ALBERT

Description: Basin of attraction of a stable equilibrium point is an effective concept for stability analysis in deterministic systems; however, it does not contain information on the external perturbations that may affect it. Here we introduce the concept of stochastic basin of attraction (SBA) by incorporating a suitable probabilistic notion of basin. We define criteria for the size of the SBA based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small noise intensity but also with noise whose amplitude is proportional or in general is a function of an order parameter. As an application of our main results, we analyze a three potential well system perturbed by two types of noise: Brownian motion and non-Gaussian α -stable Lévy motion. Our main conclusions are that the thermal fluctuations stabilize the metastable system with an asymmetric three-well potential but have the opposite effect for a symmetric one. For Lévy noise with larger jumps and lower jump frequencies ( α = 0.5 ) metastability is enhanced for both symmetric and asymmetric potentials.

Description: A large parameter mismatch can induce amplitude death in two instantaneously coupled oscillators. Alternatively, a time delay in the coupling can induce amplitude death in two identical oscillators. We unify the mechanism of quenching of oscillation in coupled oscillators, either by a large parameter mismatch or a delay coupling, by a common lag scenario that is, surprisingly, different from the conventional lag synchronization. We present numerical as well as experimental evidence of this unknown kind of lag scenario when the lag increases with coupling and at a critically large value at a critical coupling strength, amplitude death emerges in two largely mismatched oscillators. This is analogous to amplitude death in identical systems with increasingly large coupling delay. In support, we use examples of the Chua oscillator and the Bonhoeffer-van der Pol system. Furthermore, we confirm this lag scenario during the onset of amplitude death in identical Stuart-Landau system under various instantaneous coupling forms, repulsive, conjugate, and a type of nonlinear coupling.

Description: Recurrence is an ubiquitous feature which provides deep insights into the dynamics of real dynamical systems. A suitable tool for investigating recurrences is recurrence quantification analysis (RQA). It allows, e.g., the detection of regime transitions with respect to varying control parameters. We investigate the complexity of different coexisting nonlinear dynamical regimes of the plasma floating potential fluctuations at different magnetic fields and discharge voltages by using recurrence quantification variables, in particular, DET , L max , and Entropy . The recurrence analysis reveals that the predictability of the system strongly depends on discharge voltage. Furthermore, the persistent behaviour of the plasma time series is characterized by the Detrended fluctuation analysis technique to explore the complexity in terms of long range correlation. The enhancement of the discharge voltage at constant magnetic field increases the nonlinear correlations; hence, the complexity of the system decreases, which corroborates the RQA analysis.

Description: The spatial distributions of system's frequencies have significant influences on the critical coupling strengths for amplitude death (AD) in coupled oscillators. We find that the left and right critical coupling strengths for AD have quite different relations to the increasing spatial period m of the frequency distribution in coupled oscillators. The left one has a negative linear relationship with m in log-log axis for small initial frequency mismatches while remains constant for large initial frequency mismatches. The right one is in quadratic function relation with spatial period m of the frequency distribution in log-log axis. There is an optimal spatial period m 0 of frequency distribution with which the coupled system has a minimal critical strength to transit from an AD regime to reviving oscillation. Moreover, the optimal spatial period m 0 of the frequency distribution is found to be related to the system size N . Numerical examples are explored to reveal the inner regimes of effects of the spatial frequency distribution on AD.

Description: The dynamical behavior of delay-coupled networks of electrochemical reactions is investigated to explore the formation of amplitude death (AD) and the synchronization states in a parameter region around the amplitude death region. It is shown that difference coupling with odd and even numbered ring and random networks can produce the AD phenomenon. Furthermore, this AD can be restored by changing the coupling type from difference to direct coupling. The restored oscillations tend to create synchronization patterns in which neighboring elements are in nearly anti-phase configuration. The ring networks produce frozen and rotating phase waves, while the random network exhibits a complex synchronization pattern with interwoven frozen and propagating phase waves. The experimental results are interpreted with a coupled Stuart-Landau oscillator model. The experimental and theoretical results reveal that AD behavior is a robust feature of delayed coupled networks of chemical units; if an oscillatory behavior is required again, even a small amount of direct coupling could be sufficient to restore the oscillations. The restored nearly anti-phase oscillatory patterns, which, to a certain extent, reflect the symmetry of the network, represent an effective means to overcome the AD phenomenon.

