Special Issue "Complex Systems, Non-Equilibrium Dynamics and Self-Organisation"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (31 May 2017).

Special Issue Editor

Guest Editor
Dr. Gunnar Pruessner Website E-Mail
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
Interests: non-equilibrium statistical mechanics; reaction-diffusion processes; self-organised criticality; complexity; biological physics

Special Issue Information

Dear Colleagues,

Over the last two decades or so, the notion of complex systems has found its way into many different areas of science and humanities, allowing for a quantitative understanding of phenomena that were traditionally studied in a more qualitative fashion. A particularly attractive aspect of complex systems is the emergence of co-operative phenomena, or self-organisation, often driven by non-equilibrium dynamics that relies on an external (energy) source. Such systems seem to be all around us, and govern and represent all that we do and are.

Particular interest in self-organisation and non-equilibrium systems in the form of "active matter" has been generated within the biological sciences with the continued emphasis of more quantitative methods. Pattern or tissue formation may be a particularly good example of a phenomenon suitable for the present issue. Other good examples may be entropy production in sociological and financial systems or recent developments in self-organised criticality.

I would like to invite you to contribute to a Special Issue of Entropy on "Complex Systems, Non-Equilibrium Dynamics and Self-Organisation". The title is deliberately broad and I would hope to gather together a broad spectrum of contributions concerned with complex or non-equilibrium dynamics resulting in some form of organisation, order, pattern or continued entropy production.

Dr. Gunnar Pruessner
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Complexity
  • Non-equilibrium
  • Self-organisation
  • Active matter
  • Entropy-production
  • Shanon-entropy
  • Fluctuation and Correlation
  • Pattern formation
  • Self-organised Criticality

Published Papers (20 papers)

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Research

Open AccessArticle
Complex and Entropy of Fluctuations of Agent-Based Interacting Financial Dynamics with Random Jump
Entropy 2017, 19(10), 512; https://doi.org/10.3390/e19100512 - 23 Sep 2017
Cited by 3
Abstract
This paper investigates the complex behaviors and entropy properties for a novel random complex interacting stock price dynamics, which is established by the combination of stochastic contact process and compound Poisson process, concerning with stock return fluctuations caused by the spread of investors’ [...] Read more.
This paper investigates the complex behaviors and entropy properties for a novel random complex interacting stock price dynamics, which is established by the combination of stochastic contact process and compound Poisson process, concerning with stock return fluctuations caused by the spread of investors’ attitudes and random jump fluctuations caused by the macroeconomic environment, respectively. To better understand the fluctuation complex behaviors of the proposed price dynamics, the entropy analyses of random logarithmic price returns and corresponding absolute returns of simulation dataset with different parameter set are preformed, including permutation entropy, fractional permutation entropy, sample entropy and fractional sample entropy. We found that a larger λ or γ leads to more complex dynamics, and the absolute return series exhibit lower complex dynamics than the return series. To verify the rationality of the proposed compound price model, the corresponding analyses of actual market datasets are also comparatively preformed. The empirical results verify that the proposed price model can reproduce some important complex dynamics of actual stock markets to some extent. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Coupled Effects of Turing and Neimark-Sacker Bifurcations on Vegetation Pattern Self-Organization in a Discrete Vegetation-Sand Model
Entropy 2017, 19(9), 478; https://doi.org/10.3390/e19090478 - 08 Sep 2017
Abstract
Wind-induced vegetation patterns were proposed a long time ago but only recently a dynamic vegetation-sand relationship has been established. In this research, we transformed the continuous vegetation-sand model into a discrete model. Fixed points and stability analyses were then studied. Bifurcation analyses are [...] Read more.
Wind-induced vegetation patterns were proposed a long time ago but only recently a dynamic vegetation-sand relationship has been established. In this research, we transformed the continuous vegetation-sand model into a discrete model. Fixed points and stability analyses were then studied. Bifurcation analyses are done around the fixed point, including Neimark-Sacker and Turing bifurcation. Then we simulated the parameter space for both bifurcations. Based on the bifurcation conditions, simulations are carried out around the bifurcation point. Simulation results showed that Neimark-Sacker bifurcation and Turing bifurcation can induce the self-organization of complex vegetation patterns, among which labyrinth and striped patterns are the key results that can be presented by the continuous model. Under the coupled effects of the two bifurcations, simulation results show that vegetation patterns can also be self-organized, but vegetation type changed. The type of the patterns can be Turing type, Neimark-Sacker type, or some other special type. The difference may depend on the relative intensity of each bifurcation. The calculation of entropy may help understand the variance of pattern types. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980
Entropy 2017, 19(9), 437; https://doi.org/10.3390/e19090437 - 23 Aug 2017
Cited by 20
Abstract
Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts [...] Read more.
Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical data. A major force in the development and study of long memory was the late Benoit B. Mandelbrot. Here, we discuss the original motivation of the development of long memory and Mandelbrot’s influence on this fascinating field. We will also elucidate the sometimes contrasting approaches to long memory in different scientific communities. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
Open AccessArticle
The Emergence of Hyperchaos and Synchronization in Networks with Discrete Periodic Oscillators
Entropy 2017, 19(8), 413; https://doi.org/10.3390/e19080413 - 16 Aug 2017
Cited by 3
Abstract
In this paper, the emergence of hyperchaos in a network with two very simple discrete periodic oscillators is presented. Uncoupled periodic oscillators may represent, in the crudest and simplest form, periodic oscillators in nature, for example fireflies, crickets, menstrual cycles of women, among [...] Read more.
In this paper, the emergence of hyperchaos in a network with two very simple discrete periodic oscillators is presented. Uncoupled periodic oscillators may represent, in the crudest and simplest form, periodic oscillators in nature, for example fireflies, crickets, menstrual cycles of women, among others. Nevertheless, the emergence of hyperchaos in this kind of real-life network has not been proven. In particular, we focus this study on the emergence of hyperchaotic dynamics, considering that these can be mainly used in engineering applications such as cryptography, secure communications, biometric systems, telemedicine, among others. In order to corroborate that the emerging dynamics are hyperchaotic, some chaos and hyperchaos verification tests are conducted. In addition, the presented hyperchaotic coupled system synchronizes, based on the proposed coupling scheme. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Self-Organized Supercriticality and Oscillations in Networks of Stochastic Spiking Neurons
Entropy 2017, 19(8), 399; https://doi.org/10.3390/e19080399 - 02 Aug 2017
Cited by 4
Abstract
Networks of stochastic spiking neurons are interesting models in the area of theoretical neuroscience, presenting both continuous and discontinuous phase transitions. Here, we study fully-connected networks analytically, numerically and by computational simulations. The neurons have dynamic gains that enable the network to converge [...] Read more.
Networks of stochastic spiking neurons are interesting models in the area of theoretical neuroscience, presenting both continuous and discontinuous phase transitions. Here, we study fully-connected networks analytically, numerically and by computational simulations. The neurons have dynamic gains that enable the network to converge to a stationary slightly supercritical state (self-organized supercriticality (SOSC)) in the presence of the continuous transition. We show that SOSC, which presents power laws for neuronal avalanches plus some large events, is robust as a function of the main parameter of the neuronal gain dynamics. We discuss the possible applications of the idea of SOSC to biological phenomena like epilepsy and Dragon-king avalanches. We also find that neuronal gains can produce collective oscillations that coexist with neuronal avalanches. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Avalanching Systems with Longer Range Connectivity: Occurrence of a Crossover Phenomenon and Multifractal Finite Size Scaling
Entropy 2017, 19(8), 383; https://doi.org/10.3390/e19080383 - 26 Jul 2017
Cited by 1
Abstract
Many out-of-equilibrium systems respond to external driving with nonlinear and self-similar dynamics. This near scale-invariant behavior of relaxation events has been modeled through sand pile cellular automata. However, a common feature of these models is the assumption of a local connectivity, while in [...] Read more.
Many out-of-equilibrium systems respond to external driving with nonlinear and self-similar dynamics. This near scale-invariant behavior of relaxation events has been modeled through sand pile cellular automata. However, a common feature of these models is the assumption of a local connectivity, while in many real systems, we have evidence for longer range connectivity and a complex topology of the interacting structures. Here, we investigate the role that longer range connectivity might play in near scale-invariant systems, by analyzing the results of a sand pile cellular automaton model on a Newman–Watts network. The analysis clearly indicates the occurrence of a crossover phenomenon in the statistics of the relaxation events as a function of the percentage of longer range links and the breaking of the simple Finite Size Scaling (FSS). The more complex nature of the dynamics in the presence of long-range connectivity is investigated in terms of multi-scaling features and analyzed by the Rank-Ordered Multifractal Analysis (ROMA). Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Study on the Business Cycle Model with Fractional-Order Time Delay under Random Excitation
Entropy 2017, 19(7), 354; https://doi.org/10.3390/e19070354 - 12 Jul 2017
Cited by 1
Abstract
Time delay of economic policy and memory property in a real economy system is omnipresent and inevitable. In this paper, a business cycle model with fractional-order time delay which describes the delay and memory property of economic control is investigated. Stochastic averaging method [...] Read more.
Time delay of economic policy and memory property in a real economy system is omnipresent and inevitable. In this paper, a business cycle model with fractional-order time delay which describes the delay and memory property of economic control is investigated. Stochastic averaging method is applied to obtain the approximate analytical solution. Numerical simulations are done to verify the method. The effects of the fractional order, time delay, economic control and random excitation on the amplitude of the economy system are investigated. The results show that time delay, fractional order and intensity of random excitation can all magnify the amplitude and increase the volatility of the economy system. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Chaos Synchronization of Nonlinear Fractional Discrete Dynamical Systems via Linear Control
Entropy 2017, 19(7), 351; https://doi.org/10.3390/e19070351 - 11 Jul 2017
Cited by 12
Abstract
By using a linear feedback control technique, we propose a chaos synchronization scheme for nonlinear fractional discrete dynamical systems. Then, we construct a novel 1-D fractional discrete income change system and a kind of novel 3-D fractional discrete system. By means of the [...] Read more.
By using a linear feedback control technique, we propose a chaos synchronization scheme for nonlinear fractional discrete dynamical systems. Then, we construct a novel 1-D fractional discrete income change system and a kind of novel 3-D fractional discrete system. By means of the stability principles of Caputo-like fractional discrete systems, we lastly design a controller to achieve chaos synchronization, and present some numerical simulations to illustrate and validate the synchronization scheme. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Entropy Characterization of Random Network Models
Entropy 2017, 19(7), 321; https://doi.org/10.3390/e19070321 - 30 Jun 2017
Cited by 3
Abstract
This paper elaborates on the Random Network Model (RNM) as a mathematical framework for modelling and analyzing the generation of complex networks. Such framework allows the analysis of the relationship between several network characterizing features (link density, clustering coefficient, degree distribution, connectivity, etc.) [...] Read more.
This paper elaborates on the Random Network Model (RNM) as a mathematical framework for modelling and analyzing the generation of complex networks. Such framework allows the analysis of the relationship between several network characterizing features (link density, clustering coefficient, degree distribution, connectivity, etc.) and entropy-based complexity measures, providing new insight on the generation and characterization of random networks. Some theoretical and computational results illustrate the utility of the proposed framework. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Laminar-Turbulent Patterning in Transitional Flows
Entropy 2017, 19(7), 316; https://doi.org/10.3390/e19070316 - 29 Jun 2017
Cited by 4
Abstract
Wall-bounded flows experience a transition to turbulence characterized by the coexistence of laminar and turbulent domains in some range of Reynolds number R, the natural control parameter. This transitional regime takes place between an upper threshold R t above which turbulence is [...] Read more.
Wall-bounded flows experience a transition to turbulence characterized by the coexistence of laminar and turbulent domains in some range of Reynolds number R, the natural control parameter. This transitional regime takes place between an upper threshold R t above which turbulence is uniform (featureless) and a lower threshold R g below which any form of turbulence decays, possibly at the end of overlong chaotic transients. The most emblematic cases of flow along flat plates transiting to/from turbulence according to this scenario are reviewed. The coexistence is generally in the form of bands, alternatively laminar and turbulent, and oriented obliquely with respect to the general flow direction. The final decay of the bands at R g points to the relevance of directed percolation and criticality in the sense of statistical-physics phase transitions. The nature of the transition at R t where bands form is still somewhat mysterious and does not easily fit the scheme holding for pattern-forming instabilities at increasing control parameter on a laminar background. In contrast, the bands arise at R t out of a uniform turbulent background at a decreasing control parameter. Ingredients of a possible theory of laminar-turbulent patterning are discussed. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Complex Dynamics of an SIR Epidemic Model with Nonlinear Saturate Incidence and Recovery Rate
Entropy 2017, 19(7), 305; https://doi.org/10.3390/e19070305 - 27 Jun 2017
Cited by 4
Abstract
Susceptible-infectious-removed (SIR) epidemic models are proposed to consider the impact of available resources of the public health care system in terms of the number of hospital beds. Both the incidence rate and the recovery rate are considered as nonlinear functions of the number [...] Read more.
Susceptible-infectious-removed (SIR) epidemic models are proposed to consider the impact of available resources of the public health care system in terms of the number of hospital beds. Both the incidence rate and the recovery rate are considered as nonlinear functions of the number of infectious individuals, and the recovery rate incorporates the influence of the number of hospital beds. It is shown that backward bifurcation and saddle-node bifurcation may occur when the number of hospital beds is insufficient. In such cases, it is critical to prepare an appropriate amount of hospital beds because only reducing the basic reproduction number less than unity is not enough to eradicate the disease. When the basic reproduction number is larger than unity, the model may undergo forward bifurcation and Hopf bifurcation. The increasing hospital beds can decrease the infectious individuals. However, it is useless to eliminate the disease. Therefore, maintaining enough hospital beds is important for the prevention and control of the infectious disease. Numerical simulations are presented to illustrate and complement the theoretical analysis. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Node Importance Ranking of Complex Networks with Entropy Variation
Entropy 2017, 19(7), 303; https://doi.org/10.3390/e19070303 - 26 Jun 2017
Cited by 11
Abstract
The heterogeneous nature of a complex network determines the roles of each node in the network that are quite different. Mechanisms of complex networks such as spreading dynamics, cascading reactions, and network synchronization are highly affected by a tiny fraction of so-called important [...] Read more.
The heterogeneous nature of a complex network determines the roles of each node in the network that are quite different. Mechanisms of complex networks such as spreading dynamics, cascading reactions, and network synchronization are highly affected by a tiny fraction of so-called important nodes. Node importance ranking is thus of great theoretical and practical significance. Network entropy is usually utilized to characterize the amount of information encoded in the network structure and to measure the structural complexity at the graph level. We find that entropy can also serve as a local level metric to quantify node importance. We propose an entropic metric, Entropy Variation, defining the node importance as the variation of network entropy before and after its removal, according to the assumption that the removal of a more important node is likely to cause more structural variation. Like other state-of-the-art methods for ranking node importance, the proposed entropic metric is also used to utilize structural information, but at the systematical level, not the local level. Empirical investigations on real life networks, the Snake Idioms Network, and several other well-known networks, demonstrate the superiority of the proposed entropic metric, notably outperforming other centrality metrics in identifying the top-k most important nodes. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Large Scale Emerging Properties from Non Hamiltonian Complex Systems
Entropy 2017, 19(7), 302; https://doi.org/10.3390/e19070302 - 26 Jun 2017
Cited by 5
Abstract
The concept of “large scale” depends obviously on the phenomenon we are interested in. For example, in the field of foundation of Thermodynamics from microscopic dynamics, the spatial and time large scales are order of fraction of millimetres and microseconds, respectively, or lesser, [...] Read more.
The concept of “large scale” depends obviously on the phenomenon we are interested in. For example, in the field of foundation of Thermodynamics from microscopic dynamics, the spatial and time large scales are order of fraction of millimetres and microseconds, respectively, or lesser, and are defined in relation to the spatial and time scales of the microscopic systems. In large scale oceanography or global climate dynamics problems the time scales of interest are order of thousands of kilometres, for space, and many years for time, and are compared to the local and daily/monthly times scales of atmosphere and ocean dynamics. In all the cases a Zwanzig projection approach is, at least in principle, an effective tool to obtain class of universal smooth “large scale” dynamics for few degrees of freedom of interest, starting from the complex dynamics of the whole (usually many degrees of freedom) system. The projection approach leads to a very complex calculus with differential operators, that is drastically simplified when the basic dynamics of the system of interest is Hamiltonian, as it happens in Foundation of Thermodynamics problems. However, in geophysical Fluid Dynamics, Biology, and in most of the physical problems the building block fundamental equations of motions have a non Hamiltonian structure. Thus, to continue to apply the useful projection approach also in these cases, we exploit the generalization of the Hamiltonian formalism given by the Lie algebra of dissipative differential operators. In this way, we are able to analytically deal with the series of the differential operators stemming from the projection approach applied to these general cases. Then we shall apply this formalism to obtain some relevant results concerning the statistical properties of the El Niño Southern Oscillation (ENSO). Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
Open AccessArticle
Approximation of Stochastic Quasi-Periodic Responses of Limit Cycles in Non-Equilibrium Systems under Periodic Excitations and Weak Fluctuations
Entropy 2017, 19(6), 280; https://doi.org/10.3390/e19060280 - 15 Jun 2017
Cited by 2
Abstract
A semi-analytical method is proposed to calculate stochastic quasi-periodic responses of limit cycles in non-equilibrium dynamical systems excited by periodic forces and weak random fluctuations, approximately. First, a kind of 1/N-stroboscopic map is introduced to discretize the quasi-periodic torus into closed [...] Read more.
A semi-analytical method is proposed to calculate stochastic quasi-periodic responses of limit cycles in non-equilibrium dynamical systems excited by periodic forces and weak random fluctuations, approximately. First, a kind of 1/N-stroboscopic map is introduced to discretize the quasi-periodic torus into closed curves, which are then approximated by periodic points. Using a stochastic sensitivity function of discrete time systems, the transverse dispersion of these circles can be quantified. Furthermore, combined with the longitudinal distribution of the circles, the probability density function of these closed curves in stroboscopic sections can be determined. The validity of this approach is shown through a van der Pol oscillator and Brusselator. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
The Entropy of Words—Learnability and Expressivity across More than 1000 Languages
Entropy 2017, 19(6), 275; https://doi.org/10.3390/e19060275 - 14 Jun 2017
Cited by 12
Abstract
The choice associated with words is a fundamental property of natural languages. It lies at the heart of quantitative linguistics, computational linguistics and language sciences more generally. Information theory gives us tools at hand to measure precisely the average amount of choice associated [...] Read more.
The choice associated with words is a fundamental property of natural languages. It lies at the heart of quantitative linguistics, computational linguistics and language sciences more generally. Information theory gives us tools at hand to measure precisely the average amount of choice associated with words: the word entropy. Here, we use three parallel corpora, encompassing ca. 450 million words in 1916 texts and 1259 languages, to tackle some of the major conceptual and practical problems of word entropy estimation: dependence on text size, register, style and estimation method, as well as non-independence of words in co-text. We present two main findings: Firstly, word entropies display relatively narrow, unimodal distributions. There is no language in our sample with a unigram entropy of less than six bits/word. We argue that this is in line with information-theoretic models of communication. Languages are held in a narrow range by two fundamental pressures: word learnability and word expressivity, with a potential bias towards expressivity. Secondly, there is a strong linear relationship between unigram entropies and entropy rates. The entropy difference between words with and without co-textual information is narrowly distributed around ca. three bits/word. In other words, knowing the preceding text reduces the uncertainty of words by roughly the same amount across languages of the world. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Spurious Results of Fluctuation Analysis Techniques in Magnitude and Sign Correlations
Entropy 2017, 19(6), 261; https://doi.org/10.3390/e19060261 - 07 Jun 2017
Cited by 4
Abstract
Fluctuation Analysis (FA) and specially Detrended Fluctuation Analysis (DFA) are techniques commonly used to quantify correlations and scaling properties of complex time series such as the observable outputs of great variety of dynamical systems, from Economics to Physiology. Often, such correlated time series [...] Read more.
