AIAA-92-4369

Chaos in the Pitch Equation of Motion for the Gravity-Gradient Satellite

H a r r y K a r a s o p o u l o s W r i g h t L a b o r a t o r y W L / F I M G W P A F B , O H 4 5 4 3 3 - 6 5 5 3

D a v i d L . R i c h a r d s o n D e p t . A e r o s p a c e E n g i n e e r i n g U n i v e r s i t y o f C i n c i n n a t i C i n c i n n a t i , O H 4 5 2 2 1

1992 A I A A / A A S Astrodynamics Conference

A u g u s t 1 0 - 1 2 , 1 9 9 2 H i l t o n H e a d , S o u t h C a r o l i n a

F o r p e m M o n t o c o p y o r r e p u p l s h . c o n t a c t t h e A m e r i c a n i n s t i t u t e o f A e r o n a u t i c s a n d A s t r o n a u t i c s 3 7 0 L ' E n f a t i t P r o m e n a d e . S . W . . W a s h i n g t o n . O . C . 2 0 0 2 4

- «

Chaos in the Pitch Equation of Motion for the Gravity-Gradient Satellite

Harry Karasopoulos* Wright Laboratory

and David L . Richardson'

University of Cincinnati

Abstract

The nonlinear dynamics of the pitch equation of motion for a gravity-gradient satellite in am elliptical or­bit about a central body are investigated. This planar motion is shown to be either periodic, quasiperiodic, or chaotic, depending upon the values of eccentricity, satel­lite inertia ratio, and the initial conditions for pitch an­gle and its derivative with respect to the true anomaly. Bifurcation plots, Poincare maps, and Lyapunov expo­nents are numerically calculated and presented. Chaos diagrams, which are computed from Lyapunov expo­nents and are dependent upon the satellite's initial con­ditions, are also presented and may serve as a valuable satellite or orbit design tool. The sea of chaotic motion observed in the chaos diagrams has an interesting and complex structure. It is found that the instability of the pitch angle for a gravity-gradient satellite generally increases for increasing values of orbit eccentricity.

Nomenclat ure

C Constant of integration e Orbit eccentricity I r x , lyy, hz Principle moments of inertia A' Moment of inertia ratio iV Number of orbits P Integration step size constant a Pitch angle winding number P Pitch angle rate winding number A t Integration step size

•Aerospace Engineer, W L / F I M G , Wright-Patterson Air Force Base, Ohio 45433-6553, Member AIAA

'Professor, Dept. of Aerospace Engineering, University of Cincinnati, Cincinnati, OH 45221

T h i s p a p e r i s d e c l a r e d a w o r k o f t h e U . S . G o v e r n m e n t a n d i s n o t s u b j e c t t o c o p y r i g h t p r o t e c t i o n i n t h e U n i t e d S t a t e s .

fC{k) Complete elliptic integral of the first kind

A Constant of integration f Satellite true anomaly ujg Earth rotation rate (T First Lyapunov exponent 9 Satellite pitch angle measured from

the local vertical 9' Derivative of the pitch angle with

respect to true anomaly 9" Second derivative of the pitch angle

with respect to true anomady

Introduction

Gravity-gradient perturbations can have a pro­found influence upon the motion of natural and artificial satellites alike. A number of investigations have shown that gravity-gradient torques combined with tidail fric­tion can cause chaotic tumbling of irregularly shaped satellites. Hyperion, one of Saturn's moons, currently exhibits such behavior. Wisdom^ points out that every irregularly shaped satellite in the Solar System which tidally evolved into a synchronous rotation, had to tum­ble chaotically at one time.

Determination of the dynamics of an artificial satellite due to gravity-gradient torques is a very impor­tant step in the design process. There are many appli­cations (communication, observation, etc) where point­ing requirements make passive stabilization of pitch an­gle attractive for artificial satellites in orbit about the Earth. Pitch angle stability for a gravity-gradient satel­lite has received a great deal of attention, with publi­cations spanning three decades. Of particular interest, however, are studies where phase plane mappings are presented to indicate stability boundaries for the noncir-cular orbit case. ' '"*' In one of these. Modi and Brere-ton^ applied the method of harmonic balance to obtain families of periodic solutions and found that at the max­imum eccentricity for stable motion, the solution must be periodic.

1

In this paper, modern nonlinear analysis tech­niques are applied to a study of the phase space of the pitch equation of motion. Periodic, quasiperiodic, and chaotic motion are discussed through presentations of Poincare maps, bifurcation plots, Lyapunov exponents, and chaos diagrams. The closed-form solutions to the circular orbit case are also briefly exaunined for compar­ison purposes.

