hence,dc1p/dt≅0; so, c1p is the second order approximation constant. Second equation of (9) gives(12)dlpdt=np+2mpc1pnpsinlpA∗sint3+B∗sint2+C∗sint+D∗t+E∗.Rejecting second or higher order terms, we get(13)d2lpdt2=2C∗mpc1pnp+2D∗mpc1pnp−6E∗2mp2c1pnp2+4E∗2mp2c1p2npsinlp+1+2c1pC∗mpsint+D∗mpt+E∗mp,d2lpdt2+k2p2sinlp=mp1+2c1pC∗sint+D∗t+E∗.Letlp=xp,d2xp/dt2+k2p2sinxp=mpg″xp,t, where g″xp,t=1+2c1pC∗sint+D∗t+E∗, and k2p2=−2C∗mpc1p/np+2D∗mpc1p/np−6E∗2mp2c1p/np+4E∗2mp2c1p2/np. The unperturbed part of the above equation is d2xp/dt2+k2p2sinxp=0⇒dxp/dt2=2k2p2cosxp+c2p, where c2p is a constant of integration. Three types of motions are obtained for the motion of pendulum.(1)
Ifdxp/dt≠0, then motion of type 1 exists. For type 1, the solution is xp=Npt+ε2p+k2p2/Np2sinNpt+ε2p+⋯; 1/Np=1/2π∫02πdxp/c2p+2k2p2cosxp1/2, where c2p and ε2p are the arbitrary constants. For first approximation, Np=N0p; so, xp=x0p+k2p2/N0p2sinx0p, where x0p=N0pt+ε2p. This is situation of unrest.(2)
Ifdxp/dt=0at0orπ, then motion of type 2 exists. For type 2, solution is xp=λpsinp′t+λ0, where p′=2mpω02Hp,e/np3C∗+D∗+E∗2mp/np−3+22ω02Hp,e/np2,λp, λ0 are defined as the constants of integration. This is situation of libration.(3)
Type 3 whenc2p=2k2p2=−4ω02mpHp,e/np3C∗+D∗+E∗2mp/np2ω02Hp,e/np2−3. Solution is xp+π=4tan−1expk2pt+α0. The arbitrary constant is defined as α0. It is observed that as t⟶±∞,dxp/dt=0 as xp⟶±π are consequently higher subsidiaries of xp way to deal with zero. As xp ways to deal with ±π, t watches out for uncertain capacity of xp. This is case of boundless period separatrix.We presumed that because of the aftereffects of type 1, type 2, and type 3, the streamlined force assumes a huge part in changing the movement of upset to libration or to boundless period separatrix.


### 2.2. Category-II:c1p<2k1p2

For this situation, unperturbed arrangement isxp=c1psinlp+oc1p3,lp=npt+ε1, and np=k1p1−1/16c1p2+⋯, where c1p and ε1 are the discretionary constants, and lp is a contention. If there should arise an occurrence of bothered pendulum by utilizing the hypothesis of variety of boundaries, we get kp=∂/∂c1pnp∂x/∂l∂x/∂l−np∂2x/∂l2∂x/∂c1p, kp≅k1pc1p,(14)d2lpdt2=mpk1pc1p−C∗−D∗+mpE∗2k1pc1p2sinlp−mpc1pC∗sint+D∗t+E∗+mp2k1p2c1p2C∗E∗sint+D∗E∗t+E∗2,⇒d2lpdt2+k3p2sinlp=−mpc1pC∗sint+D∗t+E∗+mp2k1p2c1p2C∗E∗sint+D∗E∗t+E∗2,where k3p2=−mp/k1pc1p−C∗−D∗+mpE∗2/k1pc1p2. The unperturbed part of the equation is d2lp/dt2+k3p2sinlp=0, where lp is little, and the arrangement of above condition is lp=ek3pt+e−k3pt. It is again a condition of a pendulum, and as in a prior case, the movement is alluded as upset, libration, and boundless period separatrix.


### 2.3. Category-III:c1p=2k1p2

The unperturbed arrangement is(15)xp+π=4tan−1expk1pt+α0,where α0 is a discretionary steady. This is the situation of endless period separatrix as asymptotic forward and in reverse so as to insecure harmony. In this class, the idea of unperturbed arrangement does not change by considering the streamlined force. Close to the endless period, separatrix widened by high recurrence term into tight clamorous band for little n, and half width of disordered separatrix is given by(16)ωp=Ip−IpsIps=4πε1λ3e−πλ/2,where ɛ1 is the proportion of coefficient of closest annoying high-recurrence term to coefficient of perturbed term, and λ = Ω/ω is the proportion of recurrence distinction between full term and closest nonfull term (Ω) to recurrence of little sufficiency freedoms (ω).


