Pub Date : 2012-06-22DOI: 10.3934/DCDSB.2014.19.2809
T. Ma, Shouhong Wang
The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.
{"title":"Dynamic Transition and Pattern Formation for Chemotactic Systems","authors":"T. Ma, Shouhong Wang","doi":"10.3934/DCDSB.2014.19.2809","DOIUrl":"https://doi.org/10.3934/DCDSB.2014.19.2809","url":null,"abstract":"The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131930716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble of coupled planar oscillators. In the limit of a large number of communities, we find a low dimensional description of the level of synchronization between the communities. In this limit, we describe bifurcations between incoherence, local synchrony, and global synchrony. We compare the predictions of this simplified model with simulations of heterogeneous networks in which the internal structure of each community is preserved and find excellent agreement. Finally, we investigate synchronization in networks where several layers of communities within communities may be present.
{"title":"Synchronization of Kuramoto oscillators in networks of networks","authors":"P. S. Skardal, J. Restrepo","doi":"10.15248/PROC.1.171","DOIUrl":"https://doi.org/10.15248/PROC.1.171","url":null,"abstract":"We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble of coupled planar oscillators. In the limit of a large number of communities, we find a low dimensional description of the level of synchronization between the communities. In this limit, we describe bifurcations between incoherence, local synchrony, and global synchrony. We compare the predictions of this simplified model with simulations of heterogeneous networks in which the internal structure of each community is preserved and find excellent agreement. Finally, we investigate synchronization in networks where several layers of communities within communities may be present.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129310140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-11-24DOI: 10.12693/APhysPolA.121.B-28
A. Górski, S. Drożdż, A. Mokrzycka, A. Mokrzycka, J. Pawlik, J. Pawlik
Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. The real computational error was found to have power scaling with respect to the number of data points in the sample ($n_{tot}$). For fractals embedded in two-dimensional space the error is larger than for those embedded in one-dimensional space. For fractal functions the error is even larger. Obtained formula can give more realistic estimates for the computed generalized fractal exponents' accuracy.
{"title":"Accuracy analysis of the box-counting algorithm","authors":"A. Górski, S. Drożdż, A. Mokrzycka, A. Mokrzycka, J. Pawlik, J. Pawlik","doi":"10.12693/APhysPolA.121.B-28","DOIUrl":"https://doi.org/10.12693/APhysPolA.121.B-28","url":null,"abstract":"Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. The real computational error was found to have power scaling with respect to the number of data points in the sample ($n_{tot}$). For fractals embedded in two-dimensional space the error is larger than for those embedded in one-dimensional space. For fractal functions the error is even larger. Obtained formula can give more realistic estimates for the computed generalized fractal exponents' accuracy.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131000976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-08-09DOI: 10.1007/978-1-4614-9161-3_13
R. López-Ruiz, E. Shivanian
{"title":"A Nonlinear Map for the Decay to Equilibrium of Ideal Gases","authors":"R. López-Ruiz, E. Shivanian","doi":"10.1007/978-1-4614-9161-3_13","DOIUrl":"https://doi.org/10.1007/978-1-4614-9161-3_13","url":null,"abstract":"","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126892522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space. Also, several multi-agent economic models based on mappings, with random, deterministic or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution. An alternative approach to this problem in the framework of iterations in the space of distributions has been recently presented. Concretely, the new iteration given by $ f_{n+1}(x) = intint_{u+v>x}{f_n(u)f_n(v)over u+v} dudv.$. It is found that the exponential distribution is a stable fixed point of the former functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution (or by extension, other kind of distributions) is asymptotically obtained in different multi-agent economic models.
{"title":"Exponential wealth distribution: a new approach from functional iteration theory","authors":"R. López-Ruiz, J. López, X. Calbet","doi":"10.1051/PROC/201236015","DOIUrl":"https://doi.org/10.1051/PROC/201236015","url":null,"abstract":"Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space. Also, several multi-agent economic models based on mappings, with random, deterministic or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution. An alternative approach to this problem in the framework of iterations in the space of distributions has been recently presented. Concretely, the new iteration given by $ f_{n+1}(x) = intint_{u+v>x}{f_n(u)f_n(v)over u+v} dudv.$. It is found that the exponential distribution is a stable fixed point of the former functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution (or by extension, other kind of distributions) is asymptotically obtained in different multi-agent economic models.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114841742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the integrate-and-fire model of the cardiac pacemaker with delayed pulsatile coupling. Sufficient conditions of synchronization are obtained for identical and non-identical oscillators.
我们考虑具有延迟脉冲耦合的心脏起搏器的集火模型。得到了同振和非同振同步的充分条件。
{"title":"Synchronization of the cardiac pacemaker model with delayed pulse-coupling","authors":"M. Akhmet","doi":"10.5890/DNC.2014.03.002","DOIUrl":"https://doi.org/10.5890/DNC.2014.03.002","url":null,"abstract":"We consider the integrate-and-fire model of the cardiac pacemaker with delayed pulsatile coupling. Sufficient conditions of synchronization are obtained for identical and non-identical oscillators.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129184814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Kleidon suggested to analyze Gaia as a non-equilibrium thermodynamic system that continuously moves away from equilibrium, driven by maximum entropy production which materializes in hierarchically coupled mechanisms of energetic flows via dissipation and physical work. I relate this view with Kauffman's 'Fourth Law of Thermodynamics', which I interprete as a proposition about the accumulation of information in evolutionary processes. The concept of physical work is expanded to including work directed at the capacity to work: I offer a twofold specification of Kauffman's concept of an 'autonomous agent', one as a 'self-referential heat engine', and the other in terms of physiosemeiosis, which is a naturalized application of Peirce's theory of signs. The conjunction of these three theoretical sources, Maximum Entropy, Kauffman's Fourth Law, and physiosemeiosis, shows that the Kleidon restatement of the Gaia hypothesis is equivalent to the proposition that the biosphere is generating, processing and storing information, thus directly treating information as a physical phenomenon. There is a fundamental ontological continuity between the biological processes and the human economy, as both are seen as information processing and entropy producing systems. Knowledge and energy are not substitutes, with energy and information being two aspects of the same underlying physical process.
