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1.
Turing patterns in simplicial complexes
Shupeng Gao, Lili Chang, Matjaž Perc, Zhen Wang, 2023, izvirni znanstveni članek

Opis: The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.
Ključne besede: Turing pattern, higher-order network, nonlinear dynamics, pattern formation
Objavljeno v DKUM: 31.05.2024; Ogledov: 92; Prenosov: 0
.pdf Celotno besedilo (1,57 MB)
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2.
Double stochastic resonance in neuronal dynamics due to astrocytes
Tugba Palabas, Joaquín J. Torres, Matjaž Perc, Muhammet Uzuntarla, 2023, izvirni znanstveni članek

Opis: A continuously growing body of evidence indicates that astrocytes, which is the most abundant sub-type of glial cells in the nervous system, not only provide structural and metabolic support to neurons, but also they are essential sentinels and dynamic modulators of neuronal and synaptic functions. However, the potential constructive role of astrocytes in information processing at the neuronal and synaptic level, and especially also in the presence of different noise sources in the neural system, is yet unclear. With this in mind, we here study the phenomenon of stochastic resonance - the enhanced detection of weak signals in the presence of noise - in neuronal dynamics by means of a mathematical model that includes interactions between the neuron and the astrocyte. We show that astrocytes may evoke a second peak in the neuronal detection of weak signals in dependence on the noise intensity, which is the hallmark of double stochastic resonance. We explore in detail the mechanisms underlying this discovery, in particular focusing on the determinants of astrocytic function and their role in the emergence of the second stochastic resonance peak. Our research thus provides fundamental insights into the possible roles of astrocytes in inherently noisy neuronal information processing.
Ključne besede: stochastic resonance, astrocyte, dressed neuron, nonlinear dynamics
Objavljeno v DKUM: 30.05.2024; Ogledov: 102; Prenosov: 4
.pdf Celotno besedilo (1,70 MB)
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3.
Fluctuating number of energy levels in mixed-type lemon billiards
Črt 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: 457; Prenosov: 12
.pdf Celotno besedilo (1,40 MB)
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4.
Quantum chaos in triangular billiards
Črt Lozej, Giulio Casati, Tomaž Prosen, 2022, izvirni znanstveni članek

Opis: We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with π, whose classical dynamics is presumably mixing, and three with exactly one angle rational with π, which are presumably only weakly mixing or even nonergodic in case of right triangles. We find excellent agreement of short- and long-range spectral statistics with the Gaussian orthogonal ensemble of random matrix theory for the most irrational generic triangle, while the other cases show small but significant deviations which are attributed either to a scarring or superscarring mechanism. This result, which extends the quantum chaos conjecture to systems with dynamical mixing in the absence of hard (Lyapunov) chaos, has been corroborated by analyzing distributions of phase-space localization measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.
Ključne besede: quantum physics, quantum chaos, quantum scars, wave chaos, billiards, chaos and nonlinear dynamics, ergodic theory
Objavljeno v DKUM: 12.10.2023; Ogledov: 232; Prenosov: 28
.pdf Celotno besedilo (11,21 MB)
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5.
The dynamics of a duopoly Stackelberg game with marginal costs among heterogeneous players
Atefeh Ahmadi, Sourav Roy, Mahtab Mehrabbeik, Dibakar Ghosh, Sajad Jafari, Matjaž Perc, 2023, izvirni znanstveni članek

