Coarse Graining the Dynamics of Coupled Oscillators

Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey, United States
Physical Review Letters (Impact Factor: 7.51). 05/2006; 96(14):144101. DOI: 10.1103/PhysRevLett.96.144101
Source: PubMed


We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.

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    • "Ideally, this reparametrisation would be done adaptively as clusters form, in the same way that algorithms for numerical integration adapt as the solution varies [30]. Alternatively, if a single oscillator ‘breaks away’ [27], the methods presented here could be used on the remaining synchronous oscillators, with the variables describing the state of the rogue oscillator also fully resolved. More generally, there are systems in which it is not necessarily the state of an oscillator that is a smooth function of the heterogeneous parameter, but the parameters describing the distribution of states[37,38], and the ideas presented here could also be useful in this case. "
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    ABSTRACT: We consider a coupled, heterogeneous population of relaxation oscillators used to model rhythmic oscillations in the pre-Bötzinger complex. By choosing specific values of the parameter used to describe the heterogeneity, sampled from the probability distribution of the values of that parameter, we show how the effects of heterogeneity can be studied in a computationally efficient manner. When more than one parameter is heterogeneous, full or sparse tensor product grids are used to select appropriate parameter values. The method allows us to effectively reduce the dimensionality of the model, and it provides a means for systematically investigating the effects of heterogeneity in coupled systems, linking ideas from uncertainty quantification to those for the study of network dynamics.
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    • "The results of the EFREE simulations are compared with the results obtained by the one-dimensional electrostatic particle-in-cell (1D ES PIC) solver [12]. In addition results are compared with the results in [11] and multiscale calculations in the framework of the systems of coupled oscillators [13] [14] [15] [16] "
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    ABSTRACT: IntheEquation-freeframework, a macro-coarseprojectiveintegration method consists of two parts: the time stepper and time projection on macro scale. The first one consists of lifting, micro simulation and restriction. For extracting directly from microscopic simulations the information which would be obtained from the macro- scopic model of two-dimensional microscopic systems, the time stepper based on the one-dimensional cumulative distribution functions, the marginal cumulative and ap- propriate number of the conditional cumulative distributions, is introduced. Here this procedure is tested on the nonlinear ion acoustic wave in a plasma. The numerical micro-solver is the one dimensional electrostatic particle-in-cell code. It is shown that particle correlations related to wave structures are better preserved by the new model. The lifting step is critically related to the noise in system. The enlarged noise, rise of correlations, trapping of particles during the wave steepening can seriously violate the basic assumptions of the equation-free approach.
    Full-text · Article · Sep 2008 · Communications in Computational Physics
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    • "Both sides of the bifurcation point can be described by the same set of coarsegrained observables, which clearly summarize group level dynamical behavior of the followers before and after the bifurcation. As K decreases in the Kuramoto model, oscillators get desynchronized (Kuramoto, 1984), starting with the oscillator with the maximum value of |ω i | (the " extreme " oscillator) through a saddle-node (actually a " sniper " ) bifurcation on a limit cycle (Moon et al., 2006). We expect the same type of bifurcation to occur in this model. "
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    ABSTRACT: We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.
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