Description: We systematically study effects of external perturbations on models describing earthquake fault dynamics. The latter are based on the framework of the Burridge-Knopoff spring-block system, including the cases of a simple mono-block fault, as well as the paradigmatic complex faults made up of two identical or distinct blocks. The blocks exhibit relaxation oscillations, which are representative for the stick-slip behavior typical for earthquake dynamics. Our analysis is carried out by determining the phase response curves of first and second order. For a mono-block fault, we consider the impact of a single and two successive pulse perturbations, further demonstrating how the profile of phase response curves depends on the fault parameters. For a homogeneous two-block fault, our focus is on the scenario where each of the blocks is influenced by a single pulse, whereas for heterogeneous faults, we analyze how the response of the system depends on whether the stimulus is applied to the block having a shorter or a longer oscillation period.

Description: Network theory has proven to be a powerful tool in describing and analyzing systems by modelling the relations between their constituent objects. Particularly in recent years, a great progress has been made by augmenting “traditional” network theory in order to account for the multiplex nature of many networks, multiple types of connections between objects, the time-evolution of networks, networks of networks and other intricacies. However, existing network representations still lack crucial features in order to serve as a general data analysis tool. These include, most importantly, an explicit association of information with possibly heterogeneous types of objects and relations, and a conclusive representation of the properties of groups of nodes as well as the interactions between such groups on different scales. In this paper, we introduce a collection of definitions resulting in a framework that, on the one hand, entails and unifies existing network representations (e.g., network of networks and multilayer networks), and on the other hand, generalizes and extends them by incorporating the above features. To implement these features, we first specify the nodes and edges of a finite graph as sets of properties (which are permitted to be arbitrary mathematical objects). Second, the mathematical concept of partition lattices is transferred to the network theory in order to demonstrate how partitioning the node and edge set of a graph into supernodes and superedges allows us to aggregate, compute, and allocate information on and between arbitrary groups of nodes. The derived partition lattice of a graph, which we denote by deep graph , constitutes a concise, yet comprehensive representation that enables the expression and analysis of heterogeneous properties, relations, and interactions on all scales of a complex system in a self-contained manner. Furthermore, to be able to utilize existing network-based methods and models, we derive different representations of multilayer networks from our framework and demonstrate the advantages of our representation. On the basis of the formal framework described here, we provide a rich, fully scalable (and self-explanatory) software package that integrates into the PyData ecosystem and offers interfaces to popular network packages, making it a powerful, general-purpose data analysis toolkit. We exemplify an application of deep graphs using a real world dataset, comprising 16 years of satellite-derived global precipitation measurements. We deduce a deep graph representation of these measurements in order to track and investigate local formations of spatio-temporal clusters of extreme precipitation events.

Description: We present here two promising techniques for the application of the complex network approach to continuous spatio-temporal systems that have been developed in the last decade and show large potential for future application and development of complex systems analysis. First, we discuss the transforming of a time series from such systems to a complex network. The natural approach is to calculate the recurrence matrix and interpret such as the adjacency matrix of an associated complex network, called recurrence network. Using complex network measures, such as transitivity coefficient, we demonstrate that this approach is very efficient for identifying qualitative transitions in observational data, e.g., when analyzing paleoclimate regime transitions. Second, we demonstrate the use of directed spatial networks constructed from spatio-temporal measurements of such systems that can be derived from the synchronized-in-time occurrence of extreme events in different spatial regions. Although there are many possibilities to investigate such spatial networks, we present here the new measure of network divergence and how it can be used to develop a prediction scheme of extreme rainfall events.