Vasileios Basios

Pattern formation and fluctuation-induced transitions in protein crystallization

A kinetic model of protein crystallization accounting for the nucleation stage, the growth and competition of solid particles and the formation of macroscopic patterns is developed. Different versions are considered corresponding successively, to a continuous one-dimensional crystallization reactor, a coarse grained two-box model and a model describing the evolution of the space averaged values of fluid and solid material. The analysis brings out the high multiplicity of the patterns. It provides information on their stability as well as on the kinetics of transitions between different states under the influence of the fluctuations.

Probabilistic control of chaos: The β-adic Renyi map under control

We propose a new, probabilistic, approach for the control of chaotic systems. This approach is illustrated by a specific method, for the control of any periodic orbit of the simplest piecewise linear chaotic map, namely the β-adic Renyi map. As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the dynamics. Our method consists in the probabilistic coupling of the original system with a controlling system. The chosen periodic orbit of the original system is a global attractor for the probability densities. The generalized spectral decomposition of the associated Frobenius–Perron operator provides a spectral condition of controllability for chaotic dynamical systems.

Stabilization of discrete breathers using continuous feedback control

Localized oscillations in 1- and 2-dimensional nonlinear lattices are by now recognized as a widely occurring phenomenon with applications to many problems of physical and biological interest. When the spatial distribution of the amplitudes of these oscillations possesses a single extremum they are called discrete breathers and are known to be stable for a sizable range of the inter-particle coupling parameter α⩾0. However, when the amplitudes possess more than one (local) extremum they are called multibreathers and are generally unstable for all α>0. In this Letter, we demonstrate that it is possible to stabilize breathers and multibreathers of a 1-dimensional chain with quartic on-site potential, using a continuous feedback control (CFC) method, originally proposed by Pyragas. CFC is called conservative if it preserves the Hamiltonian nature of the system and dissipative if it introduces loss terms proportional to the velocity of the particles. As is well-known from low-dimensional examples CFC works by inducing pitchfork, period-doubling or transcritical bifurcations to the unstable periodic orbits under study. Here, we demonstrate, by computing the eigenvalues of the corresponding monodromy matrix and following the phase space oscillations of the particles that CFC stabilizes our breathers and multibreathers also via low-dimensional bifurcations.

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈106) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.

Kinetics of intermediate-mediated self-assembly in nanosized materials: A generic model

We propose in this paper a generic model of a nonstandard aggregation mechanism for self-assembly processes of a class of materials involving the mediation of intermediates consisting of a polydisperse population of nanosized particles. The model accounts for a long induction period in the process. The proposed mechanism also gives insight on future experiments aiming at a more comprehensive picture of the role of self-organization in self-assembly processes.

A method for approximating one-dimensional functions

Abstract A nontraditional approach is presented to the approximation of a one-dimensional function defined on a discrete set of points. The method is based on the application of an artificial feedforward neural network, which realizes expansion of the function considered in orthogonal Chebyshev polynomials and which calculates the expansion coefficients during the network training.

Inverse Bayesian inference as a key of consciousness featuring a macroscopic quantum logical structure

To overcome the dualism between mind and matter and to implement consciousness in science, a physical entity has to be embedded with a measurement process. Although quantum mechanics have been regarded as a candidate for implementing consciousness, nature at its macroscopic level is inconsistent with quantum mechanics. We propose a measurement-oriented inference system comprising Bayesian and inverse Bayesian inferences. While Bayesian inference contracts probability space, the newly defined inverse one relaxes the space. These two inferences allow an agent to make a decision corresponding to an immediate change in their environment. They generate a particular pattern of joint probability for data and hypotheses, comprising multiple diagonal and noisy matrices. This is expressed as a nondistributive orthomodular lattice equivalent to quantum logic. We also show that an orthomodular lattice can reveal information generated by inverse syllogism as well as the solutions to the frame and symbol-grounding problems. Our model is the first to connect macroscopic cognitive processes with the mathematical structure of quantum mechanics with no additional assumptions.

Statistical properties of time-reversible triangular maps of the square

Time reversal symmetric triangular maps of the unit square are introduced with the property that the time evolution of one of their two variables is determined by a piecewise expanding map of the unit interval. We study their statistical properties and establish the conditions under which their equilibrium measures have a product structure, i.e. factorises in a symmetric form. When these conditions are not verified, the equilibrium measure does not have a product form and therefore provides additional information on the statistical properties of theses maps. This is the case of anti-symmetric cusp maps, which have an intermittent fixed point and yet have uniform invariant measures on the unit interval. We construct the invariant density of the corresponding two-dimensional triangular map and prove that it exhibits a singularity at the intermittent fixed point.

Probabilistic control of Chaos: Chaotic maps under control

Recently, we have proposed a new probabilistic method for the control of chaotic systems [1]. In this paper, we apply our method to characteristic cases of chaotic maps (one- and two-dimensional examples). As these chaotic maps are structurally stable, they cannot be controlled using conventional control methods without significant change of the dynamics. Our method consists in the probabilistic coupling of the original system with a controlling system. This coupling can be understood as a feedback control of probabilistic nature. The chosen periodic orbit of the original system is a global attractor for the probability densities. The generalized spectral decomposition of the associated Frobenius-Perron operator provides a spectral condition of controllability for chaotic dynamical systems.

SYMBOLIC DYNAMICS GENERATED BY A COMBINATION OF GRAPHS

In this paper we investigate the growth rate of the number of all possible paths in graphs with respect to their length in an exact analytical way. Apart from the typical rates of growth, i.e. exponential or polynomial, we identify conditions for a stretched exponential type of growth. This is made possible by combining two or more graphs over the same alphabet, in order to obtain a discrete dynamical system generated by a triangular map, which can also be interpreted as a discrete nonautonomous system. Since the vertices and the edges of a graph are usually used to depict the states and transitions between states of a discrete dynamical system, the combination of two (or more) graphs can be interpreted as the driving, or perturbation, of one system by another.