Edmilson Roque

We consider the coupling between two networks, each having N nodes whose individual dynamics is modeled by a two-state master equation. The intra-network interactions are all-to-all, whereas the inter-network interactions involve only a small percentage of the total number of nodes. We demonstrate that the dynamics of the mean field for a single network has an equivalent description in terms of a Langevin equation for a particle in a double-well potential. The coupling of two networks or equivalent coupling of two Langevin equations demonstrates synchronization or anti-synchronization between two systems, depending on the sign of the interaction. The anti-synchronized behavior is explained in terms of the potential function and the inter-network interaction. The relative entropy is used to establish that the conditions for maximum information transfer between the networks are consistent with the principle of complexity management and occurs when one system is near the cri...

Author(s): N. Sherborne, K. B. Blyuss, and I. Z. Kiss

The spread of an infectious disease is known to change people's behavior, which in turn affects the spread of disease. Adaptive network models that account for both epidemic and behavioral change have found oscillations, but in an extremely narrow region of the parameter space, which contrasts with ...

[Phys. Rev. E 97, 042306] Published Mon Apr 09, 2018

Author(s): Huawei Fan, Yafeng Wang, Kai Yang, and Xingang Wang

In improving the stability of complex dynamical systems, an outstanding problem is how to achieve the desired performance at a low cost. For engineering and biological complex systems whose performance and functionality rely on the synchronous motion of their units, an important question related to ...

[Phys. Rev. E] Published Fri Mar 29, 2019

Author(s): Wanzhou Zhang, Jiayu Liu, and Tzu-Chieh Wei

In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We fin...

[Phys. Rev. E 99, 032142] Published Fri Mar 29, 2019

Author(s): Jean-Régis Angilella

The dynamics of a weakly dissipative Hamiltonian system submitted to stochastic perturbations has been investigated by means of asymptotic methods. The probability of noise-induced separatrix crossing, which drastically changes the fate of the system, is derived analytically in the case where noise ...

[Phys. Rev. E 99, 032224] Published Fri Mar 29, 2019

SIAM Journal on Applied Dynamical Systems, Volume 18, Issue 1, Page 458-480, January 2019.
Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have an identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted, and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely, acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted, and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted, and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.

SIAM Journal on Applied Dynamical Systems, Volume 18, Issue 1, Page 558-593, January 2019.
This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and overfitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of overspecified EDMD systems in feature space. Nonlinear reconstruction using partially linear multikernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto--Sivashinsky equation.

A reservoir computer is a complex nonlinear dynamical system that has been shown to be useful for solving certain problems, such as prediction of chaotic signals, speech recognition or control of robotic systems. Typically a reservoir computer is constructed by connecting a large number of nonlinear nodes in a network, driving the nodes with an input signal and using the node outputs to fit a training signal. In this work, we set up reservoirs where the edges (or connections) between all the network nodes are either +1 or 0, and proceed to alter the network structure by flipping some of these edges from +1 to -1. We use this simple network because it turns out to be easy to characterize; we may use the fraction of edges flipped as a measure of how much we have altered the network. In some cases, the network can be rearranged in a finite number of ways without changing its structure; these rearrangements are symmetries of the network, and the number of symmetries is also useful for characterizing the network. We find that changing the number of edges flipped in the network changes the rank of the covariance of a matrix consisting of the time series from the different nodes in the network, and speculate that this rank is important for understanding the reservoir computer performance.

Author(s): Hunter Gabbard, Michael Williams, Fergus Hayes, and Chris Messenger

Machine learning may improve the efficiency of the computational methods used to extract gravitational wave signals from data pipelines.

[Phys. Rev. Lett. 120, 141103] Published Fri Apr 06, 2018

Author(s): Justin P. Coon, Carl P. Dettmann, and Orestis Georgiou

We analyze complexity in spatial network ensembles through the lens of graph entropy. Mathematically, we model a spatial network as a \clr{\emph{soft} random geometric graph, i.e., a graph with two sources of randomness, namely nodes located randomly in space and links formed independently between p...

