### Algebraische Aspekte Quadratischer Dynamischer Systeme

### Helmut Figalist

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ISBN 978-3-89722-440-7

165 pages, year of publication: 2000

price: 40.50 €

The main object of this dissertation is the investigation of
quadratic dynamical systems *d**x*/*dt* = *a*
+ *T* *x* + *q*(*x*)
(*x* Î
*K*^{n}, *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 *K*^{n}
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 *d**x*/*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 *d**x*/*dt*
= *T* *x* and *d**x*/*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 *d**x*/*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
*d**x*/*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 *d**x*/*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.