1. The architecture of a school system according to the theory of dynamical systemsBoris Aberšek, Janez Bregant, 2012, original scientific article Abstract: From the experience of recent years it seems clear that the existing educational system, as a whole, is perceived as an ailing system that fails to meet the needs of a major portion of the society it serves. Every aspect of the educational process must be studied and reconsidered in the light of new and different societal expectations. In the last decades we started with one school reform after the other, but it looks now that these changes had no positive impact whatsoever. Even worse, the school system was damaged afterwards. In this paper we will explore some possibilities how to improve it without facing these consequences. The main idea is, as a result of our latest research, that mathematical methods for describing and simulating could be very useful for predicting and simulating school systems without using a long term and complex empirical experimentation. The paper introduces the correlation between the dynamical mathematical calculus and the educational school system and elaborates to some extent the basic idea of using the former to predict the latter. Keywords: architecture, mathematical modelling, school systems, dynamical systems Published in DKUM: 19.12.2017; Views: 1382; Downloads: 133 Full text (802,07 KB) This document has many files! More... |
2. Statistical Properties of Time-dependent SystemsDiego Fregolente Mendes De Oliveira, 2012, doctoral dissertation Abstract: 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 Keywords: nonlinear dynamics, dynamical systems, conservative and dissipative
systems, time-dependent systems, Fermi acceleration, billiards, kicked systems, chaos, chaotic and periodic attractors, bifurcations, boundary crisis Published in DKUM: 19.09.2012; Views: 3259; Downloads: 159 Full text (16,09 MB) |
3. Fluctuating excitability : a mechanism for self-sustained information flow in excitable arraysMatjaž Perc, 2007, original scientific article Abstract: We show that the fluctuating excitability of FitzHugh-Nagumo neurons, constituting a diffusively coupled excitable array, can induce phase slips that lead to a symmetry break yielding a preferred spreading direction of excitatory events, thus enabling persistent self-sustained and self-organized information flow in a periodic array long after a localized stimulus perturbation has sized. Possible oscillation frequencies of the information-carrying signal are expressed analytically, and necessary conditions for the phenomenon are derived. Our results suggest that cellular diversity in neural tissue is crucial for maintaining self-sustained and organized activity in the brain even in the absence of immediate stimuli, thus facilitating continuous evolution of its mechanisms for information retrieval and storage. Keywords: physics, complex systems, dynamical systems, flexibility, chaos, chaotic systems, oscillations, perturbation Published in DKUM: 07.06.2012; Views: 2283; Downloads: 97 Link to full text |
4. Spatial coherence resonance in neuronal media with discrete local dynamicsMatjaž Perc, 2006, original scientific article Abstract: We study effects of spatiotemporal additive noise on the spatial dynamics of excitable neuronal media that is locally modelled by a two-dimensional map. We focus on the ability of noise to enhance a particular spatial frequency of the media in a resonant manner. We show that there exists an optimal noise intensity for which the inherent spatial periodicity of the media is resonantly pronounced, thus marking the existence of spatial coherence resonance in the studied system. Additionally, results are discussed in view of their possible biological importance. Keywords: physics, complex systems, dynamical systems, noise, spatial dynamics, chaos, chaotic systems, chaos control, resonance Published in DKUM: 07.06.2012; Views: 2462; Downloads: 106 Link to full text |
5. Effects of small-world connectivity on noise-induced temporal and spatial order in neural mediaMatjaž Perc, 2006, original scientific article Abstract: We present an overview of possible effects of small-world connectivity on noise-induced temporal and spatial order in a two-dimensional network of excitable neural media with FitzHugh-Nagumo local dynamics. Small-world networks are characterized by a given fraction of so-called long-range couplings or shortcut links that connect distant units of the system, while all other units are coupled in a diffusive-like manner. Interestingly, already a small fraction of these long-range couplings can have wide-ranging effects on the temporal as well as spatial noise-induced dynamics of the system. Here we present two main effects. First, we show that the temporal order, characterized by the autocorrelation of a firing-rate function, can be greatly enhanced by the introduction of small-world connectivity, whereby the effect increases with the increasing fraction of introduced shortcut links. Second, we show that the introduction of long-range couplings induces disorderof otherwise ordered, spiral-wave-like, noise-induced patterns that can be observed by exclusive diffusive connectivity of spatial units. Thereby, already a small fraction of shortcut links is sufficient to destroy coherent pattern formation in the media. Although the two results seem contradictive, we provide an explanation considering the inherent scale-free nature of small-world networks, which on one hand, facilitates signal transduction and thus temporal order in the system, whilst on the other hand, disrupts the internal spatial scale of the media thereby hindering the existence of coherent wave-like patterns. Additionally, the importance of spatially versus temporally ordered neural network functioning is discussed. Keywords: physics, complex systems, dynamical systems, noise, chaos, chaotic systems, chaos control, resonance Published in DKUM: 07.06.2012; Views: 2006; Downloads: 86 Link to full text |