[Phys. Rev. E] Published Fri Apr 06, 2018

We consider discrete-time one-dimensional random dynamical systems with bounded noise, which generate an associated set-valued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set-valued dynamical system in terms of the derivatives of the so-called extremal maps. We propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system. We demonstrate that the derivative reconstructed for different parameters can be used as an early-warning signal to detect an upcoming bifurcation, and apply the algorithm to the bifurcation analysis of the stochastic return map of the Koper model, which is a three-dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed-mode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.

In the last years, Bitcoins and Blockchain technologies are gathering a wide attention from different scientific communities. Notably, thanks to widespread industrial applications and to the continuous introduction of cryptocurrencies, even the public opinion is increasing its attention towards this field. The underlying structure of these technologies constitutes one of their core concepts. In particular, they are based on peer-to-peer networks. Accordingly, all nodes lay at the same level, so that there is no place for privileged actors as, for instance, banking institutions in classical financial networks. In this work, we perform a preliminary investigation on two networks, i.e. the Bitcoin network and the Bitcoin Cash network. Notably, we aim to analyze their global structure and to evaluate if they are provided with a small-world behavior. Results suggest that the principle known as 'fittest-gets-richer', combined with a continuous increasing of connections, might constitute the mechanism leading these networks to reach their current structure. In addition, further observations open the way to new investigations into this direction.

Previous works suggest that musical networks often present the scale-free and the small-world properties. From a musician's perspective, the most important aspect missing in those studies was harmony. In addition to that, the previous works made use of outdated statistical methods. Traditionally, least-squares linear regression is utilised to fit a power law to a given data set. However, according to Clauset et al . such a traditional method can produce inaccurate estimates for the power law exponent. In this paper, we present an analysis of musical networks which considers the existence of chords (an essential element of harmony). Here we show that only 52.5% of music in our database presents the scale-free property, while 62.5% of those pieces present the small-world property. Previous works argue that music is highly scale-free; consequently, it sounds appealing and coherent. In contrast, our results show that not all pieces of music present the scale-free and the small...

Capturing both the structural and temporal aspects of interactions is crucial for many real world datasets like contact between individuals. Using the link stream formalism to capture the dynamic of the systems, we tackle the issue of activity prediction in link streams, that is to say predicting the number of links occurring during a given period of time and we present a protocol that takes advantage of the temporal and structural information contained in the link stream. Using a supervised learning method, we are able to model the dynamic of our system to improve the prediction. We investigate the behavior of our algorithm and crucial elements affecting the prediction. By introducing different categories of pair of nodes, we are able to improve the quality as well as increase the diversity of our prediction.

Author(s): Filippo Radicchi and Claudio Castellano

Many real-world systems are characterized by stochastic dynamical rules where a complex network of interactions among individual elements probabilistically determines their state. Even with full knowledge of the network structure and of the stochastic rules, the ability to predict system configurati...

[Phys. Rev. Lett.] Published Wed Apr 04, 2018

We study cluster synchronization in networks and show that the stability of all possible cluster synchronization patterns depends on a small set of Lyapunov exponents. Our approach can be applied to clusters corresponding to both orbital partitions of the network nodes (symmetry-cluster synchronization) and equitable partitions of the network nodes (input-cluster synchronization.) Our results are verified experimentally in networks of coupled opto-electronic oscillators.

We apply the phase-reduction analysis to examine synchronization properties of periodic fluid flows. The dynamics of unsteady flows are described in terms of the phase dynamics reducing the high-dimensional fluid flow to its single scalar phase variable. We characterize the phase response to impulse perturbations, which can in turn quantify the influence of periodic perturbations on the unsteady flow. These insights from the phase-based analysis uncover the condition for synchronization. In the present work, we study as an example the influence of periodic external forcing on unsteady cylinder wake. The condition for synchronization is identified and agrees closely with results from direct numerical simulations. Moreover, the analysis reveals the optimal forcing direction for synchronization. The phase-response analysis holds potential to uncover lock-on characteristics for a range of periodic flows.

In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams-Bashforth-Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams-Bashforth-Moulton method, is utilized. Four representative examples are considered.