Equat ion of Motion

This analysis assumes the satellite is influenced by an inverse-square gravitational field only; satellite energy dissipation and other torques, such as aierody-namic, magnetic, thermal bending, or solar radiation pressure, are ignored. The planar pitch equation of mo­tion is^

(H-ecosi / )r-2es ini / ( l - | -6 l ' ) + 3A'sin6lcos6i = 0 (1)

where 1 / is the true anomaly, e is orbit eccentricity, 6 is the pitch angle with respect to the local vertical (Fig­ure 1), and 6' is the derivative with respect to the true anomaly. K is a function of the principle moments of inertia of the satellite

_ L x I z z (2)

Zyy

and sufficient conditions for three-eixis stabilization are given by

lyy > Ixx > I z z (3)

Thus, values of K ranging from 0.0 to 1.0 are of interest, with the upper limit corresponding to the inertia prop­erties of a dumbbell satellite. In this analysis it is as­sumed that these conditions for three-axis stabilization are met: the satellite's major tixis is in the direction of the orbital angular momentum; and for zero pitch angle, the minor and intermediate axes are aligned with the lo­cal vertical and horizontal, respectively. This analysis is therefore independent of the satellite, whether it be the Moon, the space shuttle, or a Yugo, so long as the body's inertia properties are arranged in the assumed manner.

C ircn lar Orbit Case

The solution to the circular orbit case can be found analytically and is well k n o w n . I t is briefly re­viewed here to emphasize the form of the solutions for the elliptical orbit case in the limit as eccentricity tends to zero. The pitch equation of motion for a circular orbit (e=0) reduces to

This equation has a form essentially identical to that of a simple pendulum; its phase plane (Figure 2) has iden­tical features and differs only in a shift of it radians in the location of the stable and unstable equilibria. The stable equilibrium point at (0.0,0.0) corresponds to the synchronous state where the satellite always points to­wards the Earth. For this case the satellite never rotates relative to the Earth, and its minor moment of inertia is exactly aligned with the local vertical . The unstable equilibria correspond to the alignment of the satellite's minor moment of inertia with the local horizontal.

The Hamiltonian for the circular orbit case of the gravity-gradient satellite problem is integrable. The phase space trajectories of this two degrees of freedom problem are therefore confined to a two-dimensional torus manifold. Because this is a multiply periodic sys­tem, the values of the two frequencies (pitch and true anomaly) determine the nature of the motion. An ir­rational ratio produces quasiperiodic motion, appearing as a closed curve on a Poincare map. On the torus manifold, however, quasiperiodic motion meeuis a single orbit will ultimately uniformly cover the torus. Integer combinations of the frequencies produce periodic motion and closed orbits on the torus.

The derivative of the true anomaly with respect to time is constant for a circular orbit. Noting that

1 / = w.

and integrating once yields

9"^ + 2>Kujg'^sin'^9 ^

(5)

(6)

where C"^ is the constant of integration. It is advanta­geous to define another constant, A, such that

2 \

This allows the equation of motion to be written as

(7)

(8)

Like the simple pendulum, the pitching motion of a gravity-gradient satellite in a circular orbit may either be a libration about the local vertical (A > 1) or a tum­bling motion with a periodic part (A < 1). Solutions to these two cases are in the form of elliptic integrals of the first kind. It can be found that periodic solutions consisting of m tumbling or libration oscillations in n orbits satisfy the equations*

9"-|-3A'sintfcosfl = 0 (4)

m oscillations n orbits

m i X T V 3 K

(tumbling)

(libration) (9)

2

where IC{k) is a complete elliptic integral of the first kind, given by

/C (U = ^ k^+

f 3-5 \ V 2 - 4 - 6 j

Ife* + . . . (10)

These equations may be utilized to determine the ap­propriate conditions required for periodic motion of the pitch angle for the tumbling and libration cases. One can recursively solve for A and hence find the initial conditions required for the tumbling motion to have a specified periodic part

e ( t o ) = 0

. (<o) . LjgXsJ'iK (11)

or the initial conditions required for periodic pitch angle libration.

0 uigX^yiK

or e{to) = sin A

0 (12)

Figure 3 presents the values of K and A required for various periodic solutions of pitch angle libration and tumbling. While there exists a broad range of inertia ratio and A combinations which will produce tumbling motion, the range of combinations for libration motion are often more limited, especially for higher period mo­tion. For example, satellite tumbling with a periodic motion component of period | may occur for almost any value of K , whereas period | libration can only ex­ist for satellites having inertia ratios between about 0.75 and 1.0.

