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Axenides M, Floratos E, Nicolis S. Arnol'd cat map lattices. Phys Rev E 2023; 107:064206. [PMID: 37464658 DOI: 10.1103/physreve.107.064206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2022] [Accepted: 03/17/2023] [Indexed: 07/20/2023]
Abstract
We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems also can be understood from the equations of motion in configuration space. These describe inverted harmonic oscillators, where the runaway behavior of the potential competes with the toroidal compactification of the phase space. We highlight the spatiotemporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely, the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.
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Affiliation(s)
- Minos Axenides
- Institute for Nuclear and Particle Physics, NCSR "Demokritos", Aghia Paraskevi 15310, Greece
| | - Emmanuel Floratos
- Institute for Nuclear and Particle Physics, NCSR "Demokritos", Aghia Paraskevi 15310, Greece
- Physics Department, University of Athens, Athens, 15771 Greece
| | - Stam Nicolis
- Institut Denis Poisson, Université de Tours, Université d'Orléans, CNRS, Parc Grandmont, 37200 Tours, France
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2
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Feldmann D, Borrero-Echeverry D, Burin MJ, Avila K, Avila M. Routes to turbulence in Taylor-Couette flow. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220114. [PMID: 36907218 DOI: 10.1098/rsta.2022.0114] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2022] [Accepted: 01/06/2023] [Indexed: 06/18/2023]
Abstract
Fluid flows between rotating concentric cylinders exhibit two distinct routes to turbulence. In flows dominated by inner-cylinder rotation, a sequence of linear instabilities leads to temporally chaotic dynamics as the rotation speed is increased. The resulting flow patterns occupy the whole system and sequentially lose spatial symmetry and coherence in the transition process. In flows dominated by outer-cylinder rotation, the transition is abrupt and leads directly to turbulent flow regions that compete with laminar ones. We here review the main features of these two routes to turbulence. Bifurcation theory rationalizes the origin of temporal chaos in both cases. However, the catastrophic transition of flows dominated by outer-cylinder rotation can only be understood by accounting for the spatial proliferation of turbulent regions with a statistical approach. We stress the role of the rotation number (the ratio of Coriolis to inertial forces) and show that it determines the lower border for the existence of intermittent laminar-turbulent patterns. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (Part 2)'.
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Affiliation(s)
- Daniel Feldmann
- University of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359 Bremen, Germany
| | | | - Michael J Burin
- Department of Physics, California State University - San Marcos, San Marcos, CA 92096, USA
| | - Kerstin Avila
- University of Bremen, Faculty of Production Engineering, 28359 Bremen, Germany
- Leibniz Institute for Materials Engineering (IWT), 28359 Bremen, Germany
| | - Marc Avila
- University of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359 Bremen, Germany
- University of Bremen, MAPEX Center for Materials and Processes, 28359 Bremen, Germany
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3
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Haugland SW, Tosolini A, Krischer K. Between synchrony and turbulence: intricate hierarchies of coexistence patterns. Nat Commun 2021; 12:5634. [PMID: 34561462 PMCID: PMC8463560 DOI: 10.1038/s41467-021-25907-7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 09/07/2021] [Indexed: 11/09/2022] Open
Abstract
Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence. The 2002 discovery of so-called chimera states, states of coexisting synchronized and unsynchronized oscillators, provided a possible link between the two and definitely showed that different parts of the same ensemble can sustain qualitatively different forms of motion. Here, we demonstrate that globally coupled identical oscillators can express a range of coexistence patterns more comprehensive than chimeras. A hierarchy of such states evolves from the fully synchronized solution in a series of cluster-splittings. At the far end of this hierarchy, the states further collide with their own mirror-images in phase space - rendering the motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistence patterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asynchronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, are found to be just one step in this transition from low- to high-dimensional dynamics.
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Affiliation(s)
- Sindre W Haugland
- Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748, Garching, Germany
| | - Anton Tosolini
- Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748, Garching, Germany
| | - Katharina Krischer
- Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748, Garching, Germany.
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4
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Mohapatra S, Mondal S, Mahapatra PS. Spatiotemporal dynamics of a self-propelled system with opposing alignment and repulsive forces. Phys Rev E 2020; 102:042613. [PMID: 33212711 DOI: 10.1103/physreve.102.042613] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2020] [Accepted: 10/05/2020] [Indexed: 11/07/2022]
Abstract
Effect of concurrent alignment and repulsion is studied in the purview of a confined active matter system using a modified force-based Vicsek model. On alteration of the alignment and the repulsive force parameters, a low alignment random phase, a midrange alignment milling phase, and a high alignment oscillatory phase are identified. Based on the particle aggregations, the milling phase is further classified into three subphases, two of which are spatial patterns: one consisting of compact ring-shaped mills and the other incorporating both rings and clusters. A correlation function based on the inner product of spatial velocity fluctuations of the particles shows a high correlation length for the ringed milling and the rings-clusters hybrid milling state. On analyzing temporal velocity fluctuations of particles through chaos detection techniques, low alignment and high alignment states are indicative of chaos, while the middle order alignment is symbolic of periodicity. The extent of synchronization of the particles' motion is analyzed through a Hilbert transform-based mean frequency approach, leading to the detection of a weak chimera state in the case of the spatial structures. The ringed milling state shows a unique category of weak chimera consisting of multiple oscillator groups showcasing different synchronization frequencies coexisting with desynchronized oscillators.
