arXiv subjects

Tomasz Kapitaniak

Publications and source records attributed to Tomasz Kapitaniak.

At least 19 recordsLinked to original sources

arXiv2026

Opinion-Driven Vaccination and Epidemic Dynamics on Heterogeneous Networks

Vaccination campaigns play a pivotal role in controlling infectious diseases. Their success, however, depends not only on vaccine efficacy and availability but also significantly on public opinion and the willingness of individuals to vaccinate. This paper investigates a coupled opinion-epidemic model on heterogeneous networks, where individual opinions influence vaccination probability, and opinions themselves evolve through a combination of peer interaction and local risk perception derived from observed infection rates. Embedding the coupled dynamics in scale-free networks, particularly barabasi-Albert structures, allows us to examine the role of network heterogeneity beyond homogeneous-mixing assumptions. Using Monte Carlo simulations and a semi-analytical microscopic Markov-chain approach, we derive and numerically validate analytical expressions for the critical infection threshold and stable vaccinated population where risk perception dominated peer influence. Our results show that stronger local risk perception enhances pro-vaccination opinions and suppresses infection, while dominant peer influence can increase long-term infection levels. These findings underscore the importance of accounting for social behavior and network structure when designing effective vaccination and epidemic control strategies.

physics.soc-ph↗

arXiv2026

Dispersal-induced survival of predators in metacommunities due to transient chaos

Dispersal networks critically shape the fate of ecological communities, yet the mechanisms linking connectivity and persistence remain poorly understood. We show that an interplay between asymmetric dispersal and asynchronous dynamics across patches in a dispersal network can prevent predator extinction across broad dispersal ranges, even in identical environments in which synchrony usually drives ecosystems to collapse. Unlike classical rescue effects based on environmental heterogeneity or equilibrium states, this mechanism emerges from non-equilibrium dynamics, specifically from transient chaotic dynamics. Dispersal coupling perturbs local trajectories in patches facing extinction and reinforce chaotic motion, thereby sustaining chaotic oscillations indefinitely. Strikingly, only minimal connectivity is required: small-world networks with a few long-range links suffice to rescue predator populations. These findings reveal a counterintuitive principle that limited, well-placed connectivity can harness chaos to maintain biodiversity in fragmented landscapes.

nlin.CD↗

arXiv2026

Astrocytes: Arnol'd Tongues Generalization in Dynamical Systems' Parameter Plane

We discovered generalized structures, named astrocytes due to their shape, that constitute a defined region characterizing regular behavior within the parameter plane (PP) of dynamical systems (DSs). Morphologically, they are characterized by a branch and a soma with several vertices (arms) and sometimes with multiple periodicities. A bunch of infinite astrocytes emerge through their branches from a region, in general, of low periodicity. Astrocytes are embedded in a quasiperiodic-chaotic scenario. The soma complexity (number of vertices) determines a kind of hierarchy of the astrocytes; moreover, bunches of subsequent structures from the astrocyte have been emphasized, revealing a self-similarity property. We conducted a detailed analysis in a Zeeman laser model, but we also observed astrocytes in many other DSs. The multiperiodicity exhibited by the astrocytes in their soma gives rise to harlequin dress-like patterns and tri-, quad-, and quint-critical points, which indicate the coexistence of different higher-order periodicities. In the concave borders of the soma, a doubling cascade of quint-points emerges as a bifurcation in the PP, defining regions of ordered sequences of higher periodicity in the route to chaos.

nlin.CD↗

arXiv2026

Dynamics and non-integrability of the variable-length double pendulum: exploring chaos and periodicity via the Lyapunov refined maps

This paper extends our previous work~(Szumiński and Maciejewski, 2024), where we explored the dynamics and integrability of the double-spring pendulum. Here, we investigate the variable-length double pendulum, a three-degree-of-freedom Hamiltonian system combining features of the classic double pendulum and the swinging Atwood machine. With its intricate dynamics, this system is crucial for studying nonlinear phenomena such as high-order resonances, chaos, and bifurcations. We address the challenges posed by high-dimensional phase spaces using a novel tool, the \textit{Lyapunov refined maps}, which integrates Poincaré sections, phase-parametric diagrams, and Lyapunov exponents. This framework comprehensively analyzes periodic, quasi-periodic, and chaotic behaviors. By measuring the strength of chaos, it also offers insights into the system's dynamical structure. Additionally, we apply Morales-Ramis theory to examine integrability, leveraging the differential Galois group of variational equations to establish non-integrability conditions. The Kovacic algorithm is used to analyze the solvability of higher-dimensional differential equations, complemented by Lyapunov exponent diagrams to exclude integrable dynamics under certain parameters. Our findings advance the fundamental understanding of variable-length pendulum dynamics, offering new insights and methodologies for further research with potential applications in adaptive robotics, energy harvesting, and biomechanics. Additionally, this work represents a significant step toward proving the long-sought non-integrability of the classical double pendulum.

