scholarly journals Low dimensional behavior of large systems of globally coupled oscillators

2008 ◽  
Vol 18 (3) ◽  
pp. 037113 ◽  
Author(s):  
Edward Ott ◽  
Thomas M. Antonsen
Keyword(s):  
Coupled Oscillators ◽  
Large Systems ◽  
Low Dimensional

scholarly journals EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE

2012 ◽  
Vol 22 (09) ◽  
pp. 1250216 ◽  
Author(s):  
ENRICO GERLACH ◽  
SIEGFRIED EGGL ◽  
CHARALAMPOS SKOKOS
Keyword(s):  
Hamiltonian Systems ◽  
Coupled Oscillators ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Variational Equations ◽  
Integration Schemes ◽  
Chaos Detection ◽  
Efficient Integration ◽  
Low Dimensional ◽  
Series Expansion Method

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

scholarly journals Echo phenomena in large systems of coupled oscillators

2008 ◽  
Vol 18 (3) ◽  
pp. 037115 ◽  
Author(s):  
Edward Ott ◽  
John H. Platig ◽  
Thomas M. Antonsen ◽  
Michelle Girvan
Keyword(s):  
Coupled Oscillators ◽  
Large Systems

Phase- and Center-Manifold Reductions for Large Populations of Coupled Oscillators with Application to Non-Locally Coupled Systems

1997 ◽  
Vol 07 (04) ◽  
pp. 789-805 ◽  
Author(s):  
Yoshiki Kuramoto
Keyword(s):  
Coupled Oscillators ◽  
Center Manifold ◽  
Coupled Systems ◽  
Center Manifold Reduction ◽  
Reduction Techniques ◽  
Phase Reduction ◽  
Large Populations ◽  
Large Systems ◽  
Model Equations ◽  
Manifold Reduction

In the first half of this paper, some general ideas will be developed on how to approach mathematically large systems of coupled limit-cycle oscillators. Two representative reduction techniques, namely, the phase reduction and the center-manifold reduction will be presented for a prototypal system of biological cell assembly with periodic activity. The evolution equation derived through each reduction method is further classified into three groups according to the range of the oscillator coupling (i.e. local, global and intermediate). As a consequence, six classes of model equations are obtained. In the second half of the paper, some new results from our recent study on non-locally coupled oscillators will be reported, and the generation of anomalous turbulent fluctuations obeying a power law will be discussed in some detail.

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