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

AIAA-92-4369
Chaos in the Pitch Equation of Motion for
the Gravity-Gradient Satellite
H a r r y Karasopoulos
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 45433-6553
D a v i d L . R i c h a r d s o n
D e p t . Aerospace E n g i n e e r i n g
U n i v e r s i t y of C i n c i n n a t i
C i n c i n n a t i , O H 45221
1992 A I A A / A A S Astrodynamics
Conference
A u g u s t 10-12, 1992
H i l t o n Head, S o u t h C a r o l i n a
For p e m M o n to copy or repuplsh. contact the American institute of Aeronautics and Astronautics
370 L'Enfatit Promenade. S.W.. Washington. O.C. 20024

- «
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 or-
bit about a central body are investigated. This planar
motion is shown to be either periodic, quasiperiodic, or
chaotic, depending upon the values of eccentricity, satel-
lite inertia ratio, and the initial conditions for pitch an-
gle and its derivative with respect to the true anomaly.
Bifurcation plots, Poincare maps, and Lyapunov expo-
nents are numerically calculated and presented. Chaos
diagrams, which are computed from Lyapunov expo-
nents and are dependent upon the satellite's initial con-
ditions, 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
Irx, 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, WL/FIMG, 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 pro-
found influence upon the motion of natural and artificial
satellites alike. A number of investigations have shown
that gravity-gradient torques combined with tidail fric-
tion 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 tum-
ble chaotically at one time.
Determination of the dynamics of an artificial
satellite due to gravity-gradient torques is a very impor-
tant step in the design process. There are many appli-
cations (communication, observation, etc) where point-
ing requirements make passive stabilization of pitch an-
gle attractive for artificial satellites in orbit about the
Earth. Pitch angle stability for a gravity-gradient satel-
lite has received a great deal of attention, with publi-
cations 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 max-
imum eccentricity for stable motion, the solution must
be periodic.
1