nlin.CD↗

arXiv2024

Complexity measure of extreme events

Complexity is an important metric for appropriate characterization of different classes of irregular signals, observed in the laboratory or in nature. The literature is already rich in the description of such measures using a variety of entropy and disequilibrium measures, separately or in combination. Chaotic signal was given prime importance in such studies while no such measure was proposed so far, how complex were the extreme events when compared to non-extreme chaos. We address here this question of complexity in extreme events and investigate if we can distinguish them from non-extreme chaotic signal. The normalized Shannon entropy in combination with disequlibrium is used for our study and it is able to distinguish between extreme chaos and non-extreme chaos and moreover, it depicts the transition points from periodic to extremes via Pomeau-Manneville intermittency and, from small amplitude to large amplitude chaos and its transition to extremes via interior crisis. We report a general trend of complexity against a system parameter that increases during a transition to extreme events, reaches a maximum, and then starts decreasing. We employ three models, a nonautonomous Lienard system, 2-dimensional Ikeda map and a 6-dimensional coupled Hindmarh-Rose system to validate our proposition.

nlin.CD↗

arXiv2023

Higher order interactions in Kuramoto oscillators with inertia

How higher-order interactions influence the dynamics of second order phase oscillators? We address this question using three coupled Kuramoto phase oscillators with inertia under both pairwise and higher order interactions, finding collective states, which are absent for pair-wise interactions. Chimera states appear in an expanded parameter region of the 3-node network by comparison to the network's response in the absence of the higher-order interactions. %That attempt enables us to understand the emergence of chimera states. This small network of three nodes is investigated for different sets of phase lags and over a range of negative to positive coupling strength of pairwise as well as higher order or group interactions. It demonstrates how important are the choices of the phase lag and the sign of the coupling for the emergent dynamics of the system.

math.DS↗

arXiv2023

Transition to hyperchaos and rare large-intensity pulses in Zeeman laser

A discontinuous transition to hyperchaos is observed at discrete critical parameters in the Zeeman laser model for three well known nonlinear sources of instabilities, namely, quasiperiodic breakdown to chaos followed by interior crisis, quasiperiodic intermittency, and Pomeau-Manneville intermittency. Hyperchaos appears with a sudden expansion of the attractor of the system at a critical parameter for each case and it coincides with triggering of occasional and recurrent large-intensity pulses. The transition to hyperchaos from a periodic orbit via Pomeau-Manneville intermittency shows hysteresis at the critical point, while no hysteresis is recorded during the other two processes. The recurrent large-intensity pulses show characteristic features of extremes by their height larger than a threshold and probability of rare occurrence. The phenomenon is robust to weak noise although the critical parameter of transition to hyperchaos shifts with noise strength. This phenomenon appears as common in many low dimensional systems as reported earlier, there the emergent large-intensity events or extreme events dynamics have been recognized simply as chaotic in nature although the temporal dynamics shows occasional large deviations from the original chaotic state in many examples. We need a new metric, in the future, that would be able to classify such significantly different dynamics and distinguish from chaos.

nlin.PS↗

arXiv2022

Extreme events in a complex network: interplay between degree distribution and repulsive interaction

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and thereby confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator.

physics.soc-ph↗

arXiv2022

Transition to hyperchaos: Sudden expansion of attractor and intermittent large-amplitude events in dynamical systems

Hyperchaos is distinguished from chaos by the presence of at least two positive Lyapunov exponents instead of just one in dynamical systems. A general scenario is presented here that shows emergence of hyperchaos with a sudden large expansion of the attractor of continuous dynamical systems at a critical parameter when the temporal dynamics shows intermittent large-amplitude spiking or bursting events. The distribution of local maxima of the temporal dynamics is non-Gaussian with a tail, confirming a rare occurrence of the large-amplitude events. We exemplify our results on the sudden emergence of hyperchaos in three paradigmatic models, namely, a coupled Hindmarsh-Rose model, three coupled Duffing oscillators, and a hyperchaotic model.

nlin.AO↗

arXiv2021

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known, however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both the cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

nlin.CD↗

arXiv2021

A novel, Fourier series based method of control optimization and its application to a discontinuous capsule drive model

This paper presents a novel, Fourier series based numerical method of open-loop control optimization. Due to its flexible assumptions, it can be applied in a large variety of systems, including discontinuous ones or even so-called black boxes, whose equations are not fully known. This aspect is particularly important in mechanical systems, where friction or impact induced discontinuities are common. The paper includes a mathematical background of the new method, a detailed discussion of the algorithm and a numerical example, in which control function of a discontinuous capsule drive is optimized. It is expected that the proposed method can facilitate research in all areas where control of non-smooth, discontinuous or black box systems is crucial. In particular, authors hope that the presented algorithm is going to be used for control optimization of other capsule drives.

