Edmilson Roque

We present a numerical approach for approximating unknown Hamiltonian systems using observation data. A distinct feature of the proposed method is that it is structure-preserving, in the sense that it enforces conservation of the reconstructed Hamiltonian. This is achieved by directly approximating the underlying unknown Hamiltonian, rather than the right-hand-side of the governing equations. We present the technical details of the proposed algorithm and its error estimate in a special case, along with a practical de-noising procedure to cope with noisy data. A set of numerical examples are then presented to demonstrate the structure-preserving property and effectiveness of the algorithm.

We are interested in low-dimensional dynamics in an ensemble of coupled nonidentical generalized oscillators on the 3-sphere. The system of governing equations for such an ensemble is referred to as non-Abelian Kuramoto model in the literature. We establish an analogue (or an extension) of the Ott-Antonsen (OA) result for this model.

The olfactory system is constantly solving pattern-recognition problems by the creation of a large space to codify odour representations and optimizing their distribution within it. A model of the Olfactory Bulb was developed by Z. Li and J. J. Hopfield Li and Hopfield (1989) based on anatomy and electrophysiology. They used nonlinear simulations observing that the collective behavior produce an oscillatory frequency. Here, we show that the Subthreshold hopf bifurcation is a good candidate for modeling the bulb and the Subthreshold subcritical hopf bifurcation is a good candidate for modeling the olfactory cortex. Network topology analysis of the subcritical regime is presented as a proof of the importance of synapse plasticity for memory functions in the olfactory cortex.

We develop a general theoretical framework of semiclassical phase reduction for analyzing synchronization of quantum limit-cycle oscillators. The dynamics of quantum dissipative systems exhibiting limit-cycle oscillations are reduced to a simple, one-dimensional classical stochastic differential equation approximately describing the phase dynamics of the system under the semiclassical approximation. The density matrix and power spectrum of the original quantum system can be approximately reconstructed from the reduced phase equation. The developed framework enables us to analyze synchronization dynamics of quantum limit-cycle oscillators using the standard methods for classical limit-cycle oscillators in a quantitative way. As an example, we analyze synchronization of a quantum van der Pol oscillator under harmonic driving and squeezing, including the case that the squeezing is strong and the oscillation is asymmetric. The developed framework provides insights into the relation between quantum and classical synchronization and will facilitate systematic analysis and control of quantum nonlinear oscillators.

SIAM Journal on Applied Dynamical Systems, Volume 18, Issue 2, Page 904-938, January 2019.
We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains $D\subset \mathbb{R}^{d}$ of Euclidean space where $d\in \mathbb{N}^{+}$ is arbitrary, and are subject to Neumann boundary conditions. We assume that the elliptic part of the parabolic operator in the equations is a self-adjoint Schr\"odinger operator,bounded from below and with compact resolvent in $L^{2}(D)$. The statistical operators we consider are then trace-class operators defined from sequences of probabilities associated with the point spectrum of the elliptic part in question, which allow the distinction between pure and mixed processes. We prove in particular that the Bernstein processes of maximal entropy are those for which the associated sequences of probabilities are of Gibbs type. We illustrate our results by considering processes associated with a specific hierarchy of forward-backward heat equations defined in a two-dimensional disk.

Author(s): Gabriela Petrungaro, Koichiro Uriu, and Luis G. Morelli

Individual biological oscillators can synchronize to generate a collective rhythm. During vertebrate development, mobile cells exchange signals to synchronize a rhythmic pattern generator that makes the embryonic segments. Previous theoretical works have shown that cell mobility can enhance synchron...

[Phys. Rev. E] Published Thu May 16, 2019

Author(s): Pau Clusella and Antonio Politi

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve 𝒞, thereby maintaining the typical ordering of (identical) phase oscillators. This is achieved by developing a general formalism based on two partial differenti...

[Phys. Rev. E] Published Fri May 17, 2019

Scientists are generally subject to social pressures, including pressures to conform with others in their communities, that affect achievement of their epistemic goals. Here we analyze a network epistemology model in which agents, all else being equal, prefer to take actions that conform with those of their neighbors. This preference for conformity interacts with the agents' beliefs about which of two (or more) possible actions yields the better outcome. We find a range of possible outcomes, including stable polarization in belief and action. The model results are sensitive to network structure. In general, though, conformity has a negative effect on a community's ability to reach accurate consensus about the world.