In this paper, modern nonlinear analysis tech-
niques 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 compar-
ison purposes.
Equation 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 mo-
tion is^
(H-ecosi/)r-2esini/(l-|-6l') + 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 (Fig-
ure 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 prop-
erties of a dumbbell satellite. In this analysis it is as-
sumed 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 lo-
cal 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.
Circnlar 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 re-
viewed 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 iden-
tical 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 to-
wards 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 sys-
tem, the values of the two frequencies (pitch and true
anomaly) determine the nature of the motion. An ir-
rational 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 advanta-
geous 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 tum-
bling 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 ap-
propriate 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 mo-
tion. For example, satellite tumbling with a periodic
motion component of period | may occur for almost
any value of K , whereas period | libration can only ex-
ist for satellites having inertia ratios between about 0.75
and 1.0.
Figure 4 presents the maximum pitch angle at-
tained for various periodic solutions for the circular or-
bit 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 an-
gle. 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 spec-
ified limits increases with decreasing inertia ratio.
The maximum value of 9' attained for various peri-
odic solutions for the circular orbit libration case is given
in Figure 5. Note that for any periodic libration solu-
tion, 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.
Eccentric Orbit Case
Chaos is am extreme sensitivity to initial con-
ditions (or very small chamges in system pairameters)
which leads to exponential divergence. In the words of
I.C.Percival,^
"Chaotic motion appears in dynamicad sys-
tems when local exponential divergence of tra-
jectories is au:compamied by global confinement
in the phase spau:e. The divergence produces
a locad stretching in the phase space, but, be-
cause of the confinement, this stretching cam-
not occur without folding. Repeated folding
and refolding produces very complicated be-
havior 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 conser-
vative 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. Reg-
ular and chaotic motion occur for different initial con-
ditions and values of the system parameters, e and K .
Both types of motion may occur at a very close prox-
imity 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 ap-
plication 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 facil-
itate 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 ac-
complished 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 ef-
fects of parameter variations on a dynamical system is
the bifurcation diagram. To make a bifurcation dia-
gram, 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 sys-
tem 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 dis-
cern differences between quasiperiodic and chaotic mo-
tion. 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 coef-
ficients 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 re-
quired 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 pre-
sented 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 con-
tinuous differential equation, even for integration incre-
ments below the linearized stability limit of the inte-
gration 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 ob-
vious: 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 sur-
face of section displays chaotic motion instead of quasi-
periodic tumbling (compare with Figure 18a), demon-
strating the importance of integration accuracy.
Lyapunov Exponents
Lyapunov exponents measure the exponential di-
vergence of two trajectories with nearly identical ini-
tial 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 hav-
ing semimajor axis, 6 • Li„, and semiminor axis, 6 • Lin-
L d Li are called Lyapunov numbers and the Lya-
punov exponents are the natural logarithms of these
numbers. In general, a Lyapunov exponent greater than
zero indicates the trajectory is chaotic; a Lyapunov ex-
ponent 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 con-
traction 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" Lya-
punov 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, Lya-
punov exponents were calculated using a modified ver-
sion 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 pa-
rameters cind thus indicate the occurrence and relative
magnitude of chriotic motion for specified initial states.
In this study chcios diagrauris were numerically calcu-
lated for the gravity-gradient satellite for 0 < e < 1,
0 < < 1, and a 400x500 grid. Lyapunov exponent val-
ues 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 dia-
grams 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 magni-
tudes seemed reasonably robust. Very coarsely gridded
but highly accurate runs indicated that the border be-
tween 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 in-
ertia 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 compar-
ison. 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 in-
dicates 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 quasiperi-
odic motion dominates at eccentricities between these
values. Computation of the first Lyapunov exponent
for the periodic trajectory at 6=0.313 (Figure 11) ver-
ifies 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 or-
bits 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 dis-
appear 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 mag-
nifications 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 bi-
furcations. 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 quasiperi-
odic 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 nu-
merous trajectories can also provide insight into the pla-
nar 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 ec-
centric 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 libra-
tion 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 exami-
nation 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 por-
tion 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 discov-
ered at A=1.0 do not necessarily hold for other values of
satellite inertia ratio. Figures 19-21 are surfaces of sec-
tion for A^O.l and increasing eccentricity values. Al-
though the chaotic phase space region seems to increase
with increasing eccentricity, what happens to the libra-
tion 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 param-
eter, satellite inertia ratio, is examined. A sequence of
Poincare maps is presented in Figures 22-24 with vary-
ing 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 mo-
tion of a gravity-gradient satellite. However, discerning
global trends from local trends can be difficult. To alle-
viate 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 copies o f t h i s ps4>er 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 in-
crease 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 con-
structed 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 ini-
tial 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. Bi-
furcation diagrams demonstrate that the route to chaos
for this system is probably not through period doubling
bifurcations. Further work is needed to study the tran-
sition from regular to chaotic motion for this system and
to make the chsuDs diagram construction more efficient.
Bifurcation diagrams, Poincare maps, and chaos dia-
grams may be constructed and utilized as an engineering
tool for satellite and orbit design to avoid chaotic mo-
tion. 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 Di-
vision, Wright Laboratory for his valuable computer
graphics assistance.
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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 Sys-
tems", 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 Nonlin-
ear 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.rcurar O m i t U t x T K o n , C r o i o r Ortiit
- 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 ,
LiDrotion. Cirajiar Oroit 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
99(0)
91-
= 0.0
chaotic motion
r e g u l a r
motion
-1.0O-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
99(0)
91-
= 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
99(0)
= 0.0
periodic
chaotic.
i
0 , 3 1 0 0 J 1 1 0 . 3 1 4 0 - 3 1 50 . 3 1 2 0 3 1 3
E c c w M r t c t t y
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
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
99(0)
9i<
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
O-SOl _ , „
0.4O
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
/=1 and initial conditions (0.0, 0.0)
0.0400
3 . 0 2 0 0 -
3-0400-
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
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
EccwMrtcMy
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

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

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