Figure 4 presents the maximum pitch angle at­tained for various periodic solutions for the circular or­bit libration case. Careful design of a satellite's inertia properties and/or selection of the initial state vector is necessary to attain a small maocimum pitch libration an­gle. This figure demonstrates that the lower a satellite's inertia ratio, the larger the pointing error for a specified variance in K . In other words, the required knowledge of the accuracy of K in order to keep 6 m a x within spec­ified limits increases with decreasing inertia ratio.

The maximum value of 9' attained for various peri­odic solutions for the circular orbit libration case is given in Figure 5. Note that for any periodic libration solu­tion, the magnitude of the maximum value 0' remains less than 2. In fact, the limiting values for 9' and 6 for pitch angle libration occur as A — i - 1 and they depend only on K (Figure 6). These curves are the seperatices of the phase diagram and this plot therefore depicts the change in the form of the separatices as a function of the satellite's inertia ratio.

Eccentr i c Orbit Case

Chaos is am extreme sensitivity to initial con­ditions (or very small chamges in system pairameters) which leads to exponential divergence. In the words of I.C.Percival,^

"Chaotic motion appears in dynamicad sys­tems when local exponential divergence of tra­jectories is au:compamied by global confinement in the phase spau:e. The divergence produces a locad stretching in the phase space, but, be­cause of the confinement, this stretching cam-not occur without folding. Repeated folding and refolding produces very complicated be­havior that is described as chaotic."

This folding and stretching of phase space often leads to the popular analogy of a baker's transformation in the literature. Chaotic motion commonly occurs in conser­vative systems of two degrees of freedom or more and this problem is no exception. The Hamiltoniam for this system is non-integrable and the motion is mixed. Reg­ular and chaotic motion occur for different initial con­ditions and values of the system parameters, e and K . Both types of motion may occur at a very close prox­imity to one-another in phase space, and the transition between the two types of behavior is complex. There are a number of numerical and analytical tools which may be utilized to examine the complex behavior of nonlinear dynamical systems. This work concentrates on the ap­plication of some numerical methods, such as Poincare maps, bifurcation diagrams, and Lyapunov exponents, to study the occurrence of chsios in the gravity-gradient pitch equation.

Poincare Maps and Bifurcation Diagrams

The purpose of a Poincare map (also called an area-preserving map or a surface of section) is to facil­itate the study of a system by reducing the scope of the problem. For this particular problem, mapping the continuous three degrees of freedom state space ( 9 , 9 ' , u ) into a two degrees of freedom [ 9 , 9 ' ) plot facilitates the examination of the system's characteristics. This is ac­complished by integrating the equation of motion and creating a discrete collection of points by periodically sampling the generated values of states at a particular point in the trajectory. Thus, a Poincare map provides a sort of "stroboscopic" view of the phase space, and we collect N values of

and

9 n = e { u n ) (modulus 2ir) (13)

0'. = O ' M (14)

3

where

!/„ = nAv + J/o n = 0 ,1 ,2 , . . . /V (15)

In this problem the solutions were sampled once ecich orbit as the satellite passed through periapsis, A u = 23r. At least 10 orbits were integrated before the data was sampled in order to allow the trajectories to settle, i / Q > 205r.

Regular and chaotic motion may be observed in Poincare maps. If the winding numbers a and /?, defined as

P = - (16) m

are rational (j , k, I, m are integer) and if

e{u + air) = 0 { u ) (17)

and ^ ' ( 1 / + /?«•) = ^'(i/) (18)

the trajectory is periodic with j fixed values of 0 and / fixed values of Thus, periodic motion will appear as the greater number of j or / fixed points on the Poincare map. Quasiperiodic motion, which occurs when either winding number is irrational, produces a closed curve on the surface of section when a sufficient number of trajectory samples are mapped. Chaotic motion appears on a Poincare map as a scattering of points which, if N were large enough, would completely fill an area of the surface of section. Hence, the dimension of the Poincare maps for periodic, quasiperiodic, and chziotic motion are 0, 1, and 2, respectively.

An often applied technique for examining the ef­fects of parameter variations on a dynamical system is the bifurcation diagram. To make a bifurcation dia­gram, some system state or other measure of the motion is periodically sampled in the same manner as for the Poincare map, and then plotted as a function of the sys­tem parameters. Excellent insight into period doubling routes to chaos has been garnered through examination of such plots in many past studies. However, often more information (such as a Poincare map) is needed to dis­cern differences between quasiperiodic and chaotic mo­tion. In this study, bifurcation diagrams of ^(fn) and ^ ' W n ) were constructed for variations in eccentricity and inertia ratio.

