Least squares parameter estimation in chaotic differential equations

A recent least squares algorithm, which is designed to adapt implicit models to given sets of data, especially models given by differential equations or dynamical systems, is reviewed and used to fit the Hénon-Heiles differential equations to chaotic data sets.This numerical approach for estimating parameters in differential equation models, called theboundary value problem approach, is based on discretizing the differential equations like a boundary value problem,e.g. by a multiple shooting or collocation method, and solving the resulting constrained least squares problem with a structure exploiting generalized Gauss-Newton-Method (Bock, 1981).Dynamical systems like the Hénon-Heiles system which can have initial values and parameters that lead to positive Lyapunov exponents or phase space filling Poincaré maps give rise to chaotic time series. Various scenarios representing ideal and noisy data generated from the Hénon-Heiles system in the chaotic region are analyzedw.r.t. initial conditions, parameters and Lyapunov exponents. The original initial conditions and parameters are recovered with a given accuracy. The Lyapunov spectrum is then computed directly from the identified differential equations and compared to the spectrum of the true dynamics.presently at IWR, Universität Heidelberg, Im Neuenheimer Feld 368, D-6900 Heidelberg, Germany     

Bounded orbits in the elliptic restricted three-body problem

Special solutions of the planar rectilinear elliptic restricted 3-body problem are investigated for the limiting case e=1. Numerical integration is performed for primaries of equal masses. Starting values which define circular orbit solutions lead to bounded solutions if the initial radius a₀ is larger than 3.74 in units of the primaries' semimajor axis a. A comparison with the Eulerian two-fixedcentre problem is presented in order to understand qualitatively the characteristic features of bounded orbits and the transition to escape orbits.     

Trajectories and envelopes in the repulsive two-fixed-centre problem and in the restricted three-body problem

The envelope of iso-energetic trajectories in the (repulsive) two-fixed-centre problem is derived. Our analytical calculations finally lead to a transcendental equation, only containing elliptic integrals and the Weierstraßp function, from which the envelope is constructed. The results may serve as a simple model for the boundary layer between two colliding supersonic stellar wind flows in binary systems, in which at least one of the components has a strong radiation field.Beyond this, the effect of non-inertial forces (centrifugal and Coriolis force) due to the binary's orbital motion has been estimated by a numerical analysis within the scope of the (repulsive) restricted three-body problem.All calculations have been performed for a hot model (Wolf-Rayet/O-star) binary system with a set of parameters which might be appropriate for HD 152270. The envelope may be well approximated by a hyperboloid. The non-inertial forces slightly turn the envelope against the line connecting both stars.     

Area-preserving Poincaré mappings of the unit disk

The best way to investigate the long-time behaviour of dynamical systems is to introduce an appropriate Poincaré mapping P and study its iterates.Two cases of physical interest arise: Conservative and dissipative systems. While the latter has been considered by a great many authors, much less is known for the first one (according to Liouville's theorem, here the mapping leaves a certain measure in phase space invariant). In this paper, we concentrate our attention on compact phase spaces (or, rather, surfaces of section). This assumption is mathematically useful and physically reasonable.We consider the simplest possible (2-dimensional) systems whehre the phase space is the compact unit disk D in ². A family of simple area-preserving mappings from D onto itselves will be given and discussed in detail.It is shown that general characteristics of the dynamics are quite similar to those of e.g. the Hénon-Heiles system, while other features, as the structure of invariant curves, are different.