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Prado TL, Corso G, Dos Santos Lima GZ, Budzinski RC, Boaretto BRR, Ferrari FAS, Macau EEN, Lopes SR. Maximum entropy principle in recurrence plot analysis on stochastic and chaotic systems. Chaos 2020; 30:043123. [PMID: 32357677 DOI: 10.1063/1.5125921] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2019] [Accepted: 03/27/2020] [Indexed: 06/11/2023]
Abstract
The recurrence analysis of dynamic systems has been studied since Poincaré's seminal work. Since then, several approaches have been developed to study recurrence properties in nonlinear dynamical systems. In this work, we study the recently developed entropy of recurrence microstates. We propose a new quantifier, the maximum entropy (Smax). The new concept uses the diversity of microstates of the recurrence plot and is able to set automatically the optimum recurrence neighborhood (ϵ-vicinity), turning the analysis free of the vicinity parameter. In addition, ϵ turns out to be a novel quantifier of dynamical properties itself. We apply Smax and the optimum ϵ to deterministic and stochastic systems. The Smax quantifier has a higher correlation with the Lyapunov exponent and, since it is a parameter-free measure, a more useful recurrence quantifier of time series.
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Affiliation(s)
- T L Prado
- Departamento de Física, Universidade Federal do Paraná, Curitiba 81531-980, Brazil
| | - G Corso
- Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal 59078-970, Brazil
| | - G Z Dos Santos Lima
- Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal 59078-970, Brazil
| | - R C Budzinski
- Departamento de Física, Universidade Federal do Paraná, Curitiba 81531-980, Brazil
| | - B R R Boaretto
- Departamento de Física, Universidade Federal do Paraná, Curitiba 81531-980, Brazil
| | - F A S Ferrari
- Instituto de Engenharia, Ciência e Tecnologia, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Janaúba 39447-790, Brazil
| | - E E N Macau
- Laboratório Associado de Computação e Matemática Aplicada, Instituto Nacional de Pesquisas Espaciais, São José dos Campos 12227-010, Brazil
| | - S R Lopes
- Departamento de Física, Universidade Federal do Paraná, Curitiba 81531-980, Brazil
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Reis BMF, Gómez JMR, Pinto TSN, Stekel TRC, Magrini LA, Mendes O, Vieira LEA, Dal Lago A, Cecatto JR, Macau EEN, Palacios J, Domingues MO. Recurrence quantification analysis with wavelet denoising and the characterization of magnetic flux emergence regions in solar photosphere. Phys Rev E 2019; 100:012217. [PMID: 31499878 DOI: 10.1103/physreve.100.012217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Indexed: 06/10/2023]
Abstract
Solar systems complexity, multiscale, and nonlinearity are governed by numerous and continuous changes where the sun magnetic fields can successfully represent many of these phenomena. Thus, nonlinear tools to study these challenging systems are required. The dynamic system recurrence approach has been successfully used to deal with this kind challenge in many scientific areas, objectively improving the recognition of state changes, randomness, and degrees of complexity that are not easily identified by traditional techniques. In this work we introduce the use of these techniques in photospheric magnetogram series. We employ a combination of recurrence quantification analysis with a preprocessing denoising wavelet analysis to characterize the complexity of the magnetic flux emergence in the solar photosphere. In particular, with the developed approach, we identify regions of evolving magnetic flux and where they present a large degree of complexity, i.e., where predictability is low, intermittence is high, and low organization is present.
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Affiliation(s)
- B M F Reis
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - J M Rodríguez Gómez
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - T S N Pinto
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - T R C Stekel
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - L A Magrini
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - O Mendes
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - L E A Vieira
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - A Dal Lago
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - J R Cecatto
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - E E N Macau
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil and Federal University of São Paulo, São José dos Campos, São Paulo 12247-014, Brazil
| | - J Palacios
- Leibniz-Institut für Sonnenphysik (KIS), Freiburg im Breisgau, 79104, Germany
| | - M O Domingues
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
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Noetel J, Freitas VLS, Macau EEN, Schimansky-Geier L. Search and return model for stochastic path integrators. Chaos 2018; 28:106302. [PMID: 30384667 DOI: 10.1063/1.5040108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2018] [Accepted: 06/28/2018] [Indexed: 06/08/2023]
Abstract
We extend a recently introduced prototypical stochastic model describing uniformly the search and return of objects looking for new food sources around a given home. The model describes the kinematic motion of the object with constant speed in two dimensions. The angular dynamics is driven by noise and describes a "pursuit" and "escape" behavior of the heading and the position vectors. Pursuit behavior ensures the return to the home and the escaping between the two vectors realizes exploration of space in the vicinity of the given home. Noise is originated by environmental influences and during decision making of the object. We take symmetric α -stable noise since such noise is observed in experiments. We now investigate for the simplest possible case, the consequences of limited knowledge of the position angle of the home. We find that both noise type and noise strength can significantly increase the probability of returning to the home. First, we review shortly main findings of the model presented in the former manuscript. These are the stationary distance distribution of the noise driven conservative dynamics and the observation of an optimal noise for finding new food sources. Afterwards, we generalize the model by adding a constant shift γ within the interaction rule between the two vectors. The latter might be created by a permanent uncertainty of the correct home position. Nonvanishing shifts transform the kinematics of the searcher to a dissipative dynamics. For the latter, we discuss the novel deterministic properties and calculate the stationary spatial distribution around the home.
