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| A '''coupled [[map (mathematics)|map]] [[lattice (group)|lattice]] (CML)''' is a [[dynamical system]] that models the behavior of [[non-linear]] systems (especially [[partial differential equations]]). They are predominantly used to qualitatively study the [[Chaos theory|chaotic dynamics]] of spatially extended systems. This includes the dynamics of [[wiktionary:spatiotemporal|spatiotemporal]] [[Chaos theory|chaos]] where the number of effective [[Degrees of freedom (physics and chemistry)|degrees of freedom]] diverges as the size of the system increases.<ref>Kaneko, Kunihiko. "Overview of Coupled Map Lattices." Chaos 2, Num3(1992): 279.</ref>
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| Features of the CML are [[Discrete-time dynamical system|discrete time dynamics]], discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous [[state variable]]s.<ref>Chazottes, Jean-René, and Bastien Fernandez. Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Springer, 2004. pgs 1–4</ref> Studied systems include [[Population dynamics|populations]], [[chemical reactions]], [[convection]], [[fluid flow]] and [[biological network]]s. More recently, CMLs have been applied to computational networks <ref>Xu, Jian. Wang, Xioa Fan. " Cascading failures in scale-free coupled map lattices." IEEE International Symposium on Circuits and Systems “ ISCAS Volume 4, (2005): 3395–3398.</ref> identifying detrimental attack methods and [[cascading failure]]s.
| | Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Wordpress have every reason with it which promote wordpress development. It sorts the results of a search according to category, tags and comments. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. If you're ready to find out more information regarding [http://dinky.in/?WordpressBackupPlugin348153 wordpress dropbox backup] check out the web site. You or your web designer can customize it as per your specific needs. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. In addition, Word - Press design integration is also possible. As with a terminology, there are many methods to understand how to use the terminology. |
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| CML’s are comparable to [[cellular automata]] models in terms of their discrete features.<ref>R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press,Cambridge, England, 1997).</ref> However, the value of each site in a cellular automata network is strictly dependent on its neighbor (s) from the previous time step. Each site of the CML is only dependent upon its neighbors relative to the coupling term in the [[recurrence equation]]. However, the similarities can be compounded when considering multi-component dynamical systems.
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| ==Introduction==
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| A CML generally incorporates a system of equations (coupled or uncoupled), a finite number of variables, a global or local coupling scheme and the corresponding coupling terms. The underlying lattice can exist in infinite dimensions. Mappings of interest in CMLs generally demonstrate chaotic behavior. Such maps can be found here: [[List of chaotic maps]].
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| A logistic mapping demonstrates chaotic behavior, easily identifiable in one dimension for parameter r > 3.57 (see [[Logistic map]]). It is graphed across a small lattice and decoupled with respect to neighboring sites. The [[Recurrence relation|recurrence equation]] is [[homogeneous]]{{amblink|date=November 2013}}, albeit randomly seeded. The parameter r is updated every time step (see Figure 1, Enlarge, Summary):
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| : <math> \qquad x_{n+1} = r x_n (1-x_n) </math>
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| The result is a raw form of chaotic behavior in a map lattice. The range of the function is bounded so similar [[contour line|contours]] through the lattice is expected. However, there are no significant [[spatial correlation]]s or pertinent fronts to the chaotic behavior. No obvious order is apparent.
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| For a basic coupling, we consider a 'single neighbor' coupling where the value at any given site <math> s </math> is mapped recursively with respect to itself and the neighboring site <math> s-1 </math>. The coupling parameter <math> \epsilon = 0.5 </math> is equally weighted.
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| : <math> \qquad x_{n+1} = (\epsilon)[r x_n (1-x_n)]_s + (1-\epsilon)[r x_n (1-x_n)]_{s-1} </math>
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| Even though each native recursion is chaotic, a more solid form develops in the evolution. Elongated convective spaces persist throughout the lattice (see Figure 2).
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| {| class="wikitable" border="1"
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| |<!--column1-->[[File:Cml2e.gif|thumb|250px|center]]
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| |<!--column2-->[[File:Cml3a.gif|thumb|250px|center]]
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| |<!--column1-->Figure 1: An uncoupled logistic map lattice <br> with random seeding over forty iterations.