Numerical calculation of the bifurcation plots and Poincare maps made extensive use of an optimized coef­ficients version of a Runge-Kutta integration algorithm. This routine has good stability properties and a local truncation error of the integration step size raised to the 6th power. Typical integration step sizes applied in the numerical analyses were

A u 2ir A r = — = — (19)

with P ranging from 150 to 250. To minimize re­quired integration times, the lower vedue of P was used whenever possible - especially for the generation of the Poincaue maps which typically required more than 20,000 points for the entire plot. Results are pre­sented for varying values of the two system parameters, 0.0 < K < 1.0, and 0.0 < e < 1.0.

Studies^ ' * have shown that spurious solutions are possible due entirely to the discretization of a con­tinuous differential equation, even for integration incre­ments below the linearized stability limit of the inte­gration method. In a similar study, Lorenz^'' showed that "computational chaos" may occur in turns with quasiperiodic motion for a r2inge of step sizes before the differencing scheme finally blows up. The message is ob­vious: care is required in the selection of the integration step size. Application of P=50 for K = l . Q a n d e=0.3 created the "false" Poincare map of Figure 7. This sur­face of section displays chaotic motion instead of quasi-periodic tumbling (compare with Figure 18a), demon­strating the importance of integration accuracy.

Lyapunov Exponents

Lyapunov exponents measure the exponential di­vergence of two trajectories with nearly identical ini­tial conditions. In the calculation of the Lyapunov exponents we examine the change in dimension of a small circle of initial conditions of radius 6 in the phase space over N orbits, or iterations of the Poincare map. Through stretching and contraction of the phase space, the circle is transformed into an approximate ellipse hav­ing semimajor axis, 6 • Li„, and semiminor axis, 6 • Lin-L \d Li are called Lyapunov numbers and the Lya­punov exponents are the natural logarithms of these numbers. In general, a Lyapunov exponent greater than zero indicates the trajectory is chaotic; a Lyapunov ex­ponent equal to or less than zero means the motion is regular. In contrast to dissipative systems, the area of the phase space is conserved for Hamiltonian systems (and hence the name "area preserving maps") and the stretching in one direction is exactly countered by con­traction in the other direction. Thus, for a two degrees of freedom conservative system, we have two Lyapunov exponents of the same magnitude but with opposite sign. The largest of the exponents (which corresponds to the stretching) is called the "first" or "largest" Lya­punov exponent, <T, and this exponent alone determines the nature of the trajectory. For a conservative system the first Lyapunov exponent may only be greater than (chaotic motion) or equal (regular motion) to zero in order for phase space area to be conserved. The greater the value of <r, the greater the sensitivity and the more chaotic the trajectory.

Denoting the distances in phase space between the

4

initial conditions of two trajectories and their nth pass through periapsis as and 6„,

6n+i = ^„2'-^'' (20)

For accuracy, <r must be averaged over a large number of orbits, N , leading to the following definition

( 7 = lim - J — y i o g ^ E t l (21) n = 0 "

A number of techniques for the calculation of Lyapunov exponents appear in the literature. In this study, Lya­punov exponents were calculated using a modified ver­sion of a code given in Appendix A of Wolf e< a/.^°

Chaos Diagrams

Chaos diagrams plot the magnitude of the first Lyapunov exponent as a function of the system pa­rameters cind thus indicate the occurrence and relative magnitude of chriotic motion for specified initial states. In this study chcios diagrauris were numerically calcu­lated for the gravity-gradient satellite for 0 < e < 1, 0 < < 1, and a 400x500 grid. Lyapunov exponent val­ues are often dependent upon N (Figure 8 for example) and practical limitations on computer time restricted A' to values less than ideal. However, results from varying N in the computation of a number of trial chaos dia­grams indicated that values of Af > 50 gave preliminary results of reasonable accuracy. Although the magnitude of the first Lyapunov exponent at each point in the grid was not very accurate for N = b O , their relative magni­tudes seemed reasonably robust. Very coarsely gridded but highly accurate runs indicated that the border be­tween chaotic and regular motion was also reasonably robust for values of A used in this study.

Results

The numerical analysis methods discussed in the previous section were applied to numerous values of in­ertia ratio and eccentricity. Space limitations allow only a few of these cases to be presented here. Because the dumbbell satellite has been studied extensively in the past, an inertia ratio of K = l was used in the majority of the results presented in this paper to facilitate compar­ison. Results are also presented for the case of K = 0 . l and varying eccentricity, and for the case of constant eccentricity (e=0.2) and varying values of inertia ratio. The final section presents the chaos diagram results for 0 < e < 1 and 0 < K < 1.

