Perturbation Theory

Operator theory applied to perturbed systems

Operator theory is based on the idea of transforming a non-linear problem defined in a finite number of dimensions into a completely linear system but defined in an infinite dimensional space. This allows to solve and analyze non-linear problems using the tools we already know for linear systems. However, since it is impractical to work with infinite dimensional spaces, we have to focus on a subspace of it, and thus, instead of obtaining the exact solution of the system, we can only obtain an approximation, more accurate the better we can represent the non-linear system by a combination of a selected set of basis fuctions. One of the most famous exponents of operator theory for dynamical systems is the Koopman operator, which has been successfully applied by ART in the perturbed two body problem and the three body problem. The following figure shows the accuracy of the methodology when applied to a duffing oscillator with position q and conjugate momenta p.

 

Poincaré-Lindstedt method in the perturbed two body problem

The Poincaré-Lindstedt method is a pertrbation method that can be applied to periodic oscllators that are affected by small perturbations. The idea behind it is to generate a series of power expansions of the variables and the frequencies of the solution based on a small parameter. Then, once the system of differential equations is integrated, the perturbing frequency is selected in such a way that it cancels the effect of the secular variation of the variables involved in the problem, improving the accuracy of the analytical approximation. The Poincaré-Lindstedt method can be applied to the two body problem by transforming the motion of a satellite into a system of two oscillators. In an unperturbed system, these oscillators are independent and share the same frequency. However, when a perturbation is affecting the system, the oscillators become coupled and start presenting differences in their frequencies. An example of application of this methodology can be seen in the figure below for a near circular orbit subject to the J2 perturbation (position is defined as radial distance, latitude and inertial longitude respectively).