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| |<!--column2-->Figure 2: A CML with a single-neighbor <br> coupling scheme taken over forty iterations.
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| |}<!--end wikitable-->
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| ==History==
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| CMLs were first introduced in the mid 1980’s through a series of closely released publications.<ref>K. Kaneko, Prog. Theor. Phys. 72, 480 (1984)</ref><ref>I. waller and R. Kapral; Phys. Rev. A 30 2047 (1984)</ref><ref>J. Crutchfield, Phyisca D 10, 229 (1984)</ref><ref>S. P.Kuznetsov and A. S. Pikovsky, Izvestija VUS, Radiofizika 28, 308 (1985)</ref> Kapral used CMLs for modeling chemical spatial phenomena. Kuznetsov sought to apply CMLs to electrical circuitry by developing a [[renormalization group]] approach (similar to Feigenbaum's [[Universality (dynamical systems)|universality]] to spatially extended systems). Kaneko's focus was more broad and he is still known as the most active researcher in this area.<ref>http://chaos.c.u-tokyo.ac.jp/</ref> The most examined CML model was introduced by Kaneko in 1983 where the recurrence equation is as follows:
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| : <math> u_s^{t+1} = (1-\varepsilon)f(u_s^t)+\frac{\varepsilon}{2}\left(f(u_{s+1}^t)+f(u_{s-1}^t) \right) \ \ \ t\in \mathbb{N},\ \varepsilon \in [0,1]</math>
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| where <math> u_s^t \in {\mathbb{R}} \ , </math> and <math> f </math> is a real mapping.
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| The applied CML strategy was as follows:
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| * Choose a set of field variables on the lattice at a macroscopic level. The dimension (not limited by the CML system) should be chosen to correspond to the physical space being researched.
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| * Decompose the process (underlying the phenomena) into independent components.
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| * Replace each component by a nonlinear transformation of field variables on each lattice point and the coupling term on suitable, chosen neighbors.
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| * Carry out each unit dynamics ("procedure") successively.
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| ==Classification==
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| The CML system evolves through discrete time by a mapping on vector sequences. These mappings are a recursive function of two competing terms: an individual [[non-linear]] reaction, and a spatial interaction (coupling) of variable intensity. CMLs can be classified by the strength of this coupling parameter(s).
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| Much of the current published work in CMLs is based in weak coupled systems <ref>Lectures from the school-forum (CML 2004) held in Paris, June 21{July 2, 2004. Edited by J.-R. Chazottes and B. Fernandez. Lecture Notes in Physics, 671. Springer, Berlin (2005)</ref> where [[diffeomorphism]]s of the [[state space]] close to identity are studied. Weak coupling with [[monotonic]] ([[bistable]]) dynamical regimes demonstrate spatial chaos phenomena and are popular in neural models.<ref>Nozawa, Hiroshi. "A neural network model." Chaos 2, Num3(1992): 377.</ref> Weak coupling unimodal maps are characterized by their stable [[periodic point]]s and are used by [[gene regulatory network]] models. Space-time chaotic phenomena can be demonstrated from chaotic mappings subject to weak coupling coefficients and are popular in [[phase transition]] phenomena models.
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| Intermediate and strong coupling interactions are less prolific areas of study. Intermediate interactions are studied with respect to fronts and [[traveling wave]]s, riddled basins, riddled bifurcations, clusters and non-unique phases. Strong coupling interactions are most well known to model synchronization effects of dynamic spatial systems such as the [[Kuramoto model]].
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| These classifications do not reflect the local or global (GMLs <ref>Ho, Ming-Ching. Hung, Yao-Chen. Jiang, I-Min. "Phase synchronization in inhomogenous globally coupled map lattices. Physics Letter A. 324 (2004) 450–457. [http://www.phys.sinica.edu.tw/~statphys/publications/2004_full_text/M_C_Ho_PLA_324_450(2004).pdf]</ref>) coupling nature of the interaction. Nor do they consider the frequency of the coupling which can exist as a degree of freedom in the system.<ref>http://www.mat.uniroma2.it/~liverani/Lavori/live0803.pdf</ref> Finally, they do not distinguish between sizes of the underlying space or [[Boundary value problem|boundary condition]]s.