Results for K = l and 0 < e < 1

Bifurcation diagrams of 6 and 6' versus eccertric-ity are presented in Figure 9 for the initial conditions

(0.0, 0.0) and if=1.0. A quick glance at these plots in­dicates the onset of chaos occurs at eccentricities near 0.3. Magnification of this area (Figure 10) reveals a complex mixture of chaotic and regular motion. Period five solutions of 6 and period four solutions of ^' exist at e=0.313, chaos occurs near e=0.312, and quasiperi­odic motion dominates at eccentricities between these values. Computation of the first Lyapunov exponent for the periodic trajectory at 6=0.313 (Figure 11) ver­ifies that the first 1000 orbits of this trajectory exhibit regular motion. The Poincare map of this particular trajectory (Figure 12) was calculated for over 10,000 or­bits and serves to further verify the periodicity of this motion.

Figure 13 presents bifurcation diagrams of the area near e=0.312. Areas of chaotic motion, quasiperiodic motion, and even periodic motion appear aind then dis­appear as e increases, apparently until a globad onset of chauas is reached, somewhere about e=0.3145. This structure is repeated again and again in further mag­nifications of the area near the onset of global chaos. These plots also indicate that the route to chaos for this system is probably not through period doubling bi­furcations. Lyapunov exponent calculations verify the e=0.312 trajectory is chciotic.

The bifurcation of 9 and 9' is complex even for regular motion "far" from chaos (Figure 14), revealing a fine, interwoven structure of smooth threads which form odd corners near e=0.3012. Although quasiperi­odic motion again is dominate, period five solutions of 9 and period three solutions of 9' exist near e=0.3041.

Much of the past research in orbitad mechanics has naturally concentrated on small perturbations of eccentricity from a circular orbit. Figure 15 displays bifurcation plots for small values of eccentricity and with A'=1.0. Chaos is absent for this particular case of K = l . O and the bifurcations appeair similar to those of Figure 14.

Examination of Poincare maps composed of nu­merous trajectories can also provide insight into the pla­nar pitch dynamics of a gravity-gradient satellite. Such a Poincare map is presented in Figure 16a for K = l . O and e=0.1, and a blow-up of the region about the origin is given in Figure 16b. A chaotic region surrounds the center features, separating the pitch angle libration and tumbling regions of the phase space. These plots show that a dumbbell (A=1.0) satellite inserted into an ec­centric orbit of e=0.1 with an initial pitch amgie of zero, would have to have an approximate initial pitch rate of either 9' > 1.9 or 9' < -1.7 for tumbling to take place. The period one libration solution, which occurred at the origin for the circular orbit case, has shifted upwards to about (0.0, 0.1) in the phase space. A period two libra­tion solution exits at approximately (0.0, 1.1177) and

5

(0.0, -0.8443) in the phase space, indicating that if given either of these initial conditions a satellite would point directly towards the Earth each time it passed through perigee, but with edternating pitch rates. Closer exami­nation would reveal other periodic solutions such as the period four solution, isolated in Figure 16c, which is surrounded by chaos and occurs near (0.0, 1.860), {ir, 1.860), and ( ± f , 0.6).

Surfaces of section for A'=1.0, e=0.2 and e=0.3 are given by Figures 17 and 18. Comparison of these plots with Figure 16a shows that the size of the chaotic region, which sprang from the separatices of the circular orbit case (Figure 6), increases dramatically with increasing eccentricity. Likewise, the libration phase plane area in the neighborhood of the origin shrinks and becomes more complex with increasing eccentricity, at least for the specified value of inertia ratio (Figure 18b). The value of the first Lyapunov exponent for the chaotic por­tion of Figure 18 was found to be more than 50% greater than for the A=1.0, e=0.1 case. This result is intuitive because increasing eccentricity in the pitch equation of motion magnifies the nonlinear portions of the equation.

Results for A ^ 1

The trends in change of the Poincare plots discov­ered at A=1.0 do not necessarily hold for other values of satellite inertia ratio. Figures 19-21 are surfaces of sec­tion for A^O. l and increasing eccentricity values. Al­though the chaotic phase space region seems to increase with increasing eccentricity, what happens to the libra­tion phase space region is not clear. This difficulty in the determination of global trends in the change of the motion of the gravity-gradient satellite for changes in eccentricity also occurs when the other system param­eter, satellite inertia ratio, is examined. A sequence of Poincare maps is presented in Figures 22-24 with vary­ing K and eccentricity held constant.

Global Trends: the Chaos Diagram

Both Poincare maps and bifurcation diagrams may be constructed for a variety of realistic values of satellite inertia ratio, eccentricity, and initial states, and utilized as an engineering tool in either satellite or orbit design. Such plots may be used to study the effect of varying the system parameters (e and K ) on the mo­tion of a gravity-gradient satellite. However, discerning global trends from local trends can be difficult. To alle­viate this problem, chaos diagrams were constructed.