## 2.1. Category-I

We considerc1p>2k1p2. Forc1p>2k1p2, the value of dxp/dt never vanishes; it is either certain or negative, and the pendulum is seeming well and good or the other. For this situation, the unperturbed arrangement is(8)xp=lp+c1psinlp+oc1p2,lp=npt+ε1,c1p=k1p2np2;1np=12π∫02πdxpc1p+2k1p2cosxp1/2,where c1p and ε1 are the discretionary constants, and lp is a contention. Intermittent segment of this arrangement can be viewed as swaying about the mean condition of movement which is unrest with a period 2π/np. Half plentifulness of wavering is clearly not exactly π, and it diminishes as np increments. Here, we may see that dxp/dt≠0, and the movement is supposed to be of type I, for example, upheaval. Brown and Shook [34] proposed the theory of variation of parameters for the perturbed pendulum which gives(9)dc1pdt=mkp∂x∂lg′,dlpdt=n−mkp∂x∂c1g′,kp=∂∂c1pnp∂x∂l∂x∂l−np∂2x∂l2∂x∂c1p,since c1p=k1p2/np2. Therefore,(10)∂np∂c1p=−np2c1p,∂xp∂lp=1+c1pcoslp,∂2xp∂lp2=−c1psinlp,∂xp∂c1p=sinlp,∂2xp∂c1p∂lp=coslp.Putting the above values and writingkp=k1p, equation (9) can be written as(11)k1p=−np2c1p−npc1pcos2lp2+npc1p≅−np2c1p.Hence,dc1p/dt≅0; so, c1p is the second order approximation constant. Second equation of (9) gives(12)dlpdt=np+2mpc1pnpsinlpA∗sint3+B∗sint2+C∗sint+D∗t+E∗.Rejecting second or higher order terms, we get(13)d2lpdt2=2C∗mpc1pnp+2D∗mpc1pnp−6E∗2mp2c1pnp2+4E∗2mp2c1p2npsinlp+1+2c1pC∗mpsint+D∗mpt+E∗mp,d2lpdt2+k2p2sinlp=mp1+2c1pC∗sint+D∗t+E∗.Letlp=xp,d2xp/dt2+k2p2sinxp=mpg″xp,t, where g″xp,t=1+2c1pC∗sint+D∗t+E∗, and k2p2=−2C∗mpc1p/np+2D∗mpc1p/np−6E∗2mp2c1p/np+4E∗2mp2c1p2/np. The unperturbed part of the above equation is d2xp/dt2+k2p2sinxp=0⇒dxp/dt2=2k2p2cosxp+c2p, where c2p is a constant of integration. Three types of motions are obtained for the motion of pendulum.(1)
Ifdxp/dt≠0, then motion of type 1 exists. For type 1, the solution is xp=Npt+ε2p+k2p2/Np2sinNpt+ε2p+⋯; 1/Np=1/2π∫02πdxp/c2p+2k2p2cosxp1/2, where c2p and ε2p are the arbitrary constants. For first approximation, Np=N0p; so, xp=x0p+k2p2/N0p2sinx0p, where x0p=N0pt+ε2p. This is situation of unrest.(2)
Ifdxp/dt=0at0orπ, then motion of type 2 exists. For type 2, solution is xp=λpsinp′t+λ0, where p′=2mpω02Hp,e/np3C∗+D∗+E∗2mp/np−3+22ω02Hp,e/np2,λp, λ0 are defined as the constants of integration. This is situation of libration.(3)
Type 3 whenc2p=2k2p2=−4ω02mpHp,e/np3C∗+D∗+E∗2mp/np2ω02Hp,e/np2−3. Solution is xp+π=4tan−1expk2pt+α0. The arbitrary constant is defined as α0. It is observed that as t⟶±∞,dxp/dt=0 as xp⟶±π are consequently higher subsidiaries of xp way to deal with zero. As xp ways to deal with ±π, t watches out for uncertain capacity of xp. This is case of boundless period separatrix.We presumed that because of the aftereffects of type 1, type 2, and type 3, the streamlined force assumes a huge part in changing the movement of upset to libration or to boundless period separatrix.


## 2.2. Category-II:c1p<2k1p2

For this situation, unperturbed arrangement isxp=c1psinlp+oc1p3,lp=npt+ε1, and np=k1p1−1/16c1p2+⋯, where c1p and ε1 are the discretionary constants, and lp is a contention. If there should arise an occurrence of bothered pendulum by utilizing the hypothesis of variety of boundaries, we get kp=∂/∂c1pnp∂x/∂l∂x/∂l−np∂2x/∂l2∂x/∂c1p, kp≅k1pc1p,(14)d2lpdt2=mpk1pc1p−C∗−D∗+mpE∗2k1pc1p2sinlp−mpc1pC∗sint+D∗t+E∗+mp2k1p2c1p2C∗E∗sint+D∗E∗t+E∗2,⇒d2lpdt2+k3p2sinlp=−mpc1pC∗sint+D∗t+E∗+mp2k1p2c1p2C∗E∗sint+D∗E∗t+E∗2,where k3p2=−mp/k1pc1p−C∗−D∗+mpE∗2/k1pc1p2. The unperturbed part of the equation is d2lp/dt2+k3p2sinlp=0, where lp is little, and the arrangement of above condition is lp=ek3pt+e−k3pt. It is again a condition of a pendulum, and as in a prior case, the movement is alluded as upset, libration, and boundless period separatrix.