Opis: One of the famous economic models in game theory is the duopoly Stackelberg model, in which a leader and a follower firm manufacture a single product in the market. Their goal is to obtain the maximum profit while competing with each other. The desired dynamics for a firm in a market is the convergence to its Nash equilibrium, but the dynamics of real-world markets are not always steady and can result in unpredictable market changes that exhibit chaotic behaviors. On the other hand, to approach reality more, the two firms in the market can be considered heterogeneous. The leader firm is bounded rationale, and the follower firm is adaptable. Modifying the cost function that affects the firms' profit by adding the marginal cost term is another step toward reality. We propose a Stackelberg model with heterogeneous players and marginal costs, which exhibits chaotic behavior. This model's equilibrium points, including the Nash equilibrium, are calculated by the backward induction method, and their stability analyses are obtained. The influence of changing each model parameter on the consequent dynamics is investigated through one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents spectra, and Kaplan-Yorke dimension. Eventually, using a combination of state feedback and parameter adjustment methods, the chaotic solutions of the model are successfully tamed, and the model converges to its Nash equilibrium.
Ključne besede: nonlinear dynamics, game theory, stability analysis, public goods
Objavljeno v DKUM: 02.08.2023; Ogledov: 387; Prenosov: 38
.pdf Celotno besedilo (2,45 MB)
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6.
Triggered dynamics in a model of different fault creep regimes
Srđan Kostić, Igor Franović, Matjaž Perc, Nebojša Vasović, Kristina Todorović, 2014, izvirni znanstveni članek

Opis: The study is focused on the effect of transient external force induced by a passing seismic wave on fault motion in different creep regimes. Displacement along the fault is represented by the movement of a spring-block model, whereby the uniform and oscillatory motion correspond to the fault dynamics in post-seismic and inter-seismic creep regime, respectively. The effect of the external force is introduced as a change of block acceleration in the form of a sine wave scaled by an exponential pulse. Model dynamics is examined for variable parameters of the induced acceleration changes in reference to periodic oscillations of the unperturbed system above the supercritical Hopf bifurcation curve. The analysis indicates the occurrence of weak irregular oscillations if external force acts in the post-seismic creep regime. When fault motion is exposed to external force in the inter-seismic creep regime, one finds the transition to quasiperiodic- or chaos-like motion, which we attribute to the precursory creep regime and seismic motion, respectively. If the triggered acceleration changes are of longer duration, a reverse transition from inter-seismic to post-seismic creep regime is detected on a larger time scale.
Ključne besede: geophysics, nonlinear dynamics, chaos, earthquake
Objavljeno v DKUM: 23.06.2017; Ogledov: 1412; Prenosov: 371
.pdf Celotno besedilo (669,71 KB)
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7.
Gap junctions and epileptic seizures - two sides of the same coin?
Vladislav Volman, Matjaž Perc, Maxim Bazhenov, 2011, izvirni znanstveni članek

Opis: Electrical synapses (gap junctions) play a pivotal role in the synchronization of neuronal ensembles which also makes them likely agonists of pathological brain activity. Although large body of experimental data and theoretical considerations indicate that coupling neurons by electrical synapses promotes synchronous activity (and thus is potentially epileptogenic), some recent evidence questions the hypothesis of gap junctions being among purely epileptogenic factors. In particular, an expression of inter-neuronal gap junctions is often found to be higher after the experimentally induced seizures than before. Here we used a computational modeling approach to address the role of neuronal gap junctions in shaping the stability of a network to perturbations that are often associated with the onset of epileptic seizures. We show that under some circumstances, the addition of gap junctions can increase the dynamical stability of a network and thus suppress the collective electrical activity associated with seizures. This implies that the experimentally observed post-seizure additions of gap junctions could serve to prevent further escalations, suggesting furthermore that they are a consequence of an adaptive response of the neuronal network to the pathological activity. However, if the seizures are strong and persistent, our model predicts the existence of a critical tipping point after which additional gap junctions no longer suppress but strongly facilitate the escalation of epileptic seizures. Our results thus reveal a complex role of electrical coupling in relation to epileptiform events. Which dynamic scenario (seizure suppression or seizure escalation) is ultimately adopted by the network depends critically on the strength and duration of seizures, in turn emphasizing the importance of temporal and causal aspects when linking gap junctions with epilepsy.
Ključne besede: epilepsy, nonlinear dynamics, electrical synapses, coupling, synchronization
Objavljeno v DKUM: 19.06.2017; Ogledov: 1037; Prenosov: 408
.pdf Celotno besedilo (858,25 KB)
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8.
9.
Statistical Properties of Time-dependent Systems
Diego 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: 3181; Prenosov: 147
.pdf Celotno besedilo (16,09 MB)

10.
Nonlinear time series analysis of the human electrocardiogram
Matjaž 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: 2452; Prenosov: 112
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