George Contopoulos
Publications:
Tzemos A. C., Contopoulos G.
Integrals of Motion in Timeperiodic Hamiltonian Systems: The Case of the Mathieu Equation
2021, vol. 26, no. 1, pp. 89104
Abstract
We present an algorithm for constructing analytically approximate integrals of motion in
simple timeperiodic Hamiltonians of the form $H=H_0+
\varepsilon H_i$, where $\varepsilon$ is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different values of $\varepsilon$. We find the values of $\varepsilon_{crit}$ beyond which the orbits escape to infinity and construct integrals which are expressed as series in the perturbation parameter $\varepsilon$ and converge up to $\varepsilon_{crit}$. In the absence of resonances the invariant curves are concentric ellipses which are approximated very well by our integrals. Finally, we construct an integral of motion which describes the hyperbolic stroboscopic invariant curve of a resonant case.

Contopoulos G., Tzemos A. C.
Chaos in Bohmian Quantum Mechanics: A Short Review
2020, vol. 25, no. 5, pp. 476495
Abstract
This is a short review of the theory of chaos in Bohmian quantum mechanics
based on our series of works in this field. Our first result is the development of a generic
theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system
(in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian
trajectories and study in detail (both analytically and numerically) the different kinds of
Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect
of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and
ergodicity in qubit systems which are of great theoretical and practical interest. We find that
the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points
after a long time regardless of their initial conditions. In the case of strong entanglement most
trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend
to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the
distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary
initial configuration of particles will not tend, in general, to Born’s rule unless it is initially
satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely,
whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution
of Bohmian particles.
