On Algebraic Approach in Quadratic Systems

When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and (non)chaotic dynamics. In this paper we consider homogeneous quadratic systems via the so-called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system

Finiteness properties of preperiodic points and invariant sets for polynomial mappings

This thesis discusses about algebraic dynamic, that is, the application of algebraic number theory to dynamical systems, expecially to polynomial ones. In the first part we deal with periodic and preperiodic orbits. After showing the most classical results about finiteness properties, we discuss the problem of the maximum length of a finite orbit and give a complete classification of periodic and preperiodic orbits in case of a quadratic number field. In the second part we study fully invariant sets for polynomial mappings and state some particular finiteness properties, called "Narkiewicz's properties", focusing particularly on the relationships among them

Order reduction approaches for the algebraic Riccati equation and the LQR problem

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices

On the algebraic structure of rational discrete dynamical systems

We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties. We illustrate the phenomenon explicitly with examples covering a wide range of models

Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

The $D$-dimensional Smorodinsky-Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing $D$ auxiliary continuous variables and by reducing a 2D-dimensional harmonic oscillator Hamiltonian. The su(2D) symmetry and ${\rm w}(2D) \oplus_s {\rm sp}(4D,{\mathbb R})$ dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky-Winternitz system. The action of generators on wavefunctions is given in explicit form for D=2