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Affiliation(s)
- Siddhant Mohapatra
- Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
| | - Sirshendu Mondal
- Department of Mechanical Engineering, National Institute of Technology Durgapur, Kolkata 713209, India
| | - Pallab Sinha Mahapatra
- Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
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5
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da Costa DR, de Carvalho RE. Dynamics of a light beam suffering the influence of a dispersing mechanism with tunable refraction index. Phys Rev E 2018; 98:022224. [PMID: 30253490 DOI: 10.1103/physreve.98.022224] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2018] [Indexed: 06/08/2023]
Abstract
The dynamics of a monochromatic light beam is studied inside the oval billiard with an inner scatter circle, which can be interpreted as a cross section of a long optical fiber. The outer oval boundary acts as a perfect reflector for the light beam while the scatter circle encloses a medium with changeable refraction index. The light beam refracts when it enters inside this circle and some drastic changes in the phase space are observed. The increase of the refractive index destroys the center of stability in the phase space leading to a spread of scattered light and the islands of traps for the light propagation become more pulverized. Numerical results are presented and discussed.
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Affiliation(s)
- Diogo Ricardo da Costa
- Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas (IGCE), 13506-900 Rio Claro-SP, Brazil
| | - R Egydio de Carvalho
- Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas (IGCE), 13506-900 Rio Claro-SP, Brazil
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6
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Chen DD, Gray RA, Uzelac I, Herndon C, Fenton FH. Mechanism for Amplitude Alternans in Electrocardiograms and the Initiation of Spatiotemporal Chaos. PHYSICAL REVIEW LETTERS 2017; 118:168101. [PMID: 28474934 DOI: 10.1103/physrevlett.118.168101] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2016] [Indexed: 05/22/2023]
Abstract
It is widely believed that one major life-threatening transition to chaotic fibrillation occurs via spiral-wave breakup that is preceded by spatiotemporal dispersion of refractoriness due to alternations in the duration of the cardiac action potential (AP). However, recent clinical and experimental evidence suggests that other characteristics of the AP may contribute to, and perhaps drive, this dangerous dynamical instability. To identify the relative roles of AP characteristics, we performed experiments in rabbit hearts under conditions to minimize AP duration dynamics which unmasked pronounced AP amplitude alternans just before the onset of fibrillation. We used a simplified ionic cell model to derive a return map and a stability condition that elucidates a novel underlying mechanism for AP alternans and spiral breakup. We found that inactivation of the sodium current is key to developing amplitude alternans and is directly connected to conduction block and initiation of arrhythmias. Simulations in 2D where AP amplitude alternation led to turbulence confirm our hypothesis.
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Affiliation(s)
- Diandian Diana Chen
- School of Physics, Georgia Institute of Technology, 837 State Street NW, Atlanta, Georgia 30332, USA
| | - Richard A Gray
- Food and Drug Administration, 10903 New Hampshire Avenue, Silver Spring, Maryland 20993-0002, USA
| | - Ilija Uzelac
- School of Physics, Georgia Institute of Technology, 837 State Street NW, Atlanta, Georgia 30332, USA
| | - Conner Herndon
- School of Physics, Georgia Institute of Technology, 837 State Street NW, Atlanta, Georgia 30332, USA
| | - Flavio H Fenton
- School of Physics, Georgia Institute of Technology, 837 State Street NW, Atlanta, Georgia 30332, USA
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7
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Zhong L, Li Y, Chen Y, Hong W, Hu W, Guo Q. Chaoticons described by nonlocal nonlinear Schrödinger equation. Sci Rep 2017; 7:41438. [PMID: 28134268 PMCID: PMC5278363 DOI: 10.1038/srep41438] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2016] [Accepted: 12/20/2016] [Indexed: 12/15/2022] Open
Abstract
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).
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Affiliation(s)
- Lanhua Zhong
- Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, P.R. China
- Physical Science and Technology School, Lingnan Normal University, Zhanjiang 524048, P.R. China
| | - Yuqi Li
- Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, P.R. China
| | - Yong Chen
- Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, P.R. China
| | - Weiyi Hong
- Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, P.R. China
| | - Wei Hu
- Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, P.R. China
| | - Qi Guo
- Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, P.R. China
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Ballard CC, Esty CC, Egolf DA. Finding equilibrium in the spatiotemporal chaos of the complex Ginzburg-Landau equation. CHAOS (WOODBURY, N.Y.) 2016; 26:113101. [PMID: 27908021 DOI: 10.1063/1.4966538] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the well-established theory of equilibrium statistical mechanics.
