Critical correlations in coupled map lattices

Houlrik, Jens M.; Jensen, Mogens H.

doi:10.1016/0375-9601(92)91012-g

Cited by 18 publications

(4 citation statements)

References 8 publications

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“…where φ is the correlation length which diverges as φ ≈ δ −ν and δ is given by ǫ − ǫ c [16]. Therefore, ν can be obtained by adjusting it's value till the scaled variables Lδ ν and τ δ νz collapse onto a single curve.…”

Section: Static (Bulk) Exponentsmentioning

confidence: 99%

Evidence for directed percolation universality at the onset of spatiotemporal intermittency in coupled circle maps

2003

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We consider a lattice of coupled circle maps, a popular model for the study of mode-locked phenomena. We find that the onset of spatiotemporal intermittency (STI) in this system is analogous to directed percolation (DP), with the transition being to a unique absorbing state for low nonlinearities, and to weakly chaotic absorbing states for high nonlinearities. We find that the complete set of static exponents and spreading exponents at all critical points match those of DP very convincingly. Further, hyperscaling relations are fulfilled, leading to independent controls and consistency checks of the values of all the critical exponents. These results provide an example in support of the conjecture that the onset of STI in deterministic models belongs to the DP universality class. Nonuniversal spreading exponents are seen only for the cases where the initial state is homogeneous with symmetrically placed seeds leading to strictly symmetric spreading. However, very small departures from homogeneity are sufficient to restore the DP exponents.

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Section: Static (Bulk) Exponentsmentioning

confidence: 99%

Evidence for directed percolation universality at the onset of spatiotemporal intermittency in coupled circle maps

2003

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“…Generally, the spectrum of possibilities of spatio-temporal structures that can be generated by coupled map lattices is extremely rich and has been extensively studied in the literature, the emphasis being on the bifurcation structure [Bunimovich et al (1996), Just (1995, Amritkar et al (1993)] [Gade et al (1993), Amritkar et al (1991), Pikovsky et al (1991)], Liapunov exponents [Yang et al (1996), Torcini et al (1997b), Kaneko (1986b)] [Isola et al (1990)], traveling waves [Carretero-González (1997)] [He et al (1997)], phase transition-like phenomena [Grassberger et al (1991), Cuche et al (1997), Blank (1997), Marcq et al (1996)] [Boldrighini et al (1995), Keller et al (1992a), Houlrik et al (1992)] [Miller et al (1993), Gielis et al (2000)], the existence of smooth invariant measures [Baladi et al (1998), Jiang et al (1998a), ] [Mackey et al (1995)], synchronization [Lemaitre et al (1999)] [Bagnoli et al (1999), de San Roman et al (1998), Jiang et al (1998b)] [Wang et al (1998), Ding et al (1997)], control [Gade (1998)] [Egolf et al (1998), Parekh et al (1998), Mondragon et al (1997)] [Ohishi et al (1995)] and many other properties. Applications for coupled map systems have been pointed out for various subjects, among them hydrodynamic turbulence [Beck (1994), Hilgers et al (1997b), …”

Section: Prefacementioning

confidence: 99%

Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields

Beck¹

2002

Advanced Series in Nonlinear Dynamics

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We consider deterministic chaotic models of vacuum fluctuations on a small (quantum gravity) scale. As a suitable small-scale dynamics, nonlinear versions of strings, so-called 'chaotic strings' are introduced. These can be used to provide the 'noise' for second quantization of ordinary strings via the Parisi-Wu approach of stochastic quantization. Extensive numerical evidence is presented that the vacuum energy of chaotic strings is minimized for the numerical values of the observed standard model parameters, i.e. in this extended approach to second quantization concrete predictions for vacuum expectations of dilaton-like fields and hence on masses and coupling constants can be given. Low-energy fermion and boson masses are correctly obtained with a precison of 3-4 digits, the electroweak and strong coupling strengths with a precison of 4-5 digits. In particular, the minima of the vacuum energy yield high-precision predictions of the Higgs mass (154 GeV), of the neutrino masses (1.45 · 10 −5 eV, 2.57 · 10 −3 eV, 4.92 · 10 −2 eV) and of the GUT scale (1.73 · 10 16 GeV).use for second quantization purposes, and their physical interpretation in terms of vacuum fluctuations. In the later chapters concrete numerical results are presented and these are then related to standard model phenomenology. Sections marked with an asterisk can be omitted at a first reading, these sections deal with interesting side issues which, however, are not necessary for the logical development of the following chapters. In view of the fact that (unfortunately!) many readers may not have the time to read this book from the beginning to the end, I included a very detailed summary as a self-contained chapter 12. This summary contains the most important concepts and results of this book and is written in a self-consistent way, i.e. no knowledge of previous chapters is required.The research described in this book developed over a longer period of time at various places. I started to work on the relevant types of coupled map lattices during my stay at the Niels Bohr Institute, Copenhagen, in 1992 and continued during a stay at the University of Maryland in 1993. Some important numerical results, now described in section 7.2 and 8.5, were obtained at the RWTH Aachen in 1994 as well as during a visit to

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“…It has been argued that the transition to spatiotemporal intermittency with absorbing laminar states is a second order phase transition, and that this transition falls in the same universality class as directed percolation [12] with the laminar states being identified with the 'inactive' states and the turbulent states being identified as the 'active' or percolating states. This conjecture has become the central issue in a longstanding debate [10,13,14,15,16], which is still not completely resolved. Thus, the analysis of spatiotemporal intermittency remains a challenging theoretical problem.…”

Section: Introductionmentioning

confidence: 99%

Dynamic characterizers of spatiotemporal intermittency

Jabeen

Gupte

2005

Phys. Rev. E

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We study spatiotemporal intermittency (STI) in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes where the STI lies in the directed percolation (DP) class, as well as regimes which show pure spatial intermittency (where the temporal behavior is regular) which do not belong to the DP class. Thus both DP and non-DP behavior can be seen in the same system. The signature of DP and non-DP behavior can be seen in the dynamic characterizers, viz. the spectrum of eigenvalues of the linear stability matrix of the evolution equation, as well as in the multifractal spectrum of the eigenvalue distribution. The eigenvalue spectrum of the system in the DP regimes is continuous, whereas it shows evidence of level repulsion in the form of gaps in the spectrum in the non-DP regime. The multifractal spectrum of the eigenvalue distribution also shows the signature of DP and non-DP behavior. These results have implications for the manner in which correlations build up in extended systems.

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Critical correlations in coupled map lattices

Cited by 18 publications

References 8 publications

Evidence for directed percolation universality at the onset of spatiotemporal intermittency in coupled circle maps

Evidence for directed percolation universality at the onset of spatiotemporal intermittency in coupled circle maps

Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields

Dynamic characterizers of spatiotemporal intermittency

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