{"title":"Revisiting the Gaia Hypothesis: Maximum Entropy, Kauffman's 'Fourth Law' and Physiosemeiosis","authors":"Carsten Herrmann-Pillath","doi":"10.2139/ssrn.1762603","DOIUrl":"https://doi.org/10.2139/ssrn.1762603","url":null,"abstract":"Recently, Kleidon suggested to analyze Gaia as a non-equilibrium thermodynamic system that continuously moves away from equilibrium, driven by maximum entropy production which materializes in hierarchically coupled mechanisms of energetic flows via dissipation and physical work. I relate this view with Kauffman's 'Fourth Law of Thermodynamics', which I interprete as a proposition about the accumulation of information in evolutionary processes. The concept of physical work is expanded to including work directed at the capacity to work: I offer a twofold specification of Kauffman's concept of an 'autonomous agent', one as a 'self-referential heat engine', and the other in terms of physiosemeiosis, which is a naturalized application of Peirce's theory of signs. The conjunction of these three theoretical sources, Maximum Entropy, Kauffman's Fourth Law, and physiosemeiosis, shows that the Kleidon restatement of the Gaia hypothesis is equivalent to the proposition that the biosphere is generating, processing and storing information, thus directly treating information as a physical phenomenon. There is a fundamental ontological continuity between the biological processes and the human economy, as both are seen as information processing and entropy producing systems. Knowledge and energy are not substitutes, with energy and information being two aspects of the same underlying physical process.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130173228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-01-24DOI: 10.1002/9783527639823.CH12
Leonhard Lucken, S. Yanchuk
The subject of this paper is a system of phase-oscillators, which are globally pulse-coupled via excitatory interaction. The appearance and stability of one- and two-cluster-states is investigated for a family of unimodal phase-response-curves (PRC). The PRCs and their derivatives are assumed to be zero at the spiking point. We show that there exist stable homoclinic connections of the one-cluster state for PRCs with the maximum located shortly before the spiking point and coexisting stable two-clusters states when the maximum of the PRC is located shortly after the spike.
{"title":"Emergence of One‐ and Two‐Cluster States in Populations of Globally Pulse‐Coupled Oscillators","authors":"Leonhard Lucken, S. Yanchuk","doi":"10.1002/9783527639823.CH12","DOIUrl":"https://doi.org/10.1002/9783527639823.CH12","url":null,"abstract":"The subject of this paper is a system of phase-oscillators, which are globally pulse-coupled via excitatory interaction. The appearance and stability of one- and two-cluster-states is investigated for a family of unimodal phase-response-curves (PRC). The PRCs and their derivatives are assumed to be zero at the spiking point. We show that there exist stable homoclinic connections of the one-cluster state for PRCs with the maximum located shortly before the spiking point and coexisting stable two-clusters states when the maximum of the PRC is located shortly after the spike.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132293650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-11-12DOI: 10.1142/9789814313155_0006
P. Hovel, S.A.H. Shah, M. Dahlem, E. Scholl
We investigate the synchronization dynamics of two coupled noise-driven FitzHugh-Nagumo systems, representing two neural populations. For certain choices of the noise intensities and coupling strength, we find cooperative stochastic dynamics such as frequency synchronization and phase synchronization, where the degree of synchronization can be quantified by the ratio of the interspike interval of the two excitable neural populations and the phase synchronization index, respectively. The stochastic synchronization can be either enhanced or suppressed by local time-delayed feedback control, depending upon the delay time and the coupling strength. The control depends crucially upon the coupling scheme of the control force, i.e., whether the control force is generated from the activator or inhibitor signal, and applied to either component. For inhibitor self-coupling, synchronization is most strongly enhanced, whereas for activator self-coupling there exist distinct values of the delay time where the synchronization is strongly suppressed even in the strong synchronization regime. For cross-coupling strongly modulated behavior is found.
{"title":"Feedback-dependent control of stochastic synchronization in coupled neural systems","authors":"P. Hovel, S.A.H. Shah, M. Dahlem, E. Scholl","doi":"10.1142/9789814313155_0006","DOIUrl":"https://doi.org/10.1142/9789814313155_0006","url":null,"abstract":"We investigate the synchronization dynamics of two coupled noise-driven FitzHugh-Nagumo systems, representing two neural populations. For certain choices of the noise intensities and coupling strength, we find cooperative stochastic dynamics such as frequency synchronization and phase synchronization, where the degree of synchronization can be quantified by the ratio of the interspike interval of the two excitable neural populations and the phase synchronization index, respectively. The stochastic synchronization can be either enhanced or suppressed by local time-delayed feedback control, depending upon the delay time and the coupling strength. The control depends crucially upon the coupling scheme of the control force, i.e., whether the control force is generated from the activator or inhibitor signal, and applied to either component. For inhibitor self-coupling, synchronization is most strongly enhanced, whereas for activator self-coupling there exist distinct values of the delay time where the synchronization is strongly suppressed even in the strong synchronization regime. For cross-coupling strongly modulated behavior is found.","PeriodicalId":139082,"journal":{"name":"arXiv: Adaptation and Self-Organizing Systems","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121311807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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