Stability Theory of Synchronized Motion in Coupled-Oscillator Systems

Fujisaka, Hirokazu; Yamada, Tomoji

doi:10.1143/ptp.69.32

Cited by 1,243 publications

(463 citation statements)

References 13 publications

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“…When a mathematical model of the system is available, the TLEs can also be calculated using the master stability function technique (Fujisaka & Yamada 1983;Pecora & Carroll 1998), i.e. by linearization about the synchronized chaotic solution.…”

Section: Synchronization-transient Dynamicsmentioning

confidence: 99%

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

Murphy

Cohen

Ravoori

et al. 2010

Phil. Trans. R. Soc. A.

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We describe a flexible and modular delayed-feedback nonlinear oscillator that is capable of generating a wide range of dynamical behaviours, from periodic oscillations to high-dimensional chaos. The oscillator uses electro-optic modulation and fibre-optic transmission, with feedback and filtering implemented through real-time digital signal processing. We consider two such oscillators that are coupled to one another, and we identify the conditions under which they will synchronize. By examining the rates of divergence or convergence between two coupled oscillators, we quantify the maximum Lyapunov exponents or transverse Lyapunov exponents of the system, and we present an experimental method to determine these rates that does not require a mathematical model of the system. Finally, we demonstrate a new adaptive control method that keeps two oscillators synchronized, even when the coupling between them is changing unpredictably.

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Section: Synchronization-transient Dynamicsmentioning

confidence: 99%

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

Murphy

Cohen

Ravoori

et al. 2010

Phil. Trans. R. Soc. A.

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“…Hence, (X ,Ỹ,Z) ∼ exp Λ 0 t as t → ∞. Thus, instability exists if [19,37]. This relation reflects the competition between the chaotic separation of neighbouring trajectories (as measured by Λ 0 ), and diffusive smoothing.…”

Section: Stability Of Synchronized Statesmentioning

confidence: 97%

The Lorenz–Fermi–Pasta–Ulam experiment

Balmforth

Pasquero

Provenzale

2000

Physica D: Nonlinear Phenomena

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We consider a chain of Lorenz '63 systems connected through a local, nearest-neighbour coupling. We refer to the resulting system as the Lorenz-Fermi-Pasta-Ulam lattice because of its similarity to the celebrated experiment conducted by Fermi, Pasta and Ulam. At large coupling strengths, the systems synchronize to a global, chaotic orbit of the Lorenz attractor. For smaller coupling, the synchronized state loses stability. Instead, steady, spatially structured equilibrium states are observed. These steady states are related to the heteroclinic orbits of the system describing stationary solutions to the partial differential equation that emerges on taking the continuum limit of the lattice. Notably, these orbits connect saddle-foci, suggesting the existence of a multitude of such equilibria in relatively wide systems. On lowering the coupling strength yet further, the steady states lose stability in what appear to be always subcritical Hopf bifurcations. This can lead to a variety of time-dependent states with fixed time-averaged spatial structure. Such solutions can be limit cycles, tori or possibly chaotic attractors. "Cluster states" can also occur (though with less regularity), consisting of lattices in which the elements are partitioned into families of synchronized subsystems. Ultimately, for very weak coupling, the lattice loses its time-averaged spatial structure. At this stage, the properties of the lattice are probably chaotic and approximately scale with the lattice size, suggesting that the system is essentially an ensemble of elements that evolve largely independent of one another. The weak interaction, however, is sufficient to induce widespread coherent phases; these are ephemeral states in which the dynamics of one or more subsystems takes a more regular form. We present measures of the complexity of these incoherent lattices, and discuss the concept of a "dynamical horizon" (that is, the distance along the lattice that one subsystem can effectively influence another) and error propagation (how the introduction of a disturbance in one subsystem becomes spread throughout the lattice).

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“…The stability of globally coupled maps is well studied in the literature [11,12,13]. An ideal example to consider the stability of the driven synchronized state is a complete bipartite network.…”

Section: Linear Stability Analysismentioning

confidence: 99%

Coupled dynamics on networks

Amritkar

Jalan

2005

Physica A: Statistical Mechanics and its Applications

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We study the synchronization of coupled dynamical systems on a variety of networks. The dynamics is governed by a local nonlinear map or flow for each node of the network and couplings connecting different nodes via the links of the network. For small coupling strengths nodes show turbulent behavior but form synchronized clusters as coupling increases. When nodes show synchronized behaviour, we observe two interesting phenomena. First, there are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. Secondly, we identify two different ways of cluster formation, namely self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings.

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Stability Theory of Synchronized Motion in Coupled-Oscillator Systems

Cited by 1,243 publications

References 13 publications

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

Complex dynamics and synchronization of delayed-feedback nonlinear oscillators

The Lorenz–Fermi–Pasta–Ulam experiment

Coupled dynamics on networks

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