Analysis of chaos and capsizing of a class of nonlinear ship rolling systems under excitation of random waves.

The action of wind and waves has a significant effect on the ship's roll, which can be a source of chaos and even capsize. The influence of random wave excitation is considered in order to investigate complex dynamic behavior by analytical and numerical methods. Chaotic rolling motions are theoretically studied in detail by means of the relevant Melnikov method with or without noise excitation. Numerical simulations are used to verify and analyze the appropriate parameter excitation and noise conditions. The results show that by changing the parameters of the excitation amplitude or the noise intensity, chaos can be induced or suppressed.

Different routes to large-intensity pulses in Zeeman laser model.

In this study, we report a rich variety of large-intensity pulses exhibited by a Zeeman laser model. The instabilities in the system occur via three different dynamical processes, such as quasiperiodic intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion to chaos followed by an interior crisis. This Zeeman laser model is more capable of exploring the major possible types of instabilities when changing a specific system's parameter in a particular range. We exemplified distinct dynamical transitions of the Zeeman laser model. The statistical measures reveal the appearance of the low probability of large-intensity pulses above the qualifier threshold value. Moreover, they seem to follow an exponential decay that shows a Poisson-like distribution. The impact of noise and time delay effects have been analyzed near the transition point of the system.

Coupled pendula with varied forcing direction.

In this paper, we investigate the complex dynamics of rotating pendula arranged into a simple mechanical scheme. Three nodes forming the small network are coupled via the horizontally oscillating beam (the global coupling structure) and the springs (the local coupling), which extends the research performed previously for similar models. The pendula rotate in different directions, and depending on the distribution of the latter ones, various types of behaviors of the system can be observed. We determine the regions of the existence and co-existence of particular solutions using both the classical method of bifurcations, as well as a modern sample-based approach based on the concept of basin stability. Various types of states are presented and discussed, including synchronization patterns, coherent dynamics, and irregular motion. We uncover new schemes of solutions, showing that both rotations and oscillations can co-exist for various pendula, arranged within one common system. Our analysis includes the investigations of the basins of attraction of different dynamical patterns, as well as the study on the properties of the observed states, along with the examination of the influence of system's parameters on their behavior. We show that the model can respond in spontaneous ways and uncover unpredicted irregularities occurring for the states. Our study exhibits that the inclusion of the local coupling structure can induce complex, chimeric dynamics of the system, leading to new co-existing patterns for coupled mechanical nodes.

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 with their height larger than a threshold and the probability of a 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 by Chowdhury et al. [Phys. Rep. 966, 1-52 (2022)], 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.

Asymmetry induces critical desynchronization of power grids.

Dynamical stability of the synchronous regime remains a challenging problem for secure functioning of power grids. Based on the symmetric circular model [Hellmann et al., Nat. Commun. 11, 592 (2020)], we demonstrate that the grid stability can be destroyed by elementary violations (motifs) of the network architecture, such as cutting a connection between any two nodes or removing a generator or a consumer. We describe the mechanism for the cascading failure in each of the damaging case and show that the desynchronization starts with the frequency deviation of the neighboring grid elements followed by the cascading splitting of the others, distant elements, and ending eventually in the bi-modal or a partially desynchronized state. Our findings reveal that symmetric topology underlines stability of the power grids, while local damaging can cause a fatal blackout.

Control of chaotic systems through reservoir computing.

Chaos is an important dynamic feature, which generally occurs in deterministic and stochastic nonlinear systems and is an inherent characteristic that is ubiquitous. Many difficulties have been solved and new research perspectives have been provided in many fields. The control of chaos is another problem that has been studied. In recent years, a recurrent neural network has emerged, which is widely used to solve many problems in nonlinear dynamics and has fast and accurate computational speed. In this paper, we employ reservoir computing to control chaos in dynamic systems. The results show that the reservoir calculation algorithm with a control term can control the chaotic phenomenon in a dynamic system. Meanwhile, the method is applicable to dynamic systems with random noise. In addition, we investigate the problem of different values for neurons and leakage rates in the algorithm. The findings indicate that the performance of machine learning techniques can be improved by appropriately constructing neural networks.

