Coupled map lattice
A coupled map lattice (CML) is a
Features of the CML are
CMLs are comparable to
Introduction
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.
A logistic mapping demonstrates chaotic behavior, easily identifiable in one dimension for parameter r > 3.57:
In Figure 1, is initialized to random values across a small lattice; the values are decoupled with respect to neighboring sites. The same
For a basic coupling, we consider a 'single neighbor' coupling where the value at any given site is computed from the recursive maps both on itself and on the neighboring site . The coupling parameter is equally weighted. Again, the value of is constant across the lattice, but slightly increased with each time step.
Even though the recursion is chaotic, a more solid form develops in the evolution. Elongated convective spaces persist throughout the lattice (see Figure 2).
Figure 1: An uncoupled logistic map lattice with random seeding over forty iterations. |
Figure 2: A CML with a single-neighbor coupling scheme taken over forty iterations. |
History
CMLs were first introduced in the mid 1980s through a series of closely released publications.[5][6][7][8] 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 to spatially extended systems). Kaneko's focus was more broad and he is still known as the most active researcher in this area.[9] The most examined CML model was introduced by Kaneko in 1983 where the recurrence equation is as follows:
where and is a real mapping.
The applied CML strategy was as follows:
- 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.
- Decompose the process (underlying the phenomena) into independent components.
- Replace each component by a nonlinear transformation of field variables on each lattice point and the coupling term on suitable, chosen neighbors.
- Carry out each unit dynamics ("procedure") successively.
Classification
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
Much of the current published work in CMLs is based in weak coupled systems
Intermediate and strong coupling interactions are less prolific areas of study. Intermediate interactions are studied with respect to fronts and
These classifications do not reflect the local or global (GMLs [11]) 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.[12] Finally, they do not distinguish between sizes of the underlying space or boundary conditions.
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.[13] Similar proofs exist for weakly coupled hyperbolic maps under the same conditions.
Unique CML qualitative classes
CMLs have revealed novel qualitative universality classes in (CML) phenomenology. Such classes include:
- Spatial bifurcation and frozen chaos
- Pattern Selection
- Selection of zig-zag patterns and chaotic diffusion of defects
- Spatio-temporal intermittency
- Soliton turbulence
- Global traveling waves generated by local phase slips
- Spatial bifurcation to down-flow in open flow systems.
Visual phenomena
The unique qualitative classes listed above can be visualized. By applying the Kaneko 1983 model to the logistic map, several of the CML qualitative classes may be observed. These are demonstrated below, note the unique parameters:
Frozen Chaos | Pattern Selection | Chaotic Brownian Motion of Defect |
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. | Figure 2: Near uniform sized clusters (a = 1.71, ε = 0.4). | Figure 3: Defects exist in the system and fluctuate chaotically akin to Brownian motion (a = 1.85, ε = 0.1). |
Defect Turbulence | Spatiotemporal Intermittency I | Spatiotemporal Intermittency II |
Figure 4: Many defects are generated and turbulently collide (a = 1.895, ε = 0.1). | Figure 5: Each site transits between a coherent state and chaotic state intermittently (a = 1.75, ε = 0.6), Phase I. | Figure 6: The coherent state, Phase II. |
Fully Developed Spatiotemporal Chaos | Traveling Wave | |
Figure 7: Most sites independently oscillate chaotically (a = 2.00, ε = 0.3). | Figure 8: The wave of clusters travels at 'low' speeds (a = 1.47, ε = 0.5). |
Quantitative analysis quantifiers
Coupled map lattices being a prototype of spatially extended systems easy to simulate have represented a benchmark for the definition and introduction of many indicators of spatio-temporal chaos, the most relevant ones are
- The power spectrumin space and time
- Lyapunov spectra[14]
- Dimension density
- Kolmogorov–Sinai entropydensity
- Distributions of patterns
- Pattern entropy
- Propagation speed of finite and infinitesimal disturbance
- Mutual information and correlation in space-time
- Lyapunov exponents, localization of Lyapunov vectors
- Comoving and sub-space time Lyapunov exponents.
- Spatial and temporal Lyapunov exponents [15]
See also
- Lattice model (physics)
- Cellular automata
- Lyapunov exponent
- Stochastic cellular automata
- Rulkov map
- Chialvo map
References
- PMID 12779975.
- ^ OCLC 61030071.
- .
- OCLC 34677022.
- ISSN 0033-068X.
- ISSN 0556-2791.
- S2CID 48964208.
- ^ S. P. Kuznetsov and A. S. Pikovsky, Izvestija VUS, Radiofizika 28, 308 (1985)
- ^ "Kaneko Laboratory".
- PMID 12779987.
- ISSN 0375-9601. Archived from the original(PDF) on 1 December 2008.
- S2CID 1820988.
- S2CID 250862658.
- ISSN 0375-9601. Archived from the original(PDF) on 24 November 2016. Retrieved 23 November 2016.
- S2CID 56433838.
Further reading
- Chazottes, Jean-René; Fernandez, Bastien, eds. (2005). Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Lecture notes in physics 671. Berlin: Springer. pp. 1–4. OCLC 61030071.
- Shawn D. Pethel; Ned J. Corron; Erik Bollt (2006). "Symbolic Dynamics of Coupled Map Lattices" (PDF). Physical Review Letters. 96 (3): 034105. PMID 16486708.
- E. Atlee Jackson (1989), Perspectives of Nonlinear Dynamics: Volume 2, Cambridge University Press, 1991, ISBN 0-521-42633-2
- Schuster, H.G.; Just, W. (2005), Deterministic Chaos, John Wiley and Sons Ltd, OCLC 58054938
- "Introduction to Chaos and Nonlinear Dynamics".
External links
- "Kaneko Laboratory".
- Dynamics of coupled map lattices. Paris: Institut Henri Poincaré. 21 June – 2 July 2004. Archived from the original on 26 November 2006.
- "Institute for Complex Systems". Istituto dei Sistemi Complessi. Florence, Italy
- "Coupled Map Lattice and Globally Coupled Map".
- "A simulation and Analysis Tool for Dynamical Systems". AnT 4.669. Archived from the original on 18 July 2011.