Trajectory Theory

Abstract

The starting point for trajectory theory of the dynamical systems is a natural question which goes back to H. Poincaré and is concerned with the classical (smooth) dynamical systems. Consider two topological dynamical systems, i.e. 1-parameter groups of continuous transformations of a compact space. We call them topologically (orbitally) equivalent if there is a homeomorphism of the phase space intertwining the orbits (the trajectories) of these systems and preserving the orientation (the order of the points) on the orbits. Such a rough notion of equivalence is useful for the study of phase portraits of the dynamical systems, i.e. the structure of the partitions of phase spaces into separate trajectories. Such properties as the existence (or non-existence) of the periodic trajectories, of invariant submanifolds and so on, turn out to be stable under the above equivalence. This notion is also very useful for the investigation of the so-called rough properties of the dynamical systems (cf [Ar2]).