Impact of time varying interaction: Formation and annihilation of extreme events in dynamical systems.

This study investigates the emergence of extreme events in two different coupled systems: the FitzHugh-Nagumo neuron model and the forced Liénard system, both based on time-varying interactions. The time-varying coupling function between the systems determines the duration and frequency of their interaction. Extreme events in the coupled system arise as a result of the influence of time-varying interactions within various parameter regions. We specifically focus on elucidating how the transition point between extreme events and regular events shifts in response to the duration of interaction time between the systems. By selecting the appropriate interaction time, we can effectively mitigate extreme events, which is highly advantageous for controlling undesired fluctuations in engineering applications. Furthermore, we extend our investigation to networks of oscillators, where the interactions among network elements are also time dependent. The proposed approach for coupled systems holds wide applicability to oscillator networks.

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.

Extreme transient dynamics.

We study the extreme transient dynamics of four self-excited pendula coupled via the movable beam. A slight difference in the pendula lengths induces the appearance of traveling phase behavior, within which the oscillators synchronize, but the phases between the nodes change in time. We discuss various scenarios of traveling states (involving different pendula) and their properties, comparing them with classical synchronization patterns of phase-locking. The research investigates the problem of transient dynamics preceding the stabilization of the network on a final synchronous attractor, showing that the width of transient windows can become extremely long. The relation between the behavior of the system within the transient regime and its initial conditions is examined and described. Our results include both identical and non-identical pendula masses, showing that the distribution of the latter ones is related to the transients. The research performed in this paper underlines possible transient problems occurring during the analysis of the systems when the slow evolution of the dynamics can be misinterpreted as the final behavior.

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 associated with a node.

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 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 a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.

Multi-headed loop chimera states in coupled oscillators.

In this paper, we introduce a novel type of chimera state, characterized by the geometrical distortion of the coherent ring topology of coupled oscillators. The multi-headed loop chimeras are examined for a simple network of locally coupled pendulum clocks, suspended on the vertical platform. We determine the regions of the occurrence of the observed patterns, their structure, and possible co-existence. The representative examples of behaviors are shown, exhibiting the variety of configurations that can be observed. The statistical analysis of the solutions indicates the geometrical regions of the system with the highest probability of the chimeras' occurrence. We investigate the mechanism of the creation of the observed states, showing that the manipulation of the initial positions of chosen pendula may induce the desired patterns. Apart from the study of the isolated network, we also discuss the scenario of the movable platform, showing a possible influence of the global coupling structure on the stability of the observed states. The stability of loop chimeras is examined for varying both the amplitude and the frequency of the oscillations of the platform. We indicate the excitation parameters for which the solutions can survive as well as be destroyed. The bifurcation analysis included in the paper allows us to discuss the transitions between possible behaviors. The appearance of multi-headed loop chimeras is generalized into large networks of oscillators, showing the universal character of the observed patterns. One should expect to observe similar results also in other types of coupled oscillators, especially the mechanical ones.

Experimental chaotic synchronization for coupled double pendula.

10.1063/5.0056530.4In this paper, we experimentally verify the phenomenon of chaotic synchronization in coupled forced oscillators. The study is focused on the model of three double pendula locally connected via springs. Each of the individual oscillators can behave both periodically and chaotically, which depends on the parameters of the external excitation (the shaker). We investigate the relation between the strength of coupling between the upper pendulum bobs and the precision of their synchronization, showing that the system can achieve practical synchronization, within which the nodes preserve their chaotic character. We determine the influence of the pendula parameters and the strength of coupling on the synchronization precision, measuring the differences between the nodes' motion. The results obtained experimentally are confirmed by numerical simulations. We indicate a possible mechanism causing the desynchronization of the system's smaller elements (lower pendula bobs), which involves their motion around the unstable stationary position and possible transient dynamics. The results presented in this paper may be generalized into typical models of pendula and pendula-like coupled systems, exhibiting chaotic dynamics.