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Affiliation(s)
| | - C Clark Esty
- Department of Physics, Georgetown University, Washington, DC 20057, USA
| | - David A Egolf
- Department of Physics, Georgetown University, Washington, DC 20057, USA
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Alonso S, Bär M, Echebarria B. Nonlinear physics of electrical wave propagation in the heart: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2016; 79:096601. [PMID: 27517161 DOI: 10.1088/0034-4885/79/9/096601] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.
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Affiliation(s)
- Sergio Alonso
- Physikalisch-Technische Bundesanstalt, Abbestr. 2-12 10587, Berlin, Germany. Department of Physics, Universitat Politècnica de Catalunya, Av. Dr. Marañón 44, E-08028 Barcelona, Spain
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10
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Graziano BR, Weiner OD. Self-organization of protrusions and polarity during eukaryotic chemotaxis. Curr Opin Cell Biol 2014; 30:60-7. [PMID: 24998184 DOI: 10.1016/j.ceb.2014.06.007] [Citation(s) in RCA: 58] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2014] [Revised: 05/30/2014] [Accepted: 06/13/2014] [Indexed: 11/18/2022]
Abstract
Many eukaryotic cells regulate their polarity and motility in response to external chemical cues. While we know many of the linear connections that link receptors with downstream actin polymerization events, we have a much murkier understanding of the higher order positive and negative feedback loops that organize these processes in space and time. Importantly, physical forces and actin polymerization events do not simply act downstream of chemotactic inputs but are rather involved in a web of reciprocal interactions with signaling components to generate self-organizing pseudopods and cell polarity. Here we focus on recent progress and open questions in the field, including the basic unit of actin organization, how cells regulate the number and speed of protrusions, and 2D versus 3D migration.
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Affiliation(s)
- Brian R Graziano
- Cardiovascular Research Institute and Department of Biochemistry and Biophysics, University of California San Francisco, San Francisco, CA, USA
| | - Orion D Weiner
- Cardiovascular Research Institute and Department of Biochemistry and Biophysics, University of California San Francisco, San Francisco, CA, USA.
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11
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Shlizerman E, Rom-Kedar V. Parabolic resonance: a route to hamiltonian spatiotemporal chaos. PHYSICAL REVIEW LETTERS 2009; 102:033901. [PMID: 19257354 DOI: 10.1103/physrevlett.102.033901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2008] [Indexed: 05/27/2023]
Abstract
We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and initial profiles that lead to such intermittency are predicted by utilizing a novel geometrical description of the integrable unforced equation.
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Affiliation(s)
- Eli Shlizerman
- Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Post Office Box 26, Rehovot 76100, Israel.
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12
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Oprea I, Triandaf I, Dangelmayr G, Schwartz IB. Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection. CHAOS (WOODBURY, N.Y.) 2007; 17:023101. [PMID: 17614655 DOI: 10.1063/1.2671184] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.
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Affiliation(s)
- Iuliana Oprea
- Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA
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Zhang X, Fu M, Xiao J, Hu G. Self-organization of chaos synchronization and pattern formation in coupled chaotic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:015202. [PMID: 16907144 DOI: 10.1103/physreve.74.015202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2006] [Indexed: 05/11/2023]
Abstract
Pattern formation in spatiotemporal chaotic systems is investigated. Temporally chaotic and spatially ordered patterns are observed by varying the coupling strength. Spatial orderings emerge spontaneously due to self-organization of partial and nonlocal chaos synchronization, governed by various types of spatial symmetries. The first and secondary bifurcations from spatially disordered chaos to chaos with different levels of spatial orderings are observed and the scaling behaviors associated with these bifurcations are statistically analyzed.
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Affiliation(s)
- Xiaoming Zhang
- Department of Physics, Beijing Normal University, Beijing 100875, China
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14
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Jayaraman A, Scheel JD, Greenside HS, Fischer PF. Characterization of the domain chaos convection state by the largest Lyapunov exponent. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:016209. [PMID: 16907179 DOI: 10.1103/physreve.74.016209] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2005] [Indexed: 05/11/2023]
Abstract
Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf, [Nature 404, 733 (2000)], who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity lambda1 is not intensive for aspect ratios Gamma over the range 20<Gamma<40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism.
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Affiliation(s)
- A Jayaraman
- Department of Physics, Duke University, Durham, NC 27708, USA
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15
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Fishman MP, Egolf DA. Revealing the building blocks of spatiotemporal chaos: deviations from extensivity. PHYSICAL REVIEW LETTERS 2006; 96:054103. [PMID: 16486935 DOI: 10.1103/physrevlett.96.054103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2005] [Indexed: 05/06/2023]
Abstract
We have performed high-precision computational studies of the fractal dimension as a function of system length for spatiotemporal chaotic states of the one-dimensional complex Ginzburg-Landau equation. Our data show deviations from extensivity on a length scale consistent with the chaotic length scale, indicating that this spatiotemporal chaotic system is composed of weakly interacting building blocks, each containing about 2 degrees of freedom. Our results also suggest an explanation of some of the "windows of periodicity" found in spatiotemporal systems of moderate size.