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| Surprisingly the dynamics of CMLs have little to do with the local maps that constitute their elementary components. With each model a rigorous mathematical investigation is needed to identify a chaotic state (beyond visual interpretation). Rigorous proofs have been performed to this effect. By example: the existence of space-time chaos in weak space interactions of one-dimensional maps with strong statistical properties was proven by Bunimovich and Sinai in 1988.<ref>L.A. Bunimovich and Ya. G. Sinai. "Nonlinearity" Vol. 1 pg 491 (1988)</ref> Similar proofs exist for weakly hyperbolic maps under the same conditions.
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| ==Unique CML qualitative classes==
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| CMLs have revealed novel qualitative universality classes in (CML) phenomenology. Such classes include:
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| * [[Spatial bifurcation]] and frozen chaos
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| * Pattern Selection
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| * Selection of zig-zag patterns and chaotic diffusion of defects
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| * Spatio-temporal [[intermittency]]
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| * [[Soliton]] [[turbulence]]
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| * Global traveling waves generated by local phase slips
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| * Spatial bifurcation to down-flow in open flow systems.
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| ==Visual phenomena==
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| The unique qualitative classes listed above can be visualized. By applying the Kaneko 1983 model to the logistic <math>{f(x_n)} = 1 - ax^2</math> map, several of the CML qualitative classes may be observed. These are demonstrated below, note the unique parameters:
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| {| class="wikitable" border="1"
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| |<!--column1-->'''Frozen Chaos'''
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| |<!--column2-->'''Pattern Selection'''
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| |<!--column3-->'''Chaotic Brownian Motion of Defect'''
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| |<!--column1-->[[File:Frozenchaos logmap.JPG|200px|center]]
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| |<!--column2-->[[File:PatternSelection logmap.JPG|200px|center]]
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| |<!--column3-->[[File:BrownMotionDefect logmap.JPG|200px|center]]
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| |<!--column1-->Figure 1: Sites are divided into non-uniform clusters, where the divided patterns are regarded as attractors. Sensitivity to initial conditions exist relative to ''a'' < 1.5.
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| |<!--column2-->Figure 2: Near uniform sized clusters (''a'' = 1.71, ''ε'' = 0.4).
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| |<!--column3-->Figure 3: Defects exist in the system and fluctuate chaotically akin to Brownian motion (''a'' = 1.85, ''ε'' = 0.1).
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| |<!--column1-->'''Defect Turbulence'''
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| |<!--column2-->'''Spatiotemporal Intermittency I'''
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| |<!--column3-->'''Spatiotemporal Intermittency II'''
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| |-
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| |<!--column1-->[[File:DefectTurbulence logmap.JPG|200px|center]]
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| |<!--column2-->[[File:Spatiotemporal Intermittency logmap.JPG|200px|center]]
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| |<!--column3-->[[File:Spatiotemporal Intermittency logmap2.JPG|200px|center]]
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| |-
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| |<!--column1-->Figure 4: Many defects are generated and turbulently collide (''a'' = 1.895, ''ε'' = 0.1).
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| |<!--column2-->Figure 5: Each site transits between a coherent state and chaotic state intermittently (''a'' = 1.75, ''ε'' = 0.6), Phase I.
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| |<!--column3-->Figure 6: The coherent state, Phase II.
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| |-
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| |<!--column1-->'''Fully Developed Spatiotemporal Chaos'''
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| |<!--column2-->'''Traveling Wave'''
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| |-
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| |<!--column1-->[[File:SpatiotemporalChaos fullydevd logmap.JPG|200px|center]]
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| |<!--column2-->[[File:TravelingWave logmap.JPG|200px|center]]
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| |-
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| |<!--column1-->Figure 7: Most sites independently oscillate chaotically (''a'' = 2.00, ''ε'' = 0.3).
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| |<!--column2-->Figure 8: The wave of clusters travels at 'low' speeds (''a'' = 1.47, ''ε'' = 0.5).