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Affiliation(s)
- J Noetel
- Department of Physics, Humboldt-University at Berlin, Newtonstr. 15, D-12489 Berlin, Germany
| | - V L S Freitas
- National Institute for Space Research, 12227-010 Sao Jose dos Campos, Brazil
| | - E E N Macau
- National Institute for Space Research, 12227-010 Sao Jose dos Campos, Brazil
| | - L Schimansky-Geier
- Department of Physics, Humboldt-University at Berlin, Newtonstr. 15, D-12489 Berlin, Germany
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Abstract
We develop a prototypical stochastic model for a local search around a given home. The stochastic dynamic model is motivated by experimental findings of the motion of a fruit fly around a given spot of food but will generally describe the local search behavior. The local search consists of a sequence of two epochs. In the first the searcher explores new space around the home, whereas it returns to the home during the second epoch. In the proposed two-dimensional model both tasks are described by the same stochastic dynamics. The searcher moves with constant speed and its angular dynamics is driven by a symmetric α-stable noise source. The latter stands for the uncertainty to decide the new direction of motion. The main ingredient of the model is the nonlinear interaction dynamics of the searcher with its home. In order to determine the new heading direction, the searcher has to know the actual angles of its position to the home and of the heading vector. A bound state to the home is realized by a permanent switch of a repulsive and attractive forcing of the heading direction from the position direction corresponding to search and return epochs. Our investigation elucidates the analytic tractability of the deterministic and stochastic dynamics. Noise transforms the conservative deterministic dynamics into a dissipative one of the moments. The noise enables a faster finding of a target distinct from the home with optimal intensity. This optimal situation is related to the noise-dependent relaxation time. It is uniquely defined for all α and distinguishes between the stochastic dynamics before and after its value. For times large compared to this, we derive the corresponding Smoluchowski equation and find diffusive spreading of the searcher in the space. We report on the qualitative agreement with the experimentally observed spatial distribution, noisy oscillatory return times, and spatial autocorrelation function of the fruit fly. However, as a result of its simplicity, the model aims to reproduce the local search behavior of other units during their exploration of surrounding space and their quasiperiodic return to a home.
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Affiliation(s)
- J Noetel
- Institute of Physics, Humboldt University at Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - V L S Freitas
- National Institute for Space Research, 12227-010 São José dos Campos, São Paulo, Brazil
| | - E E N Macau
- National Institute for Space Research, 12227-010 São José dos Campos, São Paulo, Brazil.,Federal University of Sao Paulo, 12247-014 São José dos Campos, São Paulo, Brazil
| | - L Schimansky-Geier
- Institute of Physics, Humboldt University at Berlin, Newtonstraße 15, 12489 Berlin, Germany.,Berlin Bernstein Center for Computational Neuroscience, Humboldt University at Berlin, Unter den Linden 6, 10099 Berlin, Germany
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Lameu EL, Yanchuk S, Macau EEN, Borges FS, Iarosz KC, Caldas IL, Protachevicz PR, Borges RR, Viana RL, Szezech JD, Batista AM, Kurths J. Recurrence quantification analysis for the identification of burst phase synchronisation. Chaos 2018; 28:085701. [PMID: 30180612 DOI: 10.1063/1.5024324] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2018] [Accepted: 03/22/2018] [Indexed: 06/08/2023]
Abstract
In this work, we apply the spatial recurrence quantification analysis (RQA) to identify chaotic burst phase synchronisation in networks. We consider one neural network with small-world topology and another one composed of small-world subnetworks. The neuron dynamics is described by the Rulkov map, which is a two-dimensional map that has been used to model chaotic bursting neurons. We show that with the use of spatial RQA, it is possible to identify groups of synchronised neurons and determine their size. For the single network, we obtain an analytical expression for the spatial recurrence rate using a Gaussian approximation. In clustered networks, the spatial RQA allows the identification of phase synchronisation among neurons within and between the subnetworks. Our results imply that RQA can serve as a useful tool for studying phase synchronisation even in networks of networks.