We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of "real" networks obtained from preferential attachment. For $\beta\ge 2$ ($\beta$ is the number of parent nodes in the preferential attachment scheme) the networks obtained with the configuration model are completely connected (giant component equal to 100%). In both generation schemes a clear disassortativity of the small degree nodes is demonstrated from the computation of the function $k_{nn}$. We also develop an efficient rewiring method which produces tunable variations of the assortativity coefficient $r$, and we use it to obtain maximally disassortative networks having the same degree distribution of BA networks with given $\beta$. Possible applications of this method concern assortative social networks.

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the $L^2$ space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on $N$ states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as $ N \to \infty $, in which case the corresponding Fourier frequency is also a Koopman eigenfrequency. If such an RKHS extension does not exist, we can still construct an $L^2$ approximation of the eigenfunction. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.

The present work is concerned with the eqiucontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts. Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs are sensitive. That yields to different examples of non-minimal sensitive systems which are not an M-system. Moreover, some interesting examples are given which provide some facts about the sensitive property of IFSs.

The theorem of Le Calvez and Yoccoz states that there are no minimal homeomorphisms on the finite punctered 2-dimensional sphere S 2 . We show that this does not hold for other surfaces. Moreover, we discuss why the fast-conjugation-method fails in the most cases to construct such homeomorphisms. This article based on an old unpublished article (Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits) with incorrect results.

Author(s): Antonio J. Fontenele, Nivaldo A. P. de Vasconcelos, Thaís Feliciano, Leandro A. A. Aguiar, Carina Soares-Cunha, Bárbara Coimbra, Leonardo Dalla Porta, Sidarta Ribeiro, Ana João Rodrigues, Nuno Sousa, Pedro V. Carelli, and Mauro Copelli

Researchers observe a critical point—a feature indicative of a continuous phase transition—in the brain’s electrical activity as it switches from an asleep-like to an awake-like state.

[Phys. Rev. Lett. 122, 208101] Published Tue May 21, 2019

We study random products of matrices in SL_2(C) from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new proof of the Central Limit Theorem for the norm cocycle under a second moment condition, originally due to Benoist-Quint, and obtain some general regularity results for stationary measures.

In this paper, we study Markovian random iterations of maps on standard measurable spaces. We establish a one-to-one correspondence between stationary measures and a certain class of invariant measures of a Markovian random iteration, extending a similar classical result of independent and identically distributed random iterations. As an application, we prove a local synchronization property for Markovian random iterations of homeomorphisms of the circle $S^{1}$.

We consider the inverse source problem of determining a source term depending on both time and space variable for fractional and classical diffusion equations in a cylindrical domain from boundary measurements. With suitable boundary conditions we prove that some class of source terms which are independent of one space direction, can be reconstructed from boundary measurements. Actually, we prove that this inverse problem is well-posed. We establish also some results of Lipschitz stability for the recovery of source terms which we apply to the stable recostruction of time-dependent coefficients.

We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$, $f$ is in $C^r(\mathbb{T}^u,\mathbb{R}^d)$ and $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $H^s(\mathbb{T}^u\times \mathbb{R}^d)$, in particular, if $s>\frac{u+d}{2}$ then the density is $C^k$ for every $k<s-\frac{u+d}{2}$. We also exhibit a condition involving $E$ and $C$ under which this tranversality condition is valid for almost every $f$.

We present that, instead of establishing the equations of motion, one can model-freely reveal the dynamical properties of a black-box system using a learning machine. Trained only by a segment of time series of a state variable recorded at present parameters values, the dynamics of the learning machine at different training stages can be mapped to the dynamics of the target system along a particular path in its parameter space, following an appropriate training strategy that monotonously decreases the cost function. This path is important, because along that, the primary dynamical properties of the target system will subsequently emerge, in the simple-to-complex order, matching closely the evolution law of certain self-evolved systems in nature. Why such a path can be reproduced is attributed to our training strategy. This particular function of the learning machine opens up a novel way to probe the global dynamical properties of a black-box system without artificially establish the equations of motion, and as such it might have countless applications. As an example, this method is applied to infer what dynamical stages a variable star has experienced and how it will evolve in future, by using the light curve observed presently.

We investigate the existence and regularity of (locally) invariant manifolds nearby an approximately invariant set satisfying certain (geometric) hyperbolicity with respect to an abstract `generalized' dynamical system in a Banach space; such hyperbolicity is between normal hyperbolicity and partial hyperbolicity which has being studied in the finite-dimension and in some concrete PDEs. The `generalized' dynamical system is allowed to be non-smooth, non-Lipschitz, or even `non-mapping', making it applicable to both well-posed and ill-posed differential equations. As an illustration, we apply our results to study the dynamics of the whiskered tori.

We consider the Kuramoto model on sparse random networks such as the Erdős–Rényi graph or its combination with a regular two-dimensional lattice and study the dynamical scaling behavior of the model at the synchronization transition by large-scale, massively parallel numerical integration. By this method, we obtain an estimate of critical coupling strength more accurate than obtained earlier by finite-size scaling of the stationary order parameter. Our results confirm the compatibility of the correlation-size and the temporal correlation-length exponent with the mean-field universality class. However, the scaling of the order parameter exhibits corrections much stronger than those of the Kuramoto model with all-to-all coupling, making thereby an accurate estimate of the order-parameter exponent hard. We find furthermore that, as a qualitative difference to the model with all-to-all coupling, the effective critical exponents involving the order-parameter exponent, such as the eff...

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm.

Let $(A_m)_{m\in \Z}$ be a sequence of bounded linear maps acting on an arbitrary Banach space $X$ and admitting an exponential trichotomy and let $f_m:X\to X$ be a Lispchitz map for every $m\in \Z$. We prove that whenever the Lipschitz constants of $f_m$, $m\in \Z$, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \Z$, has various types of shadowing. Moreover, if $X$ is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results we study the Hyers-Ulam stability for certain difference equations and we obtain a very general version of the Grobman-Hartman's theorem for nonautonomous dynamics.

New thinking needs to emerge about how to reform publishing along lines that best meet two perennial needs of scientific communication. This paper discusses a model that addresses these two needs with respect to physics. Given the considerable barriers that its realization in pristine form faces, the model aspires merely to be a heuristic or guidepost. It provides an analytical framework for criticizing aspects of the current publishing ecosystem, helps diagnose problems in current efforts to reform it, including those emanating from the open access movement, and raises consciousness about certain emphases that could gradually enrich scholarly publishing. [VERSION 2 , 5/19/2019; SUCCESSIVE REVISIONS WILL OCCUR THROUGHOUT 2019.]

Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.

Chimera states, a symmetry-breaking spatiotemporal pattern in nonlocally coupled dynamical units, prevail in a variety of systems. However, the interaction structures among oscillators are static in most of studies on chimera state. In this work, we consider a population of agents. Each agent carries a phase oscillator. We assume that agents perform Brownian motions on a ring and interact with each other with a kernel function dependent on the distance between them. When agents are motionless, the model allows for several dynamical states including two different chimera states (the type-I and the type-II chimeras). The movement of agents changes the relative positions among them and produces perpetual noise to impact on the model dynamics. We find that the response of the coupled phase oscillators to the movement of agents depends on both the phase lag $\alpha$, determining the stabilities of chimera states, and the agent mobility $D$. For low mobility, the synchronous state transits to the type-I chimera state for $\alpha$ close to $\pi/2$ and attracts other initial states otherwise. For intermediate mobility, the coupled oscillators randomly jump among different dynamical states and the jump dynamics depends on $\alpha$. We investigate the statistical properties in these different dynamical regimes and present the scaling laws between the transient time and the mobility for low mobility and relations between the mean lifetimes of different dynamical states and the mobility for intermediate mobility.

Author(s): Franz Kaiser and Karen Alim

Phase balanced states are a highly underexplored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phas...

[Phys. Rev. E 99, 052308] Published Thu May 23, 2019

Author(s): Zhiwei He, Chenggui Yao, Jun Yu, and Meng Zhan

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pu...

[Phys. Rev. E 99, 052207] Published Mon May 13, 2019

We analyze the characteristics of the global attractor of a type of dissipative nonautonomous dynamical systems in terms of the Sacker and Sell spectrum of its linear part. The model gives rise to a pattern of nonautonomous Hopf bifurcation which can be understood as a generalization of the classical autonomous one. We pay special attention to the dynamics at the bifurcation point, showing the possibility of occurrence of Li-Yorke chaos in the corresponding attractor and hence of a high degree of unpredictability.

Understanding biological network dynamics is a fundamental issue in various scientific and engineering fields. Network theory is capable of revealing the relationship between elements and their propagation; however, for complex collective motions, the network properties often transiently and complexly change. A fundamental question addressed here pertains to the classification of collective motion network based on physically-interpretable dynamical properties. Here we apply a data-driven spectral analysis called graph dynamic mode decomposition, which obtains the dynamical properties for collective motion classification. Using a ballgame as an example, we classified the strategic collective motions in different global behaviours and discovered that, in addition to the physical properties, the contextual node information was critical for classification. Furthermore, we discovered the label-specific stronger spectra in the relationship among the nearest agents, providing physical and semantic interpretations. Our approach contributes to the understanding of principles of biological complex network dynamics from the perspective of nonlinear dynamical systems.