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Stability theory of synchronized motion in coupled-oscillator systems. (English) Zbl 1171.70306
Summary: The general stability theory of the synchronized motions of the coupled-oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state $$\Phi_{unif}$$. When the coupling strength is weakened, the coupled system may exhibit several types of non-synchronized motion. In particular, if $$\Phi_{unif}$$ is chaotic, we always get a transition from chaotic $$\Phi_{unif}$$ to a certain non-uniform state and finally the non-uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.

##### MSC:
 70K20 Stability for nonlinear problems in mechanics 80A30 Chemical kinetics in thermodynamics and heat transfer 93D99 Stability of control systems 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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