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Amplitude death in coupled robust-chaos oscillators  [PDF]
M. J. Palazzi,M. G. Cosenza
Physics , 2014, DOI: 10.1140/epjst/e2014-02296-5
Abstract: We investigate the synchronization behavior of a system of globally coupled, continuous-time oscillators possessing robust chaos. The local dynamics corresponds to the Shimizu-Morioka model where the occurrence of robust chaos in a region of its parameter space has been recently discovered. We show that the global coupling can drive the oscillators to synchronization into a fixed point created by the coupling, resulting in amplitude death in the system. The existence of robust chaos allows to introduce heterogeneity in the local parameters, while guaranteeing the functioning of all the oscillators in a chaotic mode. In this case, the system reaches a state of oscillation death, with coexisting clusters of oscillators in different steady states. The phenomena of amplitude death or oscillation death in coupled robust-chaos flows could be employed as mechanisms for stabilization and control in systems that require reliable operation under chaos.
Amplitude and phase dynamics of noisy oscillators  [PDF]
Michele Bonnin
Physics , 2015,
Abstract: A description in terms of phase and amplitude variables is given, for nonlinear oscillators subject to white Gaussian noise described by It\^o stochastic differential equations. The stochastic differential equations derived for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. If the noise intensity is small, the equations can be efficiently solved using asymptotic expansions. Formulas for the expected angular frequency, expected oscillation amplitude and amplitude variance are derived using It\^o calculus.
FRUSTRATION EFFECT ON SYNCHRONIZATION AND CHAOS IN COUPLED OSCILLATORS

Zheng Zhi-gang,

中国物理 B , 2001,
Abstract: Synchronization dynamics in an array of coupled periodic oscillators with quenched natural frequencies are discussed in the presence of homogeneous phase shifts (frustrations). Frustration-induced desynchronization and chaos are found. The torus-doubling route to chaos, toroidal chaos and torus crisis are investigated.
Amplitude death in coupled chaotic oscillators  [PDF]
Awadhesh Prasad
Physics , 2005, DOI: 10.1103/PhysRevE.72.056204
Abstract: Amplitude death can occur in chaotic dynamical systems with time-delay coupling, similar to the case of coupled limit cycles. The coupling leads to stabilization of fixed points of the subsystems. This phenomenon is quite general, and occurs for identical as well as nonidentical coupled chaotic systems. Using the Lorenz and R\"ossler chaotic oscillators to construct representative systems, various possible transitions from chaotic dynamics to fixed points are discussed.
Effects of gradient coupling on amplitude death in nonidentical oscillators  [PDF]
Weiqing Liu,Jinghua Xiao,Lixiang Li,Ye Wu,Min Lu
Physics , 2011, DOI: 10.1007/s11071-012-0325-2
Abstract: In this work, we investigate gradient coupling effect on amplitude death in an array of N cou- pled nonidentical oscillators with no-flux boundary conditions and periodic boundary conditions respectively. We find that the effects of gradient coupling on amplitude death in diffusive coupled nonidentical oscillators is quite different between those two boundaries conditions. With no-flux boundary conditions, there is a system size related critical gradient coupling $r_c$ within which the gradient coupling tends to monotonically enlarge the amplitude death domain in the parameter space. With the periodical boundary conditions, there is an optimal gradient coupling constant $r_o$ to realize largest AD domain. The gradient coupling first enlarges then decreases the amplitude death domain of diffusive coupled oscillators. The amplitude death domain of parameter space are analytically predicted for small number of gradient coupled oscillators.
Characteristics of the Wave Function of Coupled Oscillators in Semiquantum Chaos  [PDF]
Gang Wu,Jinming Dong
Physics , 2007, DOI: 10.1016/S0375-9601(02)00756-9
Abstract: Using the method of adiabatic invariants and the Born-Oppenheimer approximation, we have successfully got the excited-state wave functions for a pair of coupled oscillators in the so-called \textit{semiquantum chaos}. Some interesting characteristics in the \textit{Fourier spectra} of the wave functions and its \textit{Correlation Functions} in the regular and chaos states have been found, which offers a new way to distinguish the regular and chaotic states in quantum system.
Distributed Delays Facilitate Amplitude Death of Coupled Oscillators  [PDF]
Fatihcan M. Atay
Physics , 2003, DOI: 10.1103/PhysRevLett.91.094101
Abstract: Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.
Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators  [PDF]
M. Lakshmanan
Physics , 1997, DOI: 10.1007/BFb0113697
Abstract: In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.
Neural Computation with Rings of Quasiperiodic Oscillators  [PDF]
E. A. Rietman,R. W. Hillis
Computer Science , 2006,
Abstract: We describe the use of quasiperiodic oscillators for computation and control of robots. We also describe their relationship to central pattern generators in simple organisms and develop a group theory for describing the dynamics of these systems.
Phase and Amplitude dynamics of nonlinearly coupled oscillators  [PDF]
P. Cudmore,C. A. Holmes
Mathematics , 2014, DOI: 10.1063/1.4908604
Abstract: This paper addresses the amplitude and phase dynamics of a large system non-linear coupled, non-identical damped harmonic oscillators, which is based on recent research in coupled oscillation in optomechanics. Our goal is to investigate the existence and stability of collective behaviour which occurs due to a play-off between the distribution of individual oscillator frequency and the type of nonlinear coupling. We show that this system exhibits synchronisation