With this work we have examined what dynamical systems approach is available for the scientist to tackle the formidable problem of describing complex systems and whether conceptual works of mathematics applied to realistic dynamical systems can be practicable and useful. The underlying ideas are grounded in one subject of dynamical systems theory, the periodic orbit theory, which offers formulae to compute averages of chaotic systems with the best accuracy available. On the conceptual level, the theory manages to find a common denominator between such apparently different concepts, as periodic orbits, averages in chaotic systems, spectrum of a Green's operator (asymptotic approximation is involved in the latter).

The price to pay is severe: one has to have means to compute periodic orbits in a given dynamical system.

In this work we develop methods to partition the phase space of a complex two degree of freedom Hamiltonian system, called the planar crossed-fields in terms of periodic orbits. I also have studied extensions to three degree of freedom setting, and discussed relevance of high-dimensional complex saddles.

Finally, we have developed methodology to compute unstable invariant tori in three degree of freedom setting and applied these methods to explain trapping of trajectories in the planar carbonyl-sulfide (OCS) molecule.