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Affiliation(s)
- Matthew P Fishman
- Department of Physics, Georgetown University, Washington, DC 20057, USA
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16
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Stam CJ. Nonlinear dynamical analysis of EEG and MEG: review of an emerging field. Clin Neurophysiol 2005; 116:2266-301. [PMID: 16115797 DOI: 10.1016/j.clinph.2005.06.011] [Citation(s) in RCA: 708] [Impact Index Per Article: 37.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/02/2005] [Revised: 06/03/2005] [Accepted: 06/11/2005] [Indexed: 02/07/2023]
Abstract
Many complex and interesting phenomena in nature are due to nonlinear phenomena. The theory of nonlinear dynamical systems, also called 'chaos theory', has now progressed to a stage, where it becomes possible to study self-organization and pattern formation in the complex neuronal networks of the brain. One approach to nonlinear time series analysis consists of reconstructing, from time series of EEG or MEG, an attractor of the underlying dynamical system, and characterizing it in terms of its dimension (an estimate of the degrees of freedom of the system), or its Lyapunov exponents and entropy (reflecting unpredictability of the dynamics due to the sensitive dependence on initial conditions). More recently developed nonlinear measures characterize other features of local brain dynamics (forecasting, time asymmetry, determinism) or the nonlinear synchronization between recordings from different brain regions. Nonlinear time series has been applied to EEG and MEG of healthy subjects during no-task resting states, perceptual processing, performance of cognitive tasks and different sleep stages. Many pathologic states have been examined as well, ranging from toxic states, seizures, and psychiatric disorders to Alzheimer's, Parkinson's and Cre1utzfeldt-Jakob's disease. Interpretation of these results in terms of 'functional sources' and 'functional networks' allows the identification of three basic patterns of brain dynamics: (i) normal, ongoing dynamics during a no-task, resting state in healthy subjects; this state is characterized by a high dimensional complexity and a relatively low and fluctuating level of synchronization of the neuronal networks; (ii) hypersynchronous, highly nonlinear dynamics of epileptic seizures; (iii) dynamics of degenerative encephalopathies with an abnormally low level of between area synchronization. Only intermediate levels of rapidly fluctuating synchronization, possibly due to critical dynamics near a phase transition, are associated with normal information processing, whereas both hyper-as well as hyposynchronous states result in impaired information processing and disturbed consciousness.
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Affiliation(s)
- C J Stam
- Department of Clinical Neurophysiology, VU University Medical Centre, P.O. Box 7057, 1007 MB Amsterdam, The Netherlands.
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Chiam KH, Cross MC, Greenside HS, Fischer PF. Enhanced tracer transport by the spiral defect chaos state of a convecting fluid. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036205. [PMID: 15903544 DOI: 10.1103/physreve.71.036205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2004] [Indexed: 05/02/2023]
Abstract
To understand how spatiotemporal chaos may modify material transport, we use direct numerical simulations of the three-dimensional Boussinesq equations and of an advection-diffusion equation to study the transport of a passive tracer by the spiral defect chaos state of a convecting fluid. The simulations show that the transport is diffusive and is enhanced by the spatiotemporal chaos. The enhancement in tracer diffusivity follows two regimes. For large Péclet numbers (that is, small molecular diffusivities of the tracer), we find that the enhancement is proportional to the Péclet number. For small Péclet numbers, the enhancement is proportional to the square root of the Péclet number. We explain the presence of these two regimes in terms of how the local transport depends on the local wave numbers of the convection rolls. For large Péclet numbers, we further find that defects cause the tracer diffusivity to be enhanced locally in the direction orthogonal to the local wave vector but suppressed in the direction of the local wave vector.
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Affiliation(s)
- K-H Chiam
- Nonlinear and Statistical Physics, Mail Code 114-36, California Institute of Technology, Pasadena, CA 91125-3600, USA.
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18
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Liu J, She ZS, Guo H, Li L, Ouyang Q. Hierarchical structure description of spatiotemporal chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:036215. [PMID: 15524620 DOI: 10.1103/physreve.70.036215] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2004] [Revised: 05/24/2004] [Indexed: 05/24/2023]
Abstract
We develop a hierarchical structure (HS) analysis for quantitative description of statistical states of spatially extended systems. Examples discussed here include an experimental reaction-diffusion system with Belousov-Zhabotinsky kinetics, the two-dimensional complex Ginzburg-Landau equation, and the modified FitzHugh-Nagumon equation, which all show complex dynamics of spirals and defects. We demonstrate that the spatial-temporal fluctuation fields in the above-mentioned systems all display the HS similarity property originally proposed for the study of fully developed turbulence [Phys. Rev. Lett. 72, 336 (1994)]]. The derived values of a HS parameter beta from experimental and numerical data in various physical regimes exhibit consistent trends and characterize the degree of turbulence in the systems near the transition, and the degree of heterogeneity of multiple disorders far from the transition. It is suggested that the HS analysis offers a useful quantitative description for the complex dynamics of two-dimensional spatiotemporal patterns.
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Affiliation(s)
- Jian Liu
- State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, Peking University, Beijing 100871, People's Republic of China
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Jiang M, Wang X, Ouyang Q, Zhang H. Spatiotemporal chaos control with a target wave in the complex Ginzburg-Landau equation system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:056202. [PMID: 15244899 DOI: 10.1103/physreve.69.056202] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2003] [Indexed: 05/24/2023]
Abstract
An effective method for controlling spiral turbulence in spatially extended systems is realized by introducing a spatially localized inhomogeneity into a two-dimensional system described by the complex Ginzburg-Landau equation. Our numerical simulations show that with the introduction of the inhomogeneity, a target wave can be produced, which will sweep all spiral defects out of the boundary of the system. The effects exist in certain parameter regions where the spiral waves are absolutely unstable. A theoretical explanation is given to reveal the underlying mechanism.
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Affiliation(s)
- Minxi Jiang
- Department of Physics, Peking University, Beijing 100871, China
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20
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Chiam KH, Lai MC, Greenside HS. Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:026705. [PMID: 14525146 DOI: 10.1103/physreve.68.026705] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2003] [Indexed: 11/07/2022]
Abstract
An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Bénard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single nonstaggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.
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Affiliation(s)
- K-H Chiam
- Nonlinear and Statistical Physics, California Institute of Technology, Mail Code 114-36, Pasadena, California 91125-3600, USA
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21
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Zhang C, Zhang H, Ouyang Q, Hu B, Gunaratne GH. Transition from spiral waves to defect-mediated turbulence induced by gradient effects in a reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:036202. [PMID: 14524863 DOI: 10.1103/physreve.68.036202] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2003] [Indexed: 05/24/2023]
Abstract
The transition from spiral waves to defect-mediated turbulence was studied in a spatial open reactor using Belousov-Zhabotinsky reaction. The experimental results show a new mechanism of the transition from spirals to spatiotemporal chaos, in which the gradient effects in the three-dimensional system are essential. The transition scenario consists of two stages: first, the effects of gradients in the third dimension cause a splitting of the spiral tip and a deletion of certain wave segments, generating new wave sources; second, the waves sent by the new wave sources undergo a backfire instability, and the back waves are laterally unstable. As a result, defects are automatically generated and fill all over the system. The result of numerical simulation using the FitzHugh-Nagumo model essentially agrees with the experimental observation.
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Affiliation(s)
- Chunxia Zhang
- Department of Physics, Peking University, Beijing 100871, China
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22
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Chiam KH, Paul MR, Cross MC, Greenside HS. Mean flow and spiral defect chaos in Rayleigh-Bénard convection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056206. [PMID: 12786249 DOI: 10.1103/physreve.67.056206] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2002] [Indexed: 05/24/2023]
Abstract
We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.
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Affiliation(s)
- K-H Chiam
- Nonlinear and Statistical Physics, Mail Code 114-36, California Institute of Technology, Pasadena, CA 91125-3600, USA.
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23
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Tajima S, Greenside HS. Microextensive chaos of a spatially extended system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:017205. [PMID: 12241518 DOI: 10.1103/physreve.66.017205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2001] [Revised: 04/26/2002] [Indexed: 05/23/2023]
Abstract
By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky equation for system sizes L in the range 79 < or = L < or = 93, we show that the Lyapunov fractal dimension D scales microextensively, increasing linearly with L even for increments Delta L that are small compared to the average cell size of 9 and to various correlation lengths. This suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity.
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Affiliation(s)
- Shigeyuki Tajima
- Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA
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24
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Eguíluz VM, Hernández-García E, Piro O. Complex Ginzburg-Landau equation in the presence of walls and corners. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:036205. [PMID: 11580420 DOI: 10.1103/physreve.64.036205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2000] [Revised: 05/10/2001] [Indexed: 05/23/2023]
Abstract
We investigate the influence of walls and corners (with Dirichlet and Neumann boundary conditions) in the evolution of two-dimensional autooscillating fields described by the complex Ginzburg-Landau equation. Analytical solutions are found, and arguments provided, to show that Dirichlet walls introduce strong selection mechanisms for the wave pattern. Corners between walls provide additional synchronization mechanisms and associated selection criteria. The numerical results fit well with the theoretical predictions in the parameter range studied.
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Affiliation(s)
- V M Eguíluz
- Center for Chaos and Turbulence Studies, The Niels Bohr Institute, Blegdamsvej 17, DK2100 Copenhagen Ø, Denmark.
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25
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Zimmermann MG, Toral R, Piro O. Stochastic spatiotemporal intermittency and noise-induced transition to an absorbing phase. PHYSICAL REVIEW LETTERS 2000; 85:3612-3615. [PMID: 11030963 DOI: 10.1103/physrevlett.85.3612] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/1999] [Indexed: 05/23/2023]
Abstract
We introduce a stochastic partial differential equation capable of reproducing the main features of spatiotemporal intermittency (STI). The model also displays a noise-induced transition from STI to laminarity as the noise intensity increases. Simulations and mean-field analysis are used to characterize this transition and to give a quantitative prescription of its loci in the relevant parameter space.
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Affiliation(s)
- MG Zimmermann
- Instituto Mediterraneo de Estudios Avanzados (CSIC-UIB), Campus Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain
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26
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Abstract
Spatially extended dynamical systems exhibit complex behaviour in both space and time--spatiotemporal chaos. Analysis of dynamical quantities (such as fractal dimensions and Lyapunov exponents) has provided insights into low-dimensional systems; but it has proven more difficult to understand spatiotemporal chaos in high-dimensional systems, despite abundant data describing its statistical properties. Initial attempts have been made to extend the dynamical approach to higher-dimensional systems, demonstrating numerically that the spatiotemporal chaos in several simple models is extensive (the number of dynamical degrees of freedom scales with the system volume). Here we report a computational investigation of a phenomenon found in nature, 'spiral defect' chaos in Rayleigh-Benard convection, in which we find that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom. By studying the detailed space-time evolution of the dynamical degrees of freedom, we find that the mechanism for the generation of chaotic disorder is spatially and temporally localized to events associated with the creation and annihilation of defects.
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Abstract
Far-from-equilibrium, spatially extended chaotic systems have generally eluded analytical solution, leading researchers to consider theories based on a statistical rather than a detailed knowledge of the microscopic length scales. Building on the recent discovery of a separation of length scales between macroscopic behavior and microscopic chaos, a simple far-from-equilibrium spatially extended chaotic system has been studied computationally at intermediate, coarse-grained scales. Equilibrium properties such as Gibbs distributions and detailed balance are recovered at these scales, which suggests that the macroscopic behavior of some far-from-equilibrium systems might be understood in terms of equilibrium statistical mechanics.
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Affiliation(s)
- D A Egolf
- Center for Nonlinear Studies (MS B258), Theoretical Division and Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
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28
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Eguíluz VM, Hernández-García E, Piro O, Balle S. Frozen spatial chaos induced by boundaries. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:6571-9. [PMID: 11970576 DOI: 10.1103/physreve.60.6571] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/1999] [Indexed: 04/18/2023]
Abstract
We show that rather simple but nontrivial boundary conditions could induce the appearance of spatial chaos (that is stationary, stable, but spatially disordered configurations) in extended dynamical systems with very simple dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion equation in a two-dimensional undulated domain. Concepts from the theory of dynamical systems, and a transverse-single-mode approximation are used to describe the spatially chaotic structures.
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Affiliation(s)
- V M Eguíluz
- Instituto Mediterráneo de Estudios Avanzados IMEDEA (CSIC-UIB) E-07071 Palma de Mallorca, Spain.
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Maurits NM, Sevink GJA, Zvelindovsky AV, Fraaije JGEM. Pathway Controlled Morphology Formation in Polymer Systems: Reactions, Shear, and Microphase Separation. Macromolecules 1999. [DOI: 10.1021/ma990336x] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- N. M. Maurits
- University of Groningen, Faculty of Mathematics and Natural Sciences, Nijenborgh 4, 9747 AG Groningen, The Netherlands
| | - G. J. A. Sevink
- University of Groningen, Faculty of Mathematics and Natural Sciences, Nijenborgh 4, 9747 AG Groningen, The Netherlands
| | - A. V. Zvelindovsky
- University of Groningen, Faculty of Mathematics and Natural Sciences, Nijenborgh 4, 9747 AG Groningen, The Netherlands
| | - J. G. E. M. Fraaije
- University of Groningen, Faculty of Mathematics and Natural Sciences, Nijenborgh 4, 9747 AG Groningen, The Netherlands
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Lachaux JP, Pezard L, Garnero L, Pelte C, Renault B, Varela FJ, Martinerie J. Spatial extension of brain activity fools the single-channel reconstruction of EEG dynamics. Hum Brain Mapp 1998; 5:26-47. [DOI: 10.1002/(sici)1097-0193(1997)5:1<26::aid-hbm4>3.0.co;2-p] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
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31
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Parekh N, Ravi Kumar V, Kulkarni BD. Synchronization and control of spatiotemporal chaos using time-series data from local regions. CHAOS (WOODBURY, N.Y.) 1998; 8:300-306. [PMID: 12779733 DOI: 10.1063/1.166310] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
In this paper we show that the analysis of the dynamics in localized regions, i.e., sub-systems can be used to characterize the chaotic dynamics and the synchronization ability of the spatiotemporal systems. Using noisy scalar time-series data for driving along with simultaneous self-adaptation of the control parameter representative control goals like suppressing spatiotemporal chaos and synchronization of spatiotemporally chaotic dynamics have been discussed. (c) 1998 American Institute of Physics.
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Affiliation(s)
- Nita Parekh
- Chemical Engineering Division, National Chemical Laboratory, Pune - 411 008, India
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32
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Sadana A, Sutaria M. Influence of diffusion to fractal surfaces on the binding kinetics for antibody-antigen, analyte-receptor, and analyte-receptorless (protein) systems. Biophys Chem 1997; 65:29-44. [PMID: 9130375 DOI: 10.1016/s0301-4622(96)02253-3] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
The diffusion-limited binding kinetics of antigen-antibody, ligand-receptor, analyte-receptorless systems for biosensor applications is analyzed within a fractal framework. The analysis presented applies equally well to these types of systems. For example, for the binding of 2-(p-toluidiny)-naphthalene-6-sulfonic acid (2,6-TNS) to beta-cyclodextrin (ligand-receptor system) immobilized on a fiber-optic base inclusate biosensor, an increase in temperature from 4 to 30 degrees C leads to an increase in the fractal dimension, D1 and to a decrease in the binding rate coefficient, k1. For the binding of TRITC-labeled low-density proteins (LDL) directly to an optical fiber-based sensor (analyte-receptorless system), an increase in the LDL concentration from 5 to 50 micrograms ml-1 in solution leads to a decrease in the fractal dimension, D1 and to an increase in the binding rate coefficient, k1. Also, during the binding of human chorionic gonadotropin (hCG) to anti-hCG antibody immobilized on a HPLC column (antigen-antibody system), an increase in temperature from 4 to 50 degrees C leads to an increase in the fractal dimension, D1 and in the binding rate coefficient. k1. The different examples analyzed and presented together for the three different types of systems provide one means of a 'unified analysis,' and a method by which the forward binding rate coefficient, k1 may be controlled, that is, by changing the fractal dimension or 'disorder' on the surface. The analysis should assist in improving the stability, sensitivity, and response time of biosensors wherein different types of binding systems are utilized in the analysis method. More-or-less all of the treatment presented should be applicable to the above types of binding systems occurring in non-biosensor applications also. However, the single-fractal analysis is unable to describe the data over the full time course of some of the experiments.
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Affiliation(s)
- A Sadana
- Chemical Engineering Department, University of Mississippi, University 88677-9740, USA
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33
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Sadana A, Suturia M. Antigen-antibody binding kinetics for biosensor applications. A dual-fractal analysis. Appl Biochem Biotechnol 1997; 62:275-90. [PMID: 9170257 DOI: 10.1007/bf02788003] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The fit obtained by a dual-fractal analysis is compared with that obtained from a single-fractal analysis. In some cases, the dual-fractal analysis provides an improved fit when compared with a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (San Rafael, CA). These examples are presented. It is of interest to note that the state of disorder (or the fractal dimension) and the binding rate coefficient both increase (or decrease, a single example is presented for this case) as the reaction progresses on the biosensor surface. For example, for the binding of monoclonal antibody MAb 49 in solution to surface-immobilized antigen, a 90.4% increase in the fractal dimension (Df1 to Df2) from 1.327 to 2.527 leads to an increase in the binding rate coefficient (k1 to k2) by a factor of 9.4 from 11.74 to 110.3. The different examples analyzed and presented together provide a means by which the antigen-antibody reactions may be better controlled by noting the magnitude of the changes in the fractal dimension and in the binding rate coefficient as the reaction progresses on the biosensor surface.
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Affiliation(s)
- A Sadana
- Chemical Engineering Department, University of Mississippi 38677-9740, USA.
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Sadana A, Ram AB. Antigen-antibody binding kinetics for biosensors. Changes in the fractal dimension (surface roughness) and in the binding rate coefficient. Appl Biochem Biotechnol 1996; 60:123-38. [PMID: 8856941 DOI: 10.1007/bf02788067] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/02/2023]
Abstract
The diffusion-limited binding kinetics of antigen in solution to antibody immobilized on a biosensor surface is analyzed within a fractal framework. Changes in the fractal dimension, Df observed are in the same and in the reverse directions as the forward binding rate coefficient k. For example, an increase in the concentration of the isoenzyme human creatine kinase isoenzyme MB form (CK-MB) (antigen) solution from 0.1 to 50 ng/mL and bound to anti-CK-MB antibody immobilized on fused silica fiber rods leads to increases in the fractal dimension Df from 0.294 to 0.5080, and in the forward binding rate coefficient k from 0.1194 to 9.716, respectively. The error in the fractal dimension Df decreases with an increase in the CK-MB isoenzyme concentration in solution. An increase in the concentration of human chorionic gonadotrophin (hCG) in solution from 4000 to 6000 mIU/mL hCG and bound to anti-hCG antibody immobilized on a fluorescence capillary fill device leads to a decrease in the fractal dimension Df from 2.6806 to 2.6164, and to an increase in the forward binding rate coefficient k from 3.571 to 4.033, respectively. The different examples analyzed and presented together indicate one means by which the forward binding rate coefficient k may be controlled, that is by changing the fractal dimension or the "disorder' on the surface. The analysis should assist in helping to improve the stability, the sensitivity, and the response time of biosensors.