Figure 25 presents a color scale chaos diagram for the initial conditions of (0.0, 0.0) and for Ar=50. White indicates regular motion and other colors indicate the motion is chaotic' The magnitude of the chaos of each

I n s o m e c o p i e s o f t h i s p s 4 > e r a b l a c k a n d w h i t e v e r s i o n o f t h i s

trajectory is displayed in the color spectrum, ramging from blue (small <r), through red, to yellow (Ijirge <T). On a large sceile, increases in eccentricity tend to in­crease the chaos, or sensitivity to initiM conditions, of the system while the effect of varying K can not be easily stated. It is interesting to see how complex this chaotic sea is for such a simple system.

Conclusions

The Hamiltonian for the in-plane pitching of a gravity-gradient satellite in an elliptical orbit is non-integrable. As is the case for most conservative systems of two degrees of freedom or more, chaotic motion is common in this system. Examination of the bifurcation plots and the Poincare maps presented herein indicates that chaotic, periodic, and quasiperiodic motion exist for different initial conditions. Chaotic motion observed in this mauiner was verified with Lyapunov exponent calculations. An approximate chaos diagram was con­structed which displayed a surface map of the magnitude of the first Lyapunov exponent for a range of values of eccentricity and satellite inertia ratio for a specified ini­tial condition. Increasing values of eccentricity tended to increase the magnitude of the Lyapunov exponents (and hence increase the exponential divergence of pitch angle) causing the system to become more chaotic. Bi­furcation diagrams demonstrate that the route to chaos for this system is probably not through period doubling bifurcations. Further work is needed to study the tran­sition from regular to chaotic motion for this system and to make the chsuDs diagram construction more efficient. Bifurcation diagrams, Poincare maps, and chaos dia­grams may be constructed and utilized as an engineering tool for satellite and orbit design to avoid chaotic mo­tion. Chaos diagrams may be particularly well suited for such design work since only one plot may be required for the entire range of the system parameters of interest.

Acknowledgments

The authors would like to thank James Hayes of the Aerodynamic Heating Group, Aeromechanics Di­vision, Wright Laboratory for his valuable computer graphics assistance.

References

^ Wisdom, J . , "Chaotic Behavior in the Solar Sys­tem," Proc. R. Soc. Loud. A 413, 109-129 (1987).

2 Modi, V . J . and Brereton, R . C . , "Periodic Solutions Associated with the Gravity-Gradient-Oriented

p l o t i s u s e d i n s t e a d o f t h e c o l o r s c i d e . F o r t h i s c a s e , b l a c k i n d i c a t e s r e g u l a r m o t i o n a n d a l l o t h e r s h a d e s d e n o t e c h a o t i c m o t i o n , w i t h t h e l i g h t e r s h a d e s a s s i g n e d t o t h e l a r g e r a v a l u e s .

6

System, Part I . Analytical and Numerical Deter­mination," AIAA Journal, No. 7, July 1969, pp. 1217-1225.

* Modi, V . J . and Brereton, R . C . , "Periodic Solutions Associated with the Gravity-Gradient-Oriented System, Part I I . Stability Analysis," AIAA Jour­nal, No. 8, August 1969, pp. 1465-1468.

Anand, D.K., et al, "Gravity-Gradient Capture and Stability in am Eccentric Orbit," Journal of Space­craft, Vol. 6, No. 12, September 1969.

^ Anand, D.K., Yuhasz, R.S., and Whisnant, J.M., "Attitude Motion in an Eccentric Orbit," Journal of Spacecraft, Vol. 8, No. 8, August 1971.

* Moran, John P., "Effects of Plane Librations on the Orbital Motion of a Dumbbell Satellite," ARS Journal, 31, 1089-1096 (1961).

' Hughes, P.C.. Spacecraft Attitude Dynamics, John Wiley k Sons, Inc., New York, 1986.

* Karasopoulos, Harry A., "Pitch Dynamics for the Gravity-Gradient Satellite," Wright Laboratory Technical Report - to be published, 1992.

^ Percival, I . C . , F .R.S . , "Chaos in Hamiltonian Sys­tems", Proc. R. Soc. Loud. A 413, 131-144 (1987).

° Wolf, A., Swift, J .B . , Swinney, H.L. , and Vastano, J.A., "Determining Lyapunov Exponents from a Time Series," Physica 16D, pp. 285-317, 1985.

Moon, Francis C. , Chaotic V i b r a t i o n s : A n I n t r o ­duction for Applied Scientists and Engineers, John Wiley k Sons, Inc., New York, 1987.

12 Sweby, P.K., Yee, H.C. , and Griffiths, D.F. , "On Spurious Steady-State Solutions of Explicit Runge-Kutta Schemes," NASA T M 102819, April 1990.