Fluctuation Analysis (FA) and specially Detrended Fluctuation Analysis (DFA) are techniques commonly used to quantify correlations and scaling properties of complex time series such as the observable outputs of great variety of dynamical systems, from Economics to Physiology. Often, such correlated time series are analyzed using the magnitude and sign decomposition, i.e., by using FA or DFA to study separately the sign and the magnitude series obtained from the original signal. This approach allows for distinguishing between systems with the same linear correlations but different dynamical properties. However, here we present analytical and numerical evidence showing that FA and DFA can lead to spurious results when applied to sign and magnitude series obtained from power-law correlated time series of fractional Gaussian noise (fGn) type. Specifically, we show that: (i) the autocorrelation functions of the sign and magnitude series obtained from fGns are always power-laws; However, (ii) when the sign series presents power-law anticorrelations, FA and DFA wrongly interpret the sign series as purely uncorrelated; Similarly, (iii) when analyzing power-law correlated magnitude (or volatility) series, FA and DFA fail to retrieve the real scaling properties, and identify the magnitude series as purely uncorrelated noise; Finally, (iv) using the relationship between FA and DFA and the autocorrelation function of the time series, we explain analytically the reason for the FA and DFA spurious results, which turns out to be an intrinsic property of both techniques when applied to sign and magnitude series. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response
Entropy 2017, 19(6), 258; https://doi.org/10.3390/e19060258 - 07 Jun 2017
Cited by 2
Abstract
The formation of self-organized patterns in predator-prey models has been a very hot topic recently. The dynamics of these models, bifurcations and pattern formations are so complex that studies are urgently needed. In this research, we transformed a continuous predator-prey model with Lesie-Gower [...] Read more.
The formation of self-organized patterns in predator-prey models has been a very hot topic recently. The dynamics of these models, bifurcations and pattern formations are so complex that studies are urgently needed. In this research, we transformed a continuous predator-prey model with Lesie-Gower functional response into a discrete model. Fixed points and stability analyses were studied. Around the stable fixed point, bifurcation analyses including: flip, Neimark-Sacker and Turing bifurcation were done and bifurcation conditions were obtained. Based on these bifurcation conditions, parameters values were selected to carry out numerical simulations on pattern formation. The simulation results showed that Neimark-Sacker bifurcation induced spots, spirals and transitional patterns from spots to spirals. Turing bifurcation induced labyrinth patterns and spirals coupled with mosaic patterns, while flip bifurcation induced many irregular complex patterns. Compared with former studies on continuous predator-prey model with Lesie-Gower functional response, our research on the discrete model demonstrated more complex dynamics and varieties of self-organized patterns. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Meromorphic Non-Integrability of Several 3D Dynamical Systems
Entropy 2017, 19(5), 211; https://doi.org/10.3390/e19050211 - 10 May 2017
Abstract
In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé–Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of [...] Read more.
In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé–Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
Open AccessArticle
Fractional Diffusion in a Solid with Mass Absorption
Entropy 2017, 19(5), 203; https://doi.org/10.3390/e19050203 - 02 May 2017
Cited by 2
Abstract
The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases [...] Read more.
The space-time-fractional diffusion equation with the Caputo time-fractional derivative and Riesz fractional Laplacian is considered in the case of axial symmetry. Mass absorption (mass release) is described by a source term proportional to concentration. The integral transform technique is used. Different particular cases of the solution are studied. The numerical results are illustrated graphically. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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Open AccessArticle
Entropy in Natural Time and the Associated Complexity Measures
Entropy 2017, 19(4), 177; https://doi.org/10.3390/e19040177 - 20 Apr 2017
Cited by 3
Abstract
Natural time is a new time domain introduced in 2001. The analysis of time series associated with a complex system in natural time may provide useful information and may reveal properties that are usually hidden when studying the system in conventional time. In [...] Read more.
Natural time is a new time domain introduced in 2001. The analysis of time series associated with a complex system in natural time may provide useful information and may reveal properties that are usually hidden when studying the system in conventional time. In this new time domain, an entropy has been defined, and complexity measures based on this entropy, as well as its value under time-reversal have been introduced and found applications in various complex systems. Here, we review these applications in the electric signals that precede rupture, e.g., earthquakes, in the analysis of electrocardiograms, as well as in global atmospheric phenomena, like the El Niño/La Niña Southern Oscillation. Full article
(This article belongs to the Special Issue Complex Systems, Non-Equilibrium Dynamics and Self-Organisation)
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