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Affiliation(s)
- E L Lameu
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - S Yanchuk
- Institute of Mathematics, Technical University of Berlin, Berlin 10623, Germany
| | - E E N Macau
- National Institute for Space Research, São José dos Campos, São Paulo 12227-010, Brazil
| | - F S Borges
- Center for Mathematics, Computation, and Cognition, Federal University of ABC, São Bernardo do Campo, São Paulo 09606-045, Brazil
| | - K C Iarosz
- Department of Physics, Humboldt University, Berlin 12489, Germany
| | - I L Caldas
- Institute of Physics, University of São Paulo, São Paulo 05508-900, Brazil
| | - P R Protachevicz
- Program of Post-graduation in Science, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - R R Borges
- Department of Mathematics, Federal University of Technology-Paraná, Ponta Grossa, Paraná 84016-210, Brazil
| | - R L Viana
- Department of Physics, Federal University of Paraná, Curitiba, Paraná 80060-000, Brazil
| | - J D Szezech
- Program of Post-graduation in Science, State University of Ponta Grossa, Ponta Grossa, Paraná 84030-900, Brazil
| | - A M Batista
- Institute of Physics, University of São Paulo, São Paulo 05508-900, Brazil
| | - J Kurths
- Department of Physics, Humboldt University, Berlin 12489, Germany
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Protachevicz PR, Borges RR, Reis AS, Borges FS, Iarosz KC, Caldas IL, Lameu EL, Macau EEN, Viana RL, Sokolov IM, Ferrari FAS, Kurths J, Batista AM, Lo CY, He Y, Lin CP. Synchronous behaviour in network model based on human cortico-cortical connections. Physiol Meas 2018; 39:074006. [DOI: 10.1088/1361-6579/aace91] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Borges FS, Lameu EL, Iarosz KC, Protachevicz PR, Caldas IL, Viana RL, Macau EEN, Batista AM, Baptista MS. Inference of topology and the nature of synapses, and the flow of information in neuronal networks. Phys Rev E 2018; 97:022303. [PMID: 29548150 DOI: 10.1103/physreve.97.022303] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2017] [Indexed: 11/07/2022]
Abstract
The characterization of neuronal connectivity is one of the most important matters in neuroscience. In this work, we show that a recently proposed informational quantity, the causal mutual information, employed with an appropriate methodology, can be used not only to correctly infer the direction of the underlying physical synapses, but also to identify their excitatory or inhibitory nature, considering easy to handle and measure bivariate time series. The success of our approach relies on a surprising property found in neuronal networks by which nonadjacent neurons do "understand" each other (positive mutual information), however, this exchange of information is not capable of causing effect (zero transfer entropy). Remarkably, inhibitory connections, responsible for enhancing synchronization, transfer more information than excitatory connections, known to enhance entropy in the network. We also demonstrate that our methodology can be used to correctly infer directionality of synapses even in the presence of dynamic and observational Gaussian noise, and is also successful in providing the effective directionality of intermodular connectivity, when only mean fields can be measured.
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Affiliation(s)
- F S Borges
- Physics Institute, University of São Paulo, São Paulo, SP 05508-090, Brazil.,Center of Mathematics, Computation and Cognition, Federal University of ABC, São Bernardo do Campo, SP 09606-045, Brazil
| | - E L Lameu
- National Institute for Space Research, São José dos Campos, SP 12227-010, Brazil
| | - K C Iarosz
- Physics Institute, University of São Paulo, São Paulo, SP 05508-090, Brazil.,Institute for Complex Systems and Mathematical Biology, University of Aberdeen, SUPA, AB24 3FX, United Kingdom
| | - P R Protachevicz
- Post-Graduation in Science, State University of Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil
| | - I L Caldas
- Physics Institute, University of São Paulo, São Paulo, SP 05508-090, Brazil
| | - R L Viana
- Physics Department, Federal University of Paraná, Curitiba, PR 81531-980, Brazil
| | - E E N Macau
- National Institute for Space Research, São José dos Campos, SP 12227-010, Brazil.,Federal University of São Paulo, São José dos Campos, SP 12231-280, Brazil
| | - A M Batista
- Physics Institute, University of São Paulo, São Paulo, SP 05508-090, Brazil.,Institute for Complex Systems and Mathematical Biology, University of Aberdeen, SUPA, AB24 3FX, United Kingdom.,Post-Graduation in Science, State University of Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil.,Mathematics and Statistics Department, State University of Ponta Grossa, Ponta Grossa, PR 84030-900, Brazil
| | - M S Baptista
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, SUPA, AB24 3FX, United Kingdom
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Grzybowski JMV, Macau EEN, Yoneyama T. On synchronization in power-grids modelled as networks of second-order Kuramoto oscillators. Chaos 2016; 26:113113. [PMID: 27908000 DOI: 10.1063/1.4967850] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
This work concerns analytical results on the role of coupling strength in the phenomenon of onset of complete frequency locking in power-grids modelled as a network of second-order Kuramoto oscillators. Those results allow estimation of the coupling strength for the onset of complete frequency locking and to assess the features of network and oscillators that favor synchronization. The analytical results are evaluated using an order parameter defined as the normalized sum of absolute values of phase deviations of the oscillators over time. The investigation of the frequency synchronization within the subsets of the parameter space involved in the synchronization problem is also carried out. It is shown that the analytical results are in good agreement with those observed in the numerical simulations. In order to illustrate the methodology, a case study is presented, involving the Brazilian high-voltage transmission system under a load peak condition to study the effect of load on the syncronizability of the grid. The results show that both the load and the centralized generation might have concurred to the 2014 blackout.
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Affiliation(s)
- J M V Grzybowski
- UFFS-Federal University of Fronteira Sul, Rodovia RS 135 km 72, CEP 99.700-000 Erechim, Rio Grande do Sul, Brazil
| | - E E N Macau
- INPE-National Institute for Space Research, Av. dos Astronautas, 1758, Jd. Granja, CEP 12.227-010 São José dos Campos, Sao Paulo, Brazil
| | - T Yoneyama
- ITA-Aeronautics Institute of Technology, Pça Marechal Eduardo Gomes, 50, Vila das Acácias, CEP 12.228-615 São José dos Campos, Sao Paulo, Brazil
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Grzybowski JMV, Macau EEN, Yoneyama T. On the formulation and solution of the isochronal synchronization stability problem in delay-coupled complex networks. Chaos 2012; 22:033152. [PMID: 23020491 DOI: 10.1063/1.4753921] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We present a new framework to the formulation of the problem of isochronal synchronization for networks of delay-coupled oscillators. Using a linear transformation to change coordinates of the network state vector, this method allows straightforward definition of the error system, which is a critical step in the formulation of the synchronization problem. The synchronization problem is then solved on the basis of Lyapunov-Krasovskii theorem. Following this approach, we show how the error system can be defined such that its dimension can be the same as (or smaller than) that of the network state vector.
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Affiliation(s)
- J M V Grzybowski
- Technological Institute of Aeronautics-ITA, Praça Marechal Eduardo Gomes, 50, Vila das Acácias, São José dos Campos, SP CEP 12.228-900, Brazil.
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Campanharo ASLO, Ramos FM, Macau EEN, Rosa RR, Bolzan MJA, Sá LDA. Searching chaos and coherent structures in the atmospheric turbulence above the Amazon forest. Philos Trans A Math Phys Eng Sci 2008; 366:579-89. [PMID: 17698463 DOI: 10.1098/rsta.2007.2118] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
In this work, the possible chaotic nature of the atmospheric turbulence above a densely forested area in the Amazon region is investigated. To this end, we use high-resolution temperature data obtained during a micrometeorological measurement campaign in the Brazilian Amazonia. Estimates of the correlation dimension (D(2)=3.50+/-0.05) and of the largest Lyapunov exponent (lambda(1)=0.050+/-0.002) suggest the existence of chaos in the atmospheric boundary layer. Our findings indicate that this low-dimensional chaotic dynamics is associated with the presence of the coherent structures within the boundary layer right above the canopy top and not with the atmospheric turbulence per se, as previously claimed.
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Affiliation(s)
- A S L O Campanharo
- Instituto Nacional de Pesquisas Espaciais (INPE), 12227-010 São José dos Campos, São Paulo, Brazil.
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