6. Stochastic resonance in soft matter systems : combined effects of static and dynamic disorderMatjaž Perc, Marko Gosak, Samo Kralj, 2008, original scientific article Abstract: We study the impact of static and dynamic disorder on the phenomenon of stochastic resonance (SR) in a representative soft matter system. Due to their extreme susceptibility to weak perturbations, soft matter systems appear to be excellent candidates for the observation of SR. Indeed, we derive generic SR equations from a polymer-stabilized ferroelectric liquid crystal (LC) cell, which is a typical soft matter representative constituting one of the basic components in several electro-optic applications. We generalize these equations further in order to study an even broader class of qualitatively different systems, especially disclosing the influence of different types of static disorder and interaction ranges amongst LC molecules on the SR response. We determine the required conditions for the observation of SR in the examined system, and moreover, reveal that a random field type static disorder yields qualitatively different responses with respect to random dilution, random bond and spin glass universality classes. In particular, while the latter three decrease the level of dynamic disorder (Gaussian noise) warranting the optimal response, the former evokes exactly the opposite effect, hence increasing the optimal noise level that is needed to resonantly fine-tune the system's response in accordance with the weak deterministic electric field. These observations are shown to be independent of the system size and range of interactions, thus implying their general validity and potentially wide applicability also within other similar settings. We argue that soft matter systems might be particularly adequate as a base for different SR-based sensitive detectors and thus potent candidates for additional theoretical as well as experimental research in the presently outlined direction. Keywords: dynamic systems, stochastic processes, stochastic resonance, nonlinear dynamical systems, soft-matter systems, static disorder, dynamic disorder Published in DKUM: 07.06.2012; Views: 2007; Downloads: 85 Link to full text |
7. Extended Lindstedt-Poincare method for non-stationary resonances of dynamical systems with cubic nonlinearitRudi Pušenjak, 2008, original scientific article Abstract: This paper presents the extended Lindstedt-Poincare (EL-P) method, which applies multiple time variables to treat non-stationary oscillations arising in dynamical systems with cubic nonlinearities due to the slowly varied excitation parameters. The method is applied extensively in research of non-stationary vibrations of clamped-hinged beams. Recognizing the aperiodic nature of non-stationary oscillations, the new formulation is presented by adding an additional, slow time scale beside time scales of the nonlinear system, which generally correspond to the incommensurate nonlinear frequencies of the response. Using this concept, a generalized approach of the study to the passage through fundamental, superharmonic and subharmonic resonances is presented in the paper. Effects of slowly varying excitation frequency and slowly varying excitation amplitude on the non-stationary oscillations are studied with the computation of deviations from the stationary response. Although the method is formulated for N-dof dynamical systems having weak cubic nonlinearities, it is applied for non-stationary vibrations, where two-mode shape approximation of damped and undamped clamped-hinged beam, respectively, is used and the simultaneous appearance of internal resonance is taken into account. Stability analysis of stationary solutions is performed and comparisons of stationary resonance curves by results obtained with the incremental harmonic balance (IHB) method show good agreement. The passage through the fundamental resonance of damped and undamped clamped-hinged beam, respectively, is investigated in detail. Keywords: dynamical systems with cubic nonlinearities, nonlinear oscillations, nonstationary nonlinear oscillations, time scales, excitation frequency, resonance, Lindstedt-Poincare method Published in DKUM: 01.06.2012; Views: 2114; Downloads: 36 Link to full text |
8. Encyclopedia of complexity and systems sciencedictionary, encyclopaedia, lexicon, manual, atlas, map Abstract: Encyclopedia of Complexity and Systems Science provides an authoritative single source for understanding and applying the concepts of complexity theory together with the tools and measures for analyzing complex systems in all fields of science and engineering. The science and tools of complexity and systems science include theories of self-organization, complex systems, synergetics, dynamical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural networks, cellular automata, adaptive systems, and genetic algorithms. Examples of near-term problems and major unknowns that can be approached through complexity and systems science include: The structure, history and future of the universe; the biological basis of consciousness; the integration of genomics, proteomics and bioinformatics as systems biology; human longevity limits; the limits of computing; sustainability of life on earth; predictability, dynamics and extent of earthquakes, hurricanes, tsunamis, and other natural disasters; the dynamics of turbulent flows; lasers or fluids in physics, microprocessor design; macromolecular assembly in chemistry and biophysics; brain functions in cognitive neuroscience; climate change; ecosystem management; traffic management; and business cycles. All these seemingly quite different kinds of structure formation have a number of important features and underlying structures in common. These deep structural similarities can be exploited to transfer analytical methods and understanding from one field to another. This unique work will extend the influence of complexity and system science to a much wider audience than has been possible to date. Keywords: cellular automata, complex networks, computational nanoscience, ecological complexity, ergodic theory, fractals, game theory, granular computing, graph theory, intelligent systems, perturbation theory, quantum information science, system dynamics, traffic management, chaos, climate modelling, complex systems, dynamical sistems, fuzzy theory systems, nonlinear systems, soft computing, stochastic processes, synergetics, self-organization, systems biology, systems science Published in DKUM: 01.06.2012; Views: 2813; Downloads: 126 Link to full text |
9. Stochastic resonance on excitable small-world networks via a pacemakerMatjaž Perc, 2007, original scientific article Abstract: We show that the correlation between the frequency of subthreshold pacemaker activity and the response of an excitable array is resonantly dependent on the intensity of additive spatiotemporal noise. Thereby, the effect of the underlying network, defining the interactions among excitable units, largely depends on the coupling strength. Only for intermediate coupling strengths is the small world property able to enhance the stochastic resonance, whereas for smaller and larger couplings the impact of the transition from diffusive to random networks is less profound. Thus, the optimal interplay between a localized source of weak rhythmic activity and the response of the whole array demands a delicate balance between the strength of excitation transfer and the effectiveness of the network structure to support it. Keywords: stochastic resonance, small-world networks, cardiology, neurophysiology, nonlinear dynamical systems, spatiotemporal phenomena Published in DKUM: 31.05.2012; Views: 2198; Downloads: 150 Link to full text |