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The relationship between network structure and dynamics is one of the most extensively investigated problems in the theory of complex systems of recent years. Understanding this relationship is of relevance to a range of disciplines - from neuroscience to geomorphology. A major strategy of investigating this relationship is the quantitative comparison of a representation of network architecture (structural connectivity, SC) with a (network) representation of the dynamics (functional connectivity, FC). Here, we show that one can distinguish two classes of functional connectivity - one based on simultaneous activity (co-activity) of nodes, the other based on sequential activity of nodes. We delineate these two classes in different categories of dynamical processes - excitations, regular and chaotic oscillators - and provide examples for SC/FC correlations of both classes in each of these models. We expand the theoretical view of the SC/FC relationships, with conceptual instances of the SC and the two classes of FC for various application scenarios in geomorphology, ecology, systems biology, neuroscience and socio-ecological systems. Seeing the organisation of dynamical processes in a network either as governed by co-activity or by sequential activity allows us to bring some order in the myriad of observations relating structure and function of complex networks.

In 1695 the theory of fractional calculus was introduced, but it only developed as a pure mathematical branch. Currently several research groups have focused on the control, the implementation of filters, PID controllers, synchronization, the implementation of circuits of chaotic systems of fractional order, etc. Currently, the number of applications of fractional calculus is increasing rapidly, these mathematical phenomena have allowed us to describe and model a real object more accurately than the classical "integer" methods. Along with the development of the fractional calculation, it was shown that many fractional-order nonlinear dynamic systems behave in a chaotic manner. This is the type of non-linear systems that are addressed in this research topic with the focus on derivatives of arbitrary order, where numerical simulations of chaotic behavior are presented in non-linear, fractional-order autonomous models. The case studies are six chaotic oscillators of fractional order; The systems of Lorenz, Rӧssler, Financiero, Lui, Chen and Lü, whose attractors are obtained by applying the definitions of the Grünwald-Letnikov definitions and the predictive corrective method of Adams-Bashforth-Moulton.

In this paper, a new technique is introduced for chaos secure data communication. In this approach, in addition to the usually used techniques for data encryption, the concept of carrier encryption is introduced to increase the security level of the secure communication scheme. To fulfill this objective, at the transmitting end, two chaotic oscillators are coupled, and a set of inequality time dependent constraints with time dependent bounds is imposed on the generated chaotic signals. Moreover, to increase system complexity and its security level, the imposed set of constraints and their bounds are allowed to be changeable from one time period to another during the transmission process. As a result, the patterns of the generated chaotic signals are completely changed and the chaotic oscillator is completely encrypted. At the receiving end, the newly developed Constrained Smoothed Regularized Least Square (CSRLS) observer is used to synchronize the received constrained chaotic signals and hence retrieve the transmitted data. Using such an approach, the quality of the received information, measured by the Bit Error Rate (BER), is highly improved due to the superior performance of the developed CSRLS observer. The stability of the observer is analyzed, and simulation results are presented to show the efficiency and effectiveness of the proposed secure communication scheme.

In the present work, we present a new algorithm for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch embedded in heavy white Gaussian noise. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. To recover the phase at low signal-to-noise ratio, a nonlinear adaptive denoising algorithm based on finding sinusoidal fits to the local neighbourhood of the reconstructed phase space is used. Application of the proposed approach to master-slave Rössler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling in highly noisy uni-directionally coupled chaotic oscillators, especially in the case of weak and moderate coupling.