1 * Yee, H.C., Sweby, P.K., and Griffiths, D.F. , "A Study of Spurious Asymptotic Numerical Solutions of Nonlinear Differential Equations by the Nonlin­ear Dynamics Approach," Lecture Notes in Physics #371, 12th International Conference on Numerical Methods in Fluid Dynamics, 1991.

11 Lorenz, Edward N., "Computational Chaos - A Prelude to Computational Instability", Physica D 35 (1989), 299- 317.

pcriapau

apoapsis

Figure 1 - Orbit Geometry

Figure 2 - Phase Plane for the Circular Orbit

. 7

X for Soeofiea re-caic o^.^uons L i m i t m o . . . k j e s o f j ; ; a n a » t o r t - e n o o i c M o t i o n

^ t c t i A n o e u o r a t i o n o n o K y r t w i q - C . r c u r a r O m i t U t x T K o n , C r o i o r O r t i i t

- 2 I ' ' ' ' • ' • ' ' ^ ' ' - 1 0 0 - 8 0 - 8 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0 ^ 0 0

»™ (A«») Figure 3 - Circular Orbit Case: Values of A for Periodic Figure 6 - Circular Orbit Case: Limiting Values of 6' Libration and Tumbling and 9 for Periodic Libration Solutions

Maximum Pitch Angle ror Henoaic Solutions -Fito»" PIIUUM%' Scetton I U 1 . 0 ,

L i D r o t i o n . C i r a j i a r O r o i t 4 . 0 0

Figure 4 - Circular Orbit Case: Maximum 9 for Periodic Figure 7 - False Poincare Plot at A=1.0, e=0.3 Libration Solutions

K NiOTtMrefOiMIs

Figure 5 - Circular Orbit Case: Maximum 9' for Periodic Figure 8 - Example of Lyapunov Exponent Dependence Libration Solutions upon V at A = l , e=0.3, and initial conditions (0.0, 0.0)

8

K = 1 0 9 ( 0 ) = a o

9 9 ( 0 ) 9 1 -

= 0.0

chaotic motion

r e g u l a r motion

- 1 . 0 O - i 1 < 1 " i ' I • I f I • ' " i o i M 0 . 1 0 ojo a a o a w O M O « O . T O a n O . M L O G

EcciMileHy

Figure 9a - Pitch Angle Bifurcation Diagram at A = l and initial conditions (0.0, 0.0)

BlturcaUon Otognm 2 . 5 0

2 . 0 0

1 . 5 0

1 . 0 0

dB OJSO

0 . 0 0

• 0 . 5 0

- 1 . 0 0

- 1 . 5 0

-2.00

l< = 1.0 9 ( 0 ) = 0.0

9 9 ( 0 ) 9 1 -

= 0.0

ni T t I I I r — — r — r - — i - t

0 . 0 0 0 . 1 0 0 . 2 0 o j o a n a s o a . 6 0 C T O O . W a . M i . o o Eceanttlclty

Figure 9b - Pitch Angle Rate Bifurcation Diagram at A'=l and initial conditions (0.0, 0.0)

1 . 0 ( h

K = 1,0 9 ( 0 ) = 0.0

0.0(7

-oso

9 9 ( 0 ) = 0.0

periodic

chaotic.

i

0 , 3 1 0 0 J 1 1 0 . 3 1 4 0 - 3 1 5 0 . 3 1 2 0 3 1 3 E c c w M r t c t t y

Figure 10a - Pitch Angle Bifurcation Diagram at A = l and initial conditions (0.0, 0.0)

0 3 1 0 0 3 1 1 0 3 1 2 0 3 1 3 0 3 1 4 0 3 1 5 E c o M M r i c t t y

Figure 10b - Pitch Angle Rate Bifurcation Diagram at A = l and initial conditions (0.0, 0.0)

0 . 1 O

0 . 0 3

9 ( 0 ) = 0.0

•"CI „ „

•0.0S

-o.xo 0 .

Figure 11 e=0.313

0 3 0 1

0 . 4 0

0 3 0

dB dv 03O

0 . 1 0

0 3 0

3 . 1 0

5 0 0 . 1 0 0 0 N u m i M r o l O r M s

First Lyapunov Exponent at A=1.0,

K = 1.0 9 ( 0 ) = 0 0

9 9 ( 0 ) 9 i<

0 0

3 . 1 0 3 3 5 0 3 0 B

0 . 0 5 0 . 1 0

Figure 12 - Poincare Plot at A=1.0, e=0.313, and initial conditions (0.0, 0.0)