Quantum theory provides an extensive framework for the description of the equilibrium properties of quantum matter. Yet experiments in quantum simulators have now opened up a route towards the generation of quantum states beyond this equilibrium paradigm. While these states promise to show properties not constrained by equilibrium principles, such as the equal a priori probability of the microcanonical ensemble, identifying the general properties of nonequilibrium quantum dynamics remains a major challenge, especially in view of the lack of conventional concepts such as free energies. The theory of dynamical quantum phase transitions attempts to identify such general principles by lifting the concept of phase transitions to coherent quantum real-time evolution. This review provides a pedagogical introduction to this field. Starting from the general setting of nonequilibrium dynamics in closed quantum many-body systems, we give the definition of dynamical quantum phase tra...

Consider a dynamical system ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn001.gif] given by ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn002.gif] ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn003.gif] , where E is a linear expanding map of ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn004.gif] , C is a linear contracting map of ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn005.gif] and f is in ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn006.gif] . We provide sufficient conditions for E that imply the existence of an open set ##IMG## [http://ej.iop.org/images/0951-7715/31/5/2057/nonaaaaf6ieqn007.gif] of pairs ##IMG## [http://ej.iop.or...]

Author(s): Alberto Saa

In the light of the recently proposed scenario of asymmetry-induced synchronization (AISync), in which dynamical uniformity and consensus in a distributed system would demand certain asymmetries in the underlying network, we investigate here the influence of some regularities in the interlayer conne...

[Phys. Rev. E 97, 042304] Published Thu Apr 05, 2018

A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling.

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott-Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.

A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual synchronization between a pair of collectively oscillating networks of excitable and oscillatory FitzHugh-Nagumo elements with random coupling is studied.

We consider the issue of deriving superintegrable systems with position dependent mass (PDM) in two dimensions from certain known superintegrable systems using the recently introduced method of master symmetries and complex factorization by M. Ranada \cite{Rana1,Rana2,Rana3,Rana4}. We introduce a noncanonical transformation to map the Hamiltonian of the PDM systems to that of ordinary unit mass systems. We observe a duality between these systems. We also study Tsiganov's method \cite{Tsiganov1,Tsiganov2,Tsiganov3,Tsiganov4} to derive polynomial integrals of motion using addition theorems for the action-angle variables using famous Chebyshev's theorem on binomial differentials. We compare Tsiganov's method of generating an additional integral of motion with that of Ranada's master symmetry method.

By a small-size complex network of coupled chaotic Hindmarsh-Rose circuits, we study experimentally the stability of network synchronization to the removal of shortcut links. It is shown that the removal of a single shortcut link may destroy either completely or partially the network synchronization. Interestingly, when the network is partially desynchronized, it is found that the oscillators can be organized into different groups, with oscillators within each group being highly synchronized but are not for oscillators from different groups, showing the intriguing phenomenon of cluster synchronization. The experimental results are analyzed by the method of eigenvalue analysis, which implies that the formation of cluster synchronization is crucially dependent on the network symmetries. Our study demonstrates the observability of cluster synchronization in realistic systems, and indicates the feasibility of controlling network synchronization by adjusting network topology.

Author(s): Vanesa Avalos-Gaytán, Juan A. Almendral, I. Leyva, F. Battiston, V. Nicosia, V. Latora, and S. Boccaletti

Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by ex...

[Phys. Rev. E 97, 042301] Published Wed Apr 04, 2018

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$.

We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation $f$ of $f_0$ with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

Synchronisation and stability under periodic oscillatory driving are well-understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counter-intuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronisation where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilisation phenomenon is numerically observed. Our findings help support the case that in general, deterministic non-autonomous perturbation is a very good candidate for stabilising complex dynamics.

Author(s): Alessio Lerose, Jamir Marino, Bojan Žunkovič, Andrea Gambassi, and Alessandro Silva

We investigate the robustness of a dynamical phase transition against quantum fluctuations by studying the impact of a ferromagnetic nearest-neighbor spin interaction in one spatial dimension on the nonequilibrium dynamical phase diagram of the fully connected quantum Ising model. In particular, we ...

[Phys. Rev. Lett. 120, 130603] Published Fri Mar 30, 2018