Chaos in Complex Astrophysical Systems
Departamento de Fisica, Faculdade de Ciencias
da Universidade de Lisboa,
Campo Grande, Edificio C8, 1749-016 Lisboa, Portugal.
Area Departamental de Matematica, ESTIG/IPB,
Rua Afonso III, no. 1, 7800-050 Beja, Portugal.
The dynamics of some astrophysical systems, such as star clusters
(N = 102-106) or planetary systems (N = 10),
can be modeled
by a set of differential equations known as the N-body gravitational
problem. The impossibility of solving analytically the general problem
when N>2  requires the use of numerical integration when studying
The sensitivity of the trajectories of the stars to small changes
(perturbations) on the initial conditions was first reported by Miller .
The improved computational power now available allowed a sistematic study
of the typical time scale of this instability in terms of the characteristic
crossing time of the system .
In this work we present results obtained from numerical integration of
the N-body gravitational problem and the associated variational equations
of motion, using the NNEWTON package . We use the ``Lyapunov
Characteristic Indicator''  to estimate the time scale of the
instability associated with the exponential growth of perturbations
(variations) on the initial conditions.
Our preliminary results are in good agreement with those of  and
show a simple relation between the time scale of the instability, the number
of particles and the crossing time of the system.
Poincare, H. (1892): Les Methodes Nouvelles de la
Mecanique Celeste. Gauthier-Villars, Paris, Vol. (I).
Miller, R. H. (1964): Irreversibility in Small Stellar
Dynamics Systems. Astrophysical Journal, 140, 250.
Kandrup, H. E. (1994): Chaos, Regularity, and Noise in
Self-Gravitating Systems. An invited plenary talk at:
The Seventh Marcel Grossmann Meeting.
 Pereira, N. S. A. (2001): Master's Thesis. Faculty of
Sciences, University of Lisbon, Portugal.
Heggie, D. C. (1991): Chaos in the N-Body Problem of Stellar
Dynamics. Predictability, Stability, and Chaos in N-Body Dynamical
Systems, Edited by A. E. Roy, Plenum Press, New York.
Goodman, J., Heggie, D. C., Hut, P. (1993): On the Exponential
Instability of N-Body Systems. The Astrophysical Journal,