One of the most challenging problems in the study of complex dynamical systems is to find the statistical interdependencies among the system components. Granger causality (GC) represents one of the most employed approaches, based on modeling the system dynamics with a linear vector autoregressive (VAR) model and on evaluating the information flow between two processes in terms of prediction error variances. In its most advanced setting, GC analysis is performed through a state-space (SS) representation of the VAR model that allows to compute both conditional and unconditional forms of GC by solving only one regression problem. While this problem is typically solved through Ordinary Least Square (OLS) estimation, a viable alternative is to use Artificial Neural Networks (ANNs) implemented in a simple structure with one input and one output layer and trained in a way such that the weights matrix corresponds to the matrix of VAR parameters. In this work, we introduce an ANN combined with SS models for the computation of GC. The ANN is trained through the Stochastic Gradient Descent L1 (SGD-L1) algorithm, and a cumulative penalty inspired from penalized regression is applied to the network weights to encourage sparsity. Simulating networks of coupled Gaussian systems, we show how the combination of ANNs and SGD-L1 allows to mitigate the strong reduction in accuracy of OLS identification in settings of low ratio between number of time series points and of VAR parameters. We also report how the performances in GC estimation are influenced by the number of iterations of gradient descent and by the learning rate used for training the ANN. We recommend using some specific combinations for these parameters to optimize the performance of GC estimation. Then, the performances of ANN and OLS are compared in terms of GC magnitude and statistical significance to highlight the potential of the new approach to reconstruct causal coupling strength and network topology even in challenging conditions of data paucity. The results highlight the importance of of a proper selection of regularization parameter which determines the degree of sparsity in the estimated network. Furthermore, we apply the two approaches to real data scenarios, to study the physiological network of brain and peripheral interactions in humans under different conditions of rest and mental stress, and the effects of the newly emerged concept of remote synchronization on the information exchanged in a ring of electronic oscillators. The results highlight how ANNs provide a mesoscopic description of the information exchanged in networks of multiple interacting physiological systems, preserving the most active causal interactions between cardiovascular, respiratory and brain systems. Moreover, ANNs can reconstruct the flow of directed information in a ring of oscillators whose statistical properties can be related to those of physiological networks.

We show that interactions of inherently chaotic oscillators can lead to coexistence of regular oscillatory regimes and chaotic oscillations in the rings of coupled oscillators provided that the level of interaction between the oscillators exceeds a threshold value. The transformation of the initially chaotic dynamics into the regular dynamics in a number of the coupled oscillators is shown to result from suppression of chaos by separation of certain oscillation periods from the continuous spectra, which are characteristic of chaotic oscillations.

Josephson junction devices play a significant role in various physical nonlinear systems because of their complex characteristics. Chaotic phenomenon in various types of Josephson junction devices has been widely reported, but many of those literature studies exempted the analysis into multistability and megastability features of the device. In this work, we investigate the network behaviour using a type of Josephson junction-memristor (JJM) device considering the feedback flux effects while modelling. We have considered both AC- and DC-type external excitation currents, and while considering the AC excitation, the system shows megastability (Ramakrishnan et al. 2020). When analysing the lattice layer network constructed with JJM excited by DC bias current, the network shows a turbulent behaviour thus forming spiral waves. This was not the case when we applied AC bias current for which the network showed a much pattern-like formation confirming localised areas of energy distribution. This energy distribution is due to the homogeneous states of the local nodes which are correlated by the respective periodicity plots. When we apply AC bias current with very low frequency, the network shows small areas of local spirals which are soon dissipated by the inhomogeneous nodes nearby. Thus, we could show that the external bias current plays an important role in the collective performance of the Josephson junction devices. © 2021 Balamurali Ramakrishnan et al.

In this chapter, non-autonomous oscillators are considered as mappings from input functions of time to a circle’s worth of functions giving the state as a function of time. This view is justified using the theory of normally hyperbolic submanifolds of dynamical systems. It illuminates the phenomena of phase-locking, synchronisation and chimera; it allows an extension of the concept of coupling; and it allows a hierarchical aggregation treatment of synchronisation in networks of oscillators. The view extends to excitable and chaotic oscillators.

We study the dynamics of a multilayer network of chaotic oscillators subject to an amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster and chimera states. Here we consider a network with two layers of Roessler chaotic oscillators as well as applications to multilayer networks of chaotic jerk and Lienard oscillators. Intralayer coupling is considered to be all to all in the case of Roessler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Lienard, the interlayer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in depth study on the case of a network of Roessler oscillators using master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster and phase synchronization with amplification. For the case of Roessler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification and complete synchronization with amplification., 13 pages, 8 figures. To appear in Chaos: An Interdisciplinary Journal of Nonlinear Science

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

In some of the real-world networks, the interacting agents synchronize in groups or clusters, not in a complete way; and also, many networks can be modeled by multi-layer networks where nodes are coupled in several independent networks. In this work, we deal with the problem of synchronizing a pair of layers of networks via a double distributed control. Specifically, these controllers depends on the degree of nodes. Our first aim is synchronizing just some nodes of each layer, and then as a second goal, we apply a control to the already synchronized nodes of each layer to assure that they follow the same dynamics in both layers.