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Affiliation(s)
- A Sadana
- Chemical Engineering Department, University of Mississippi, University 38677-9740, USA
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35
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Montagne R, Hernández-García E. Winding Number Instability in the Phase-Turbulence Regime of the Complex Ginzburg-Landau Equation. PHYSICAL REVIEW LETTERS 1996; 77:267-270. [PMID: 10062408 DOI: 10.1103/physrevlett.77.267] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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36
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37
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O'Hern CS, Egolf DA, Greenside HS. Lyapunov spectral analysis of a nonequilibrium Ising-like transition. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:3374-3386. [PMID: 9964646 DOI: 10.1103/physreve.53.3374] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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38
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Abstract
Two special magnetic resonance imaging techniques were applied to the Rayleigh/Bénard problem of thermal convection for the first time. The methods were tested using a water cell with horizontal bottom and top covers kept at different temperatures with a downward gradient. Using Fourier encoding velocity imaging (FEVI) a five-dimensional image data set was recorded referring to two space dimensions of slice-selective images and all three components of the local velocity vector. On this basis, the fields of the velocity components or of the velocity magnitude were evaluated quantitatively and rendered as gray shade images. Furthermore the convection rolls were visualized with the aid of two- or three-dimensional multistripe/multiplane tagging imaging pulse sequences based on two or three DANTE combs for the space directions to be probed. Movies illustrating the fluid motions by convection in all three space dimensions were produced. It is demonstrated that the full spatial information of the convection rolls is accessible with microscopic resolution of typically 100 x 100 x 100 microns3. This resolution is effectively limited by flow displacements in the echo time, which should be well within the voxel dimension. The main perspective of this work is that the combined application of FEVI and multistripe/multiplane tagging imaging permits quantitative examinations of thermal convection for arbitrary boundary conditions and with imposed through-flow apart from the direct visualization of convective flow in the form of movies.
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Affiliation(s)
- J Weis
- Sektion Kernresonanzspektroskopie, Universität Ulm, Germany
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39
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Hildebrand M, Bär M, Eiswirth M. Statistics of Topological Defects and Spatiotemporal Chaos in a Reaction-Diffusion System. PHYSICAL REVIEW LETTERS 1995; 75:1503-1506. [PMID: 10060314 DOI: 10.1103/physrevlett.75.1503] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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40
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Egolf DA, Greenside HS. Characterization of the transition from defect to phase turbulence. PHYSICAL REVIEW LETTERS 1995; 74:1751-1754. [PMID: 10057748 DOI: 10.1103/physrevlett.74.1751] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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41
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Bar M, Hildebrand M, Eiswirth M, Falcke M, Engel H, Neufeld M. Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities. CHAOS (WOODBURY, N.Y.) 1994; 4:499-508. [PMID: 12780126 DOI: 10.1063/1.166028] [Citation(s) in RCA: 40] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Among heterogeneously catalyzed chemical reactions, the CO oxidation on the Pt(110) surface under vacuum conditions offers probably the greatest wealth of spontaneous formation of spatial patterns. Spirals, fronts, and solitary pulses were detected at low surface temperatures (T<500 K), in line with the standard phenomenology of bistable, excitable, and oscillatory reaction-diffusion systems. At high temperatures (T greater, similar 540 K), more surprising features like chemical turbulence and standing waves appeared in the experiments. Herein, we study a realistic reaction-diffusion model of this system, with respect to the latter phenomena. In particular, we deal both with the influence of global coupling through the gas phase on the oscillatory reaction and the possibility of wave instabilities under excitable conditions. Gas-phase coupling is shown to either synchronize the oscillations or to yield turbulence and standing structures. The latter findings are closely related to clustering in networks of coupled oscillators and indicate a dominance of the global gas-phase coupling over local coupling via surface diffusion. In the excitable regime wave instabilities in one and two dimensions have been discovered. In one dimension, pulses become unstable due to a vanishing of the refractory zone. In two dimensions, turbulence can also emerge due to spiral breakup, which results from a violation of the dispersion relation.
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Affiliation(s)
- M. Bar
- Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, GermanyAG "Dissipative Strukturen," Institut fur Theoretische Physik der TU Berlin, Rudower Chaussee 5, D-12489 Berlin, GermanyInstitut fur Theoretische Physik der Universitat Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
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