\ 9

BifiratlonOtaanM 0.001

0.0*

0 . 0 O

3 . 0 4 -

periodic chaotic 3 3 0

0 . 3 1 1 4 0 3 1 1 5 0 3 1 1 6 0 3 1 1 7 0 3 1 1 8 0 3 1 1 9 0 . 3 1 2 0 0 . 3 1 2 1 E c c o i N i k J I y

Figure 13a - Pitch Angle Bifurcation Diagram at A = l and initial conditions (0.0, 0.0) O-SOl _ , „

0 . 4 O

0 3 0

4 0

dv 0 3 0

0 . 1 0

0 . 0 0

9(0) = 0,0

' " o ' s i U 0 . 3 1 1 5 0 3 1 1 6 0 3 1 1 7 0 3 1 1 8 0 3 1 1 9 0 . 3 1 2 0 0 . 3 1 2 1 E c c t n b l c t t y

Figure 13b - Pitch Angle Rate Bifurcation Diagram at / \=1 and initial conditions (0.0, 0.0)

0 . 0 4 0 0

3 . 0 2 0 0 -

3 - 0 4 0 0 -

3 - 0 0 0 0 :

K = 1 .0

9 ( 0 ) = 0 . 0

9 9 ( 0 ) 9>-

= 0 . 0

a 3 9 W O J O M 0 3 0 1 0 0 3 0 3 0 0 3 0 3 0 0 3 0 4 0 a 3 0 » 0 3 0 «

Figure 14a - Pitch Angle Bifurcation Diagram at A'=l and initial conditions (0.0, 0.0)

di/

0 3 0 0 0 0 3 0 1 0 0 3 0 2 0 0 3 0 3 0 0 3 0 4 0 0 3 0 6 0 O J O H E c c w M r t c M y

Figure 14b - Pitch Angle Rate Bifurcation Diagram at A'=l and initial conditions (0.0, 0.0)

0 . 0 9 0 )

0 . 0 2 0 0 3 3 0 0 . 0 4 0 E c c s n l r l c i t y

0 3 6 0

Figure 15a- Pitch Angle Bifurcation Diagram for Small Eccentricity at /<'=1 and initial conditions (0.0, 0.0)

0 3 1 0 )

0 3 0 9

Z 0.

3 3 0 5

3 3 1 0 . 0 2 0 0 . 0 3 0 0 . 0 4 0

E c c s n l r i c l l y 0 3 6 0

Figure 15b - Pitch Angle Rate Bifurcation Diagram for Small Eccentricity at K = l and (0.0, 0.0)

10

e

Figure 16a- Pomcare Plot at K = l , e - 0 . 1 e

Figure 17 - Poincare Plot at K = l , e=0.2

P o l n e m ' S M O O I I K > 1 . 0 , ta.1 1 . 5 0

3J0X

2 . 5 0

2 . 0 0

1 . 5 0

1 . 0 0

0 3 0

dB — O M dv

3 . 5 0

- 1 . 0 O

- 1 . 5 0

- 2 . 0 0

- 2 . 5 0

P o i n e a n ' S M i o n K > 1 3 , taOJ

- ^ - - ^

\

0 . 0 0

Figure 16b - Blow-up of tbe Poincare Plot at A'=l, e=0.1

Figure 18a- Poincare Plot at A = l , e=0.3

2 . 0 0

dB dv

1 . 0 0

OM

Figure 16c - Isolated Periodic Solution of tbe Poincare Plot at A = l , e=0.1

0 3 0

dB dv

0 . 0 0

•OJOTir

P o l n c a r a ' S M U o n I U 1 J ) , k O J

oxb e

0XI7X

Figure 18b - Blow-up of tbe PoincMe Plot at K e=0.3

11

Figure 19 - Constant K, Variable e Series Figure 22 - Variable K, Constant e Series Poincare Plot at A:=0.1, e=0.05 Poincare Plot at A:=0.1, e=0.2

Figure 20 - Constant K, Variable e Series Poincare Plot at A=0.1, e=0.1

P o l n c a r a ' S M I o n K i O . 1 , c s O J

Figure 21 - Constant K, Variable e Series Poincare Plot at K = 0 . \ e=0.3

Figure 23 - Variable K, Constant e Series Poincare Plot at A:=0.25, e=0.2

P o l n c a r a ' S a c t l o n K E . 7 5 . a s a 2

Figure 24 - Variable K, Constcint e Series Poincare Plot at A:=0.75, e=0.2

12

Choos Plot with Lyopunov Exponents Eccentricity vs Inertia Ratio

0 0 . 1 0 .2 0 . 3 0 . 4 0 . 5 0 .6 0 . 7 0 . 8 0 . 9

I nertia Ratio, K

Figure 25 - Chaos Diagram at initial conditions (0.0, 0.0) and N = 5 0

13