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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.

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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