Ordered slow and fast dynamics of unsynchronized coupled phase oscillators.

Slow and fast dynamics of unsynchronized coupled nonlinear oscillators is hard to extract. In this paper, we use the concept of perpetual points to explain the short duration ordering in the unsynchronized motions of the phase oscillators. We show that the coupled unsynchronized system has ordered slow and fast dynamics when it passes through the perpetual point. Our simulations of single, two, three, and 50 coupled Kuramoto oscillators show the generic nature of perpetual points in the identification of slow and fast oscillations. We also exhibit that short-time synchronization of complex networks can be understood with the help of perpetual motion of the network.

Neuron-like spiking and bursting in Josephson junctions: A review.

The superconducting Josephson junction shows spiking and bursting behaviors, which have similarities with neuronal spiking and bursting. This phenomenon had been observed long ago by some researchers; however, they overlooked the biological similarity of this particular dynamical feature and never attempted to interpret it from the perspective of neuronal dynamics. In recent times, the origin of such a strange property of the superconducting junction has been explained and such neuronal functional behavior has also been observed in superconducting nanowires. The history of this research is briefly reviewed here with illustrations from studies of two junction models and their dynamical interpretation in the sense of biological bursting.

Collective behavior in a two-layer neuronal network with time-varying chemical connections that are controlled by a Petri net.

In this paper, we propose and study a two-layer network composed of a Petri net in the first layer and a ring of coupled Hindmarsh-Rose neurons in the second layer. Petri nets are appropriate platforms not only for describing sequential processes but also for modeling information circulation in complex systems. Networks of neurons, on the other hand, are commonly used to study synchronization and other forms of collective behavior. Thus, merging both frameworks into a single model promises fascinating new insights into neuronal collective behavior that is subject to changes in network connectivity. In our case, the Petri net in the first layer manages the existence of excitatory and inhibitory links among the neurons in the second layer, thereby making the chemical connections time-varying. We focus on the emergence of different types of collective behavior in the model, such as synchronization, chimeras, and solitary states, by considering different inhibitory and excitatory tokens in the Petri net. We find that the existence of only inhibitory or excitatory tokens disturbs the synchronization of electrically coupled neurons and leads toward chimera and solitary states.

Transient chimera-like states for forced oscillators.

Chimera states occur widely in networks of identical oscillators as has been shown in the recent extensive theoretical and experimental research. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, we consider a star network, in which N identical peripheral end nodes are connected to the central hub node. We find that if a single node exhibits transient chaotic behavior in the whole network, the pattern of transient chimeralike state, which persists for a significant amount of time, is created. As a proof of the concept, we examine the system of N double pendula (peripheral end nodes) located on the periodically oscillating platform (central hub). We show that such transient chimeralike states can be observed in simple experiments with mechanical oscillators, which are controlled by elementary dynamical equations. Our finding suggests that transient chimeralike states are observable in networks relevant to various real-world systems.

Effect of the policy and consumption delay on the amplitude and length of business cycle.

In this paper, the amplitude and the length of the business cycle are investigated. It is the first time the length of the business cycle based on the Goodwin model (one classical business cycle model) is discussed. The effect of the time delay of the economic policy and consumption on the amplitude and the length of the business cycle is studied. Meanwhile, the memory property of making economic policy is also considered. The theoretical amplitude of the business cycle is obtained by multiple-scale methods. The transitions of the amplitude induced by memory property and time delay are analyzed. How the economic parameter and random excitation affect the length of the business cycle is proposed. Based on the results, we can find that the time delay of both economic policy and consumption can induce the transitions. Moreover, the memory property of economic policy will change the critical value of the parameters when the transitions occur. In one typical induced investment function, the length of the business cycle is determined only by the autonomous investment and consumption. However, the length of the business cycle is not mainly affected by the autonomous investment and consumption in some other typical induced investment function. This states that the type of induced investment function has a very important role in determining the length of the business cycle.

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 extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

Prediction of bifurcations by varying critical parameters of COVID-19.

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated.