1. Fluctuating number of energy levels in mixed-type lemon billiardsČrt Lozej, Dragan Lukman, Marko Robnik, 2021, izvirni znanstveni članek Opis: In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B = 0.1953, 0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry-Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as - E, in agreement with the theoretical prediction by Steiner. Ključne besede: nonlinear dynamics, quantum chaos, mixed-type systems, energy level statistics, lemon billiards, billiards Objavljeno v DKUM: 13.10.2023; Ogledov: 472; Prenosov: 12 Celotno besedilo (1,40 MB) Gradivo ima več datotek! Več... |
2. On algebraic approach in quadratic systemsMatej Mencinger, 2011, pregledni znanstveni članek Opis: When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and (non)chaotic dynamics. In this paper we consider homogeneous quadratic systems via the so-called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system. Ključne besede: quadratic systems, nonlinear systems Objavljeno v DKUM: 14.06.2017; Ogledov: 1048; Prenosov: 388 Celotno besedilo (2,74 MB) Gradivo ima več datotek! Več... |
3. Evolution under alliance-specific cyclical invasion rates : lecture, presented at the Satellite Workshop at ECCS 07, Dresden, 4th-5th October (Evolution of of noise), 2007Matjaž Perc, 2007, prispevek na konferenci brez natisa Ključne besede: hrup, intenziteta hrupa, dilema zapornika, kaos, nelinearni dinamični sistemi, noise, spatiotemporal noise, intensity, prisoner's dilemma, chaos, nonlinear dynamic systems Objavljeno v DKUM: 10.07.2015; Ogledov: 1615; Prenosov: 43 Povezava na celotno besedilo |
4. Cooperation out of noise : lecture, presented at the Satellite Workshop at ECCS 07, Dresden, 4th-5th OctoberMatjaž Perc, 2007, prispevek na konferenci brez natisa Ključne besede: hrup, intenziteta hrupa, dilema zapornika, kaos, nelinearni dinamični sistemi, noise, spatiotemporal noise, intensity, prisoner's dilemma, chaos, nonlinear dynamic systems Objavljeno v DKUM: 10.07.2015; Ogledov: 1648; Prenosov: 27 Povezava na celotno besedilo |
5. Dynamical and statistical properties of time-dependent one-dimensional nonlinear Hamilton systemsDimitrios Andresas, 2015, doktorska disertacija Opis: We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the
adiabatic invariant (action) of the system for large time t→∞. We also show analytically that the mean energy and the variance increase as powers of A(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of $A(t)$), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time.
The main results are published in two papers:
Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and
Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014). Ključne besede: one-dimensional nonlinear Hamiltonian systems, adiabatic invariant, parametric kick, periodic driving, linear driving, energy distribution, WKB method, action Objavljeno v DKUM: 02.03.2015; Ogledov: 3236; Prenosov: 124 Celotno besedilo (11,07 MB) |
6. 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 v DKUM: 19.09.2012; Ogledov: 3204; Prenosov: 148 Celotno besedilo (16,09 MB) |
7. Nonlinear time series analysis of the human electrocardiogramMatjaž Perc, 2005, izvirni znanstveni članek Opis: We analyse the human electrocardiogram with simple nonlinear time series analysis methods that are appropriate for graduate as well as undergraduate courses. In particular, attention is devoted to the notions of determinism and stationarity in physiological data. We emphasize that methods of nonlinear time series analysis can be successfully applied only if the studied data set originates from a deterministic stationary system. After positively establishing the presence of determinism and stationarity in the studied electrocardiogram, we calculate the maximal Lyapunov exponent, thus providing interesting insights into the dynamics of the human heart. Moreover, to facilitate interest and enable the integration of nonlinear time series analysis methods into the curriculum at an early stage of the educational process, we also provide user-friendly programs for each implemented method. Ključne besede: dynamic systems, chaotic systems, nonlinear dynamics, electrocardiogram, human electrocardiogram, nonlinear analyses Objavljeno v DKUM: 07.06.2012; Ogledov: 2467; Prenosov: 114 Povezava na celotno besedilo |
8. Visualizing the attraction of strange attractorsMatjaž Perc, 2005, izvirni znanstveni članek Opis: We describe a simple new method that provides instructive insights into the dynamics of chaotic time-continuous systems that yield strange attractors as solutions in the phase space. In particular, we show that the norm of the vector field component that is orthogonal to the trajectory is an excellent quantity for visualizing the attraction of strange attractors, thus promoting the understanding of their formation and overall structure. Furthermore, based on the existence of zero orthogonal field strengths in planes that form low-dimensional strange attractors, we also provide an innovative explanation for the origin of chaotic behaviour. For instructive purposes, we first apply the method to a simple limit cycle attractor, and then analyse two paradigmatic mathematical models for classical time-continuous chaos. To facilitate the use of our method in graduate as well as undergraduate courses, we also provide user-friendly programs in which the presented theory is implemented. Ključne besede: dynamic systems, chaotic systems, nonlinear dynamics, attractors, strange attractors Objavljeno v DKUM: 07.06.2012; Ogledov: 2398; Prenosov: 99 Povezava na celotno besedilo |
9. Introducing nonlinear time series analysis in undergraduate coursesMatjaž Perc, 2006, strokovni članek Opis: This article is written for undergraduate students and teachers who would like to get familiar with basic nonlinear time series analysis methods. We present a step-by-step study of a simple example and provide user-friendly programs that allow an easy reproduction of presented results. In particular, we study an artificial time series generated by the Lorenz system. The mutual information and false nearest neighbour method are explained in detail, and used to obtain the best possible attractor reconstruction. Subsequently, the times series is tested for stationarity and determinism, which are both important properties that assure correct interpretation of invariant quantities that can be extracted from the data set. Finally, as the most prominent invariant quantity that allows distinguishing between regular and chaotic behaviour, we calculate the maximal Lyapunov exponent. By following the above steps, we are able to convincingly determine that the Lorenz system is chaotic directly from the generated time series, without the need to use the differential equations. Throughout the paper, emphasis on clear-cut guidance and a hands-on approach is given in order to make the reproduction of presented results possible also for undergraduates, and thus encourage them to get familiar with the presented theory. Ključne besede: nonlinear systems, nonlinear time series analyses, physics education Objavljeno v DKUM: 07.06.2012; Ogledov: 2004; Prenosov: 36 Povezava na celotno besedilo |
10. Thoughts out of noiseMatjaž Perc, 2006, izvirni znanstveni članek Opis: We study the effects of additive Gaussian noise on the behaviour of a simple spatially extended system, which is locally modelled by a nonlinear two-dimensional iterated map describing neuronal dynamics. In particular, we focus on the ability of noise to induce spatially ordered patterns, i. e. the so-called noise-induced pattern formation. For intermediate noise intensities, the spatially extended system exhibits ordered circular waves, thereby clearly manifesting the constructive role of random perturbations. The emergence of observed noise-induced patterns is explained with simple arguments that are obtained by analysing the typical spatial scale of patterns evoked by various diffusion coefficients. Since discrete-time systems are straightforward to implement and require modest computational capabilities, the present study describes one of the most fascinating and visually compelling examples of noise-induced self-organization in nonlinear systems in an accessible way for graduate or even advanced undergraduate students attending a nonlinear dynamics course. Ključne besede: dynamic systems, chaotic systems, nonlinear dynamics, nonlinear systems, noise, nonlinear analyses Objavljeno v DKUM: 07.06.2012; Ogledov: 1638; Prenosov: 40 Povezava na celotno besedilo |