We study the structure and properties of the chimera states in the ensembles of chaotic oscillators with nonlocal coupling. It is shown that the phase and amplitude chimera states in the ensembles of chaotic oscillators with nonhyperbolic and hyperbolic attractors can be obtained using the models in the forms of two-dimensional Henon and Lozi maps. The mechanisms of birth, the structure, and the lifetime of the phase and amplitude chimeras and the regime of solitary states are studied. The chimera states in two coupled ensembles of chaotic maps are considered. The possibility of realizing a new type of the chimera structure, namely, the chimera consisting of the solitary states is demonstrated. The effects of the external and mutual synchronizations of the chimera states are described by an example of two coupled ensembles of logistic maps with nonlocal coupling. A qualitative analogy of the obtained results with the classical effect of synchronization of periodic self-sustained oscillations is discussed.

Graphical abstract, Memristor is a non-linear circuit element in which voltage-current relationship is determined by the previous values of the voltage and current, generally the history of the circuit. The nonlinearity in this component can be considered as a fractional-order form, which yields a fractional memristor (fracmemristor). In this paper, a fractional-order memristor in a chaotic oscillator is applied, while the other electronic elements are of integer order. The fractional-order range is determined in a way that the circuit has chaotic solutions. Also, the statistical and dynamical features of this circuit are analyzed. Tools like Lyapunov exponents and bifurcation diagram show the existence of multistability and antimonotonicity, two less common properties in chaotic circuits.

Spiral wave chimeras with spatially extended unsynchronized cores have been reported in nonlocally coupled phase oscillators and chaotic oscillators. Here, we investigate the two-dimensional nonlocally coupled FitzHugh–Nagumo oscillators with open boundary conditions. Rich dynamics of spiral wave chimeras is discovered numerically. Besides outwardly propagating and inwardly propagating single-spiral wave chimeras, we find the double-spiral wave chimera with two spiral chimeras rotating with each other and the breakup of spiral wave chimeras. We also find a transition between a normal spiral wave with a phase singularity at its center and spiral wave chimera.

One of the important applications of chaotic oscillators is their employment as sources of entropy for True Random Bit Generators (TRBGs). In this work, we introduce high-speed TRBGs realized on a modular Field Programmable Gate Array (FPGA) hardware platform using two different switching-type chaotic oscillators. While both oscillators are autonomous, one is 3-D and the other is 4-D. This enables us to investigate and compare the advantages/disadvantages of higher dimensional chaotic oscillators on the throughput, hardware requirements and security of the generated bit-streams. For that purpose, two different implementations are described for each TRBG; a throughput-optimized architecture and a resource-optimized architecture that utilizes fewer FPGA blocks. In both cases, high speed is achieved by concatenating all state-space variables of each chaos generator into one variable. Furthermore, a new postprocessing method that enables the generated bit-streams to pass all NIST 800.22 statistical tests is introduced. Experimental results show that the throughput-optimized TRBG architecture, based on the 4-D system, can exceed 1882 Mbit/s. However, the resource-optimized TRBG architecture, based on the 3-D system, is the best in terms of FPGA resources and overall Figure of Merit.

Quick recovery of the information signals in secure communications restricts the hacking duration. The short synchronization convergence time is a crucial parameter for faster recovery. This paper develops a novel nonlinear finite-time synchronization control algorithm. This controller accomplishes the global finite-time multi-switching synchronization between two externally perturbed chaotic systems in the drive–response system synchronization scheme. The proposed controller assures the global convergence of the error dynamics in finite-time based on the Lyapunov theory. This implicitly guarantees the global stability of the closed loop. This paper considers Lp(0

This paper presents a digitally programmable multi-direction fully integrated chaotic oscillator. Unlike the conventional chaotic oscillators, the proposed digitally programmable multi-direction chaotic oscillator is fully integrated in one single chip, and it achieves lower supply voltage, lower power dissipation and smaller chip area. Moreover, by controlling the digitally programmable MOS switches turning on and off, the presented chaotic oscillator can provide chaotic oscillation in three different directions. The proposed digitally programmable multi-direction fully integrated chaos oscillator is verified with Cadence IC Design Tools. The post-layout simulation results demonstrate that the chaotic oscillator consumes 99.5 mW from ±2.5 V supply voltage, and it takes a compact chip area of 0.177 mm2. The integrated chaos oscillator has a wide range of practical application prospects in chaotic communications or other applications demanding portable chaos systems.