Algebraische Aspekte Quadratischer Dynamischer Systeme

The main object of this dissertation is the investigation of quadratic dynamical systems dx/dt = a + T x + q(x) (x Î Kn, a constant, T linear, q homogeneous and polynomial of degree 2) by algebraic methods. It is a well known fact that each polynomial homogeneous map of degree 2 defines a corresponding bilinear map which makes the underlying vector space Kn to a commutative nonassociative algebra. This basic idea goes back to L. Markus and many other authors used it. In the present work three different aspects of quadratic dynamical systems are treated.

In the chapter 'Orbital Symmetries' systems of the form dx/dt = T x + q(x) are investigated. If it is possible to describe the relationship between the linear vector field T and the quadratic vector field q with the help of an infinitesimal orbital symmetry, we show that the solution of the system can be determined by solving the two systems dx/dt = T x and dx/dt = q(x). Different criteria are developed when these ideas can be successfully applied and some examples are given. A broader concept is to look at common infinitesimal orbital symmetries of the linear and the quadratic vector field. In this case we can prove that we still get invariant sets of the quadratic system.

In the chapter about the Poincaré sphere the behaviour of quadratic systems dx/dt = T x + q(x) in the plane at infinity is described. Therefore we use a projection from the plane to the unit sphere which goes back to Poincaré. The points which lie on the equator of the sphere can be identified with the points at infinity. The fix points on the equator corresponds to the nilpotent and the idempotent elements of the above mentioned algebra. The behaviour of the system in an environment of such an fix point is clarified. We classify all quadratic systems by their behaviour at infinity. Some interesting results of boundness and unboundness of the solutions of a quadratic system are proved and all bounded systems are determined. All results are formulated in an invariant way so that for applying them it is not necessary to transform the system to a special form.

In the last chapter we look at homogeneous systems of the form dx/dt = q(x) + m(x)x where m shall be a linear form. S. Walcher has shown that for such systems a (in general wrong) parameterization of the orbits can be found by integration if the solution of the system dx/dt = q(x) is known. The idea of the chapter is the following: If the algebra which corresponds to q has enough nice properties then it should be possible to determine the above parameterization. It is shown that for power-associative algebras and nilpotent algebras this works quite well. In both cases we get a parameterization by elementary functions. In Jordan algebra we demonstrate how the structure of the algebra can help to solve the involved integration problems. In the end we investigate whether it is possible to get from the parameterization the exact solution of the system. It is shown that in general one could not expect more than the founded parameterization. This illustrates the usefullness of the demonstrated method because we get the whole phase portrait of a system which we are not able to solve exactly.