## 2.3. Category-III:c1p=2k1p2

The unperturbed arrangement is(15)xp+π=4tan−1expk1pt+α0,where α0 is a discretionary steady. This is the situation of endless period separatrix as asymptotic forward and in reverse so as to insecure harmony. In this class, the idea of unperturbed arrangement does not change by considering the streamlined force. Close to the endless period, separatrix widened by high recurrence term into tight clamorous band for little n, and half width of disordered separatrix is given by(16)ωp=Ip−IpsIps=4πε1λ3e−πλ/2,where ɛ1 is the proportion of coefficient of closest annoying high-recurrence term to coefficient of perturbed term, and λ = Ω/ω is the proportion of recurrence distinction between full term and closest nonfull term (Ω) to recurrence of little sufficiency freedoms (ω).


## 3. Spin-Orbit Phase Space

Utilizing Poincare surface of the segment by taking a gander at directions stroboscopically with period 2π, the segment is drawn with dη/dν versus v at each periapse section. On account of semi-intermittent direction, focuses are contained in smooth bends, while for clamorous directions, they seem to the top off region in the stage space in an arbitrary way. Since direction indicated by η is identical to the direction signified by π + η, we have, consequently, confined the span from 0 to π.


## 4. Results and Discussion

Poincare map, surface of section, and Lyapunov exponents have been plotted for Earth’s artificial satellite Resourcesat-1. For the satellite, it is assumed that semimajor axis =a = 7.195∗103 km, flightiness = e = 0.001, tendency = i = 98.69°, and angular velocity = Ω = 1.034∗10−3 rad/sec. The effect of mass parameter (n) and aerodynamic torque parameter (ɛ) is studied on the nonstraight wavering of a satellite in an elliptic circle. Poincare maps, surface of section, Lyapunov exponents, and time series for different values of mass parameter and aerodynamic toque parameter are plotted as described in tables and figures. Table 1 gives the details of figures for Earth−Resourcesat−1for fixed values ofA∗,B∗,C∗,D∗,E∗,ε,ande, and the variation of values of n from 0 ≤ n ≤ 1 is shown in Figures 2–5. Table 2 gives description of figures for the Earth-Resourcesat-1 system at fixed values of parameters, n, and e, and the variation of values of ɛ from 0 ≤ ɛ ≤ 0.5 which are plotted is shown in Figures 6–9. From the plots, it is observed that regular curves disintegrate, and this disintegration increases as ɛ increases and curves behaves chaotically but remains almost same.Table 1
Earth-Resourcesat-1 system for fixed values ofA∗=3.36E+09,B∗=1596.387,C∗=−103631,D∗=−4E+08,E∗=200.1022,e=0.001,ε, and variation in n.
Figure no.nɛGraphical behaviour of Poincare mapGraphical behaviour of Lyapunov exponent20.00010.000000000000000001Regular curves disintegrate as ɛ increasesChaotic30.00000000000001Chaotic40.90.000000000000000001Periodic50.00000000000001Periodic and chaoticFigure 2
Forn=0.0001,ε=0.000000000000000001,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 3
Forn=0.0001,ε=0.00000000000001,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 4
Forn=0.9,ε=0.000000000000000001,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 5
Forn=0.9,ε=0.00000000000001,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Table 2
Earth-Resourcesat-1 system for fixed values ofA∗=3.36E+09,B∗=1596.387,C∗=−103631,D∗=−4E+08,E∗=200.1022,e=0.001,n,and variation inε.
Figure no.ɛnGraphical behaviour of Poincare mapGraphical behaviour of Lyapunov exponent60.00000000000000000000000050.00001Curves behave chaotically but remains almost sameChaotic70.0000000000000000000050.005Chaotic80.4Periodic and chaotic90.8PeriodicFigure 6
Forn=0.00001,ε=0.0000000000000000000000005,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 7
Forn=0.005,ε=0.000000000000000000005,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 8
Forn=0.4,ε=0.000000000000000000005,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)Figure 9
Forn=0.8,ε=0.000000000000000000005,A∗=3.36E+09,D∗=−4E+08,B∗=1596.387,C∗=−103631,E∗=200.1022,ande=0.001. (a) Poincare map, (b) Poincare surface of section, (c) Lyapunov exponent, and (d) time series.
(a)(b)(c)(d)


## 5. Conclusion

From these investigations, we conclude that the streamlined force assumes an extremely huge function in changing the movement of insurgency into movement of libration or endless period separatrix. Likewise, we see that standard movement changes into a turbulent one for certain estimations of the streamlined force boundary and mass boundaryn. Half width of disordered separatrices assessed by Chirikov’s basis is not influenced by the streamlined force. It was seen that counterfeit satellite’s turn circle stage space is overwhelmed by a chaotic zone which increments further because of the streamlined force. It is additionally seen that normal bends begin breaking down because of the streamlined force and mass boundary, and this deterioration increments as the streamlined force and mass boundary increments. It is concluded that aerodynamic torque and n change regular movement to the chaotic motion.



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*Source: 6658051-2021-02-10.xml*