2.

Statistical Properties of Time-dependent SystemsDiego Fregolente Mendes De Oliveira, 2012, doktorska disertacija

**Opis:** In the dissertation I have dealt with time-dependent (nonautonomous) systems,
the conservative (Hamiltonian) as well as dissipative, and investigated their dynamical
and statistical properties. In conservative (Hamiltonian) time-dependent systems the
energy is not conserved, whilst the Liouville theorem about the conservation of the phase
space volume still applies. We are interested to know, whether the system can gain
energy, and whether this energy can grow unbounded, up to infinity, and we are
interested in the system's behaviour in the mean, as well as its statistical
properties. An example of such a system goes back to the 1940s, when Fermi proposed
the acceleration of cosmic rays (in the first place protons) upon the collisions
with moving magnetic domains in the interstellar medium of our Galaxy,
and in other galaxies. He then proposed a simple mechanical one-dimensional model,
the so-called Fermi-Ulam Model (FUM), where a point particle is moving between two
rigid walls, one being at rest and the other one oscillating. If the oscillation
is periodic and smooth, it turned out in a nontrivial way, which is, in the
modern era of understanding the chaotic dynamical systems, well understood,
namely that the unbounded increasing of the energy (the so-called Fermi
acceleration) is not possible, due to the barriers in form of invariant tori,
which partition the phase space into regions, between which the transitions are
not possible. The research has then been extended to other simple dyanamical
systems, which have complex dynamics. The first was so-called bouncer model,
in which a point particle bounces off the oscillating platform in a gravitational
field. In this simple system the Fermi acceleration is possible. Later
the research was directed towards two-dimensional billiard systems. It turned
out that the Fermi acceleration is possible in all such systems, which are at least
partially chaotic (of the mixed type), or even in a system that is integrable
as static, namely in case of the elliptic billiard. (The circle billiard
is an exception, because it is always integrable, as the angular momentum
is conserved even in time-dependent case.) The study of time-dependent systems
has developed strongly worldwide around the 1990s, in particular in 2000s,
and became one of the central topics in nonlinear dynamics. It turned out,
quite generally, but formal and implicit, in the sense of mathematical
existence theorems, that in nonautonomous Hamilton systems the energy can
grow unbounded, meaning that the system ``pumps" the energy from the environment
with which it interacts. There are many open questions: how does the energy
increase with time, in particular in the mean of some representative ensemble
of initial conditions (typically the phase space of two-dimensional time-dependent
billiards is four-dimensional.) It turned out that almost everywhere the power laws
apply, empirically, based on the numerical calculations, but with various
acceleration exponents. If the Fermi acceleration is not posssible, like e.g.
in the FUM, due to the invariant tori, then after a certain time of acceleration
stage the crossover into the regime of saturation takes place, whose
characteristics also follow the power laws. One of the central themes in
the dissertation is the study of these power laws, their critical exponents,
analytical relationships among them, using the scaling analysis
(Leonel, McClintock and Silva, Phys. Rev. Lett. 2004). Furthermore, the central
theme is the question, what happens, if, in a nonautonomous Hamilton system
which exhibits Fermi acceleration, we introduce dissipation, either
at the collisions with the walls (collisional dissipation) or
during the free motion (in-flight dissipation, due to the viscosity
of the fluid or the drag force etc.). Dissipation typically transforms
the periodic points into point attractors and chaotic components into
chaotic attractors. The Fermi acceleration is always suppressed.
We are interested in the phase portraits of

**Ključne besede:** nonlinear dynamics, dynamical systems, conservative and dissipative
systems, time-dependent systems, Fermi acceleration, billiards, kicked systems, chaos, chaotic and periodic attractors, bifurcations, boundary crisis

**Objavljeno:** 19.09.2012; **Ogledov:** 1618; **Prenosov:** 50

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