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E-mail Alerts. RSS Feeds. SIAM J. Control Optim. Related Databases. Web of Science You must be logged in with an active subscription to view this. Keywords multi-agent systems , networked systems , controllability , automorphism group , equitable partitions , agreement dynamics , algebraic graph theory. Publication Data. ISSN print : Publisher: Society for Industrial and Applied Mathematics. Cited by Irrelevance of linear controllability to nonlinear dynamical networks. Nature Communications 10 Physica A: Statistical Mechanics and its Applications , Automatica , Journal of Risk Research , International Journal of Control , Journal of Systems Science and Complexity 32 :4, International Journal of Robust and Nonlinear Control 29 :9, International Journal of Control 3 , Science China Information Sciences 62 New Journal of Physics 21 :4, International Journal of Robust and Nonlinear Control 29 :5, Swarm Intelligence 13 :1, Applied Mathematics and Computation , SIAM Review 61 :2, Physical Review E 98 Automatica 97 , Journal of Complex Networks 6 :5, Entropy 20 :9, Automatica 95 , Journal of the Franklin Institute , International Journal of Control 91 :5, Neurocomputing , Automatica 89 , International Journal of Control, Automation and Systems 16 :1, International Journal of Robust and Nonlinear Control 28 :1, Science China Information Sciences 61 Optimal Control Applications and Methods 39 :1, Scientific Reports 7 International Journal of Robust and Nonlinear Control 27 , Journal of Systems Science and Complexity 30 :6, Physical Review Letters Journal of Mathematical Analysis and Applications :1, Science China Information Sciences 60 Automatica 82 , Mathematical Models and Methods in Applied Sciences 27 , Automatica 80 , Automatica 77 , International Journal of Control, Automation and Systems 15 :1, International Journal of Robust and Nonlinear Control 27 :1, Dynamical Systems 32 :1, Active Particles, Volume 1, Filip J.

Automatica 74 , Automatica 73 , International Journal of Systems Science 47 , Reviews of Modern Physics 88 Scientific Reports 6 Physics Reports , Automatica 69 , European Journal of Control 30 , EPL Europhysics Letters :6, Communications in Theoretical Physics 65 :5, Asian Journal of Control 18 :3, Royal Society Open Science 3 :4, International Journal of Control 89 :1, They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.

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This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases. Features of the CML are discrete time dynamics , discrete underlying spaces lattices or networks , and real number or vector , local, continuous state variables.

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More recently, CMLs have been applied to computational networks [3] identifying detrimental attack methods and cascading failures. CMLs are comparable to cellular automata models in terms of their discrete features. 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. 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.

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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. The same recurrence relation is applied at each lattice point, although the parameter r is slightly increased with each time step.

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The result is a raw form of chaotic behavior in a map lattice. However, there are no significant spatial correlations or pertinent fronts to the chaotic behavior. No obvious order is apparent.

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  7. Even though the recursion is chaotic, a more solid form develops in the evolution. Elongated convective spaces persist throughout the lattice see Figure 2.

    Dynamical Systems 1976: v.2: International Symposium Proceedings: Vol 2

    CMLs were first introduced in the mid s through a series of closely released publications. 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. 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.

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    8. CMLs can be classified by the strength of this coupling parameter s. Much of the current published work in CMLs is based in weak coupled systems [10] where diffeomorphisms 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. Space-time chaotic phenomena can be demonstrated from chaotic mappings subject to weak coupling coefficients and are popular in phase transition phenomena models. Intermediate and strong coupling interactions are less prolific areas of study.

      Intermediate interactions are studied with respect to fronts and traveling waves , riddled basins, riddled bifurcations, clusters and non-unique phases.

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      Strong coupling interactions are most well known to model synchronization effects of dynamic spatial systems such as the Kuramoto model. These classifications do not reflect the local or global GMLs [12] 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.

      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 Such classes include:.

      The unique qualitative classes listed above can be visualized. These are demonstrated below, note the unique parameters:.