math.OC↗

arXiv2021

Chimera states for directed networks

We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny {\it chimera islands} arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for hundred oscillators (cyclic century), the islands merge into a single {\it chimera continent} which incorporates the world of chimeras of different configuration. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as it decreases.

nlin.CD↗

arXiv2020

Minimal Chimera States in Phase-Lag Coupled Mechanical Oscillators

We obtain experimental chimera states in the minimal network of three identical mechanical oscillators (metronomes), by introducing phase-lagged all-to-all coupling. For this, we have developed a real-time model-in-the-loop coupling mechanism that allows for flexible and online change of coupling topology, strength and phase-lag. The chimera states manifest themselves as a mismatch of average frequency between two synchronous and one desynchronized oscillator. We find this kind of striking chimeric behavior is robust in a wide parameter region. At other parameters, however, chimera state can lose stability and the system behavior manifests itself as a heteroclinic switching between three saddle-type chimeras. Our experimental observations are in a qualitative agreement with the model simulation.

nlin.AO↗

arXiv2020

Routes to extreme events in dynamical systems: Dynamical and Statistical Characteristics

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau-Manneville intermittency and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system to illustrate how the instabilities work to originate as extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

nlin.CD↗

arXiv2019

Extreme events in a network of heterogeneous Josephson junctions

We report rare and recurrent large spiking events in a heterogeneous network of superconducting Josephson junctions (JJ) connected through a resistive load and driven by a radio-frequency (rf) current in addition to a constant bias. The intermittent large spiking events show characteristic features of extreme events (EE) since they are larger than a statistically defined significant height. Under the influence of repulsive interactions and an impact of heterogeneity of damping parameters, the network splits into three sub-groups of junctions, one in incoherent rotational, another in coherent librational motion and a third sub-group originating EE. We are able to scan the whole population of junctions with their distinctive individual dynamical features either in EE mode or non-EE mode in parameter space. EE migrates spatially from one to another sub-group of junctions depending upon the repulsive strength and the damping parameter. For a weak repulsive coupling, all the junctions originate frequent large spiking events, in rotational motion when the average inter-spike-interval (ISI) is small, but it increases exponentially with repulsive interaction; it largely deviates from its exponential growth at a break point where EE triggers in a sub-group of junctions. The probability density of inter-event-intervals (IEI) in the subgroup exhibits a Poisson distribution. EE originates via bubbling instability of in-phase synchronization.

nlin.AO↗

arXiv2018

Network-induced multistability: Lossy coupling and exotic solitary states

The stability of synchronised networked systems is a multi-faceted challenge for many natural and technological fields, from cardiac and neuronal tissue pacemakers to power grids. In the latter case, the ongoing transition to distributed renewable energy sources is leading to a proliferation of dynamical actors. The desynchronization of a few or even one of those would likely result in a substantial blackout. Thus the dynamical stability of the synchronous state has become a focus of power grid research in recent years. In this letter we uncover that the non-linear stability against large perturbations is dominated and threatened by the presence of \textit{solitary states} in which individual actors desynchronise. Remarkably, when taking physical losses in the network into account, the back-reaction of the network induces new {\it exotic} solitary states in the individual actors, and the stability characteristics of the synchronous state are dramatically altered. These novel effects will have to be explicitly taken into account in the design of future power grids, and their existence poses a challenge for control. While this letter focuses on power grids, the form of the coupling we explore here is generic, and the presence of new states is very robust. We thus strongly expect the results presented here to transfer to other systems of coupled heterogeneous Newtonian oscillators.

nlin.AO↗

arXiv2017

Solitary states for coupled oscillators

We demonstrate that solitary states can be widely observed for networks of coupled oscillators with local, non-local and global couplings, and they preserve in both thermodynamic and Hamiltonian limits. We show that depending on units' and network's parameters, different types of solitary states occur, characterized by the number of isolated oscillators and the disposition in space. The creation of solitary states through the homoclinic bifurcation is described and the regions of co-existence of obtained states and typical examples of dynamics have been identified. Our analysis suggests that solitary states can be observed in a wide class of networks relevant to various real-world systems.

nlin.CD↗

arXiv2017

Perpetual points: New tool for localization of co-existing attractors in dynamical systems

Perpetual points (PPs) are special critical points for which the magnitude of acceleration describing dynamics drops to zero, while the motion is still possible (stationary points are excluded), e.g. considering the motion of the particle in the potential field, at perpetual point it has zero acceleration and non-zero velocity. We show that using PPs we can trace all the stable fixed points in the system, and that the structure of trajectories leading from former points to stable equilibria may be similar to orbits obtained from unstable stationary points. Moreover, we argue that the concept of perpetual points may be useful in tracing unexpected attractors (hidden or rare attractors with small basins of attraction). We show potential applicability of this approach by analysing several representative systems of physical significance, including the damped oscillator, pendula and the Henon map. We suggest that perpetual points may be a useful tool for localization of co-existing attractors in dynamical systems.

nlin.CD↗