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| |}<!--end wikitable-->
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| ==Quantitative analysis quantifiers==
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| Coupled map lattices being a prototype of spatially extended systems easy to simulate have represented a benchmark
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| for the definition and introduction of many indicators of spatio-temporal chaos, the most relevant ones are
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| * The [[power spectrum]] in space and time
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| * [[Lyapunov exponent#Definition of the Lyapunov spectrum|Lyapunov spectra]]<ref>[http://www.fi.isc.cnr.it/users/antonio.politi/Reprints/052.pdf Lyapunov Spectra of Coupled Map Lattices, S. Isola, A. Politi, S. Ruffo, and A. Torcini]</ref>
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| * Dimension density
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| * [[Kolmogorov–Sinai entropy]] density
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| * Distributions of patterns
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| * Pattern entropy
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| * Propagation speed of finite and infinitesimal disturbance
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| * [[Mutual information]] and correlation in space-time
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| * [[Lyapunov exponent]]s, localization of [[Lyapunov vector]]s
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| * Comoving and sub-space time [[Lyapunov exponent]]s.
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| * Spatial and temporal [[Lyapunov exponent]]s <ref>S. Lepri, A. Politi and A. Torcini
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| [http://xxx.lanl.gov/abs/chao-dyn/9504005 Chronotopic Lyapunov Analysis: (I) a Detailed Characterization of 1D Systems],
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| J. Stat. Phys., 82 5/6 (1996) 1429.</ref>
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| ==See also==
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| *[[Cellular automata]]
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| *[[Lyapunov exponent]]
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| *[[Stochastic cellular automata]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| {{refbegin|2}}
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| * {{cite book
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| | author = Google Library
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| | archiveurl = http://books.google.com/books?id=a63Q8DhKA44C&dq=coupled+map+lattices&source=gbs_summary_s&hl=en
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| | title = Dynamics of Coupled Map Lattices
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| | publisher = Springer
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| | url = http://books.google.com/?id=a63Q8DhKA44C&dq=coupled+map+lattices
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| | archivedate = 2008-03-29
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| | isbn = 978-3-540-24289-5
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| | year = 2005
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| }}
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| * {{Cite web
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| | author = Shawn D. Pethel, Ned J. Corron, and Erik Bollt
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| | archiveurl = http://people.clarkson.edu/~bolltem/Papers/PhysRevLett_96_034105PethelCorronBollt.pdf
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| | title = Symbolic Dynamics of Coupled Map Lattices
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| | publisher = Physical Review Letters
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| | url = http://dx.dio.org/10.1103/PhysRevLett.96.034105
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| | archivedate = 2008-03-29
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| }}
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| * {{Citation
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| | author = E. Atlee Jackson
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| | url = http://books.google.com/books?id=M2E0AAAAIAAJ&source=gbs_ViewAPI
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| | title = Perspectives of Nonlinear Dynamics: Volume 2
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| | publisher = Cambridge University Press, 1991
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| | ISBN = 0-521-42633-2
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| }}
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| * {{Citation
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| | author = H.G, Schuster and W. Just
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| | url = http://www.whsmith.co.uk/CatalogAndSearch/ProductDetails.aspx?productID=9783527404155
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| | title = Deterministic Chaos
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| | publisher = John Wiley and Sons Ltd, 2005
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| | ISBN = 3-527-40415-5
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| }}
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| * [http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/ Introduction to Chaos and Nonlinear Dynamics]
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| {{refend}}
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| ==External links==
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| * [http://chaos.c.u-tokyo.ac.jp/ Kaneko Laboratory]
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| * [http://www.cpht.polytechnique.fr/cpth/cml2004/ Institut Henri Poincaré, Paris, June 21 – July 2, 2004]
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| * [http://www.fi.isc.cnr.it/ Istituto dei Sistemi Complessi], [[Florence]], [[Italy]]
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| * [http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/CMLGCM/ Java CML/GML web-app]
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| * [http://ant4669.de/ AnT 4.669 – A simulation and Analysis Tool for Dynamical Systems]
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| {{Chaos theory}}
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| {{DEFAULTSORT:Coupled Map Lattice}}
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| [[Category:Nonlinear systems]]
| |
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Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Wordpress have every reason with it which promote wordpress development. It sorts the results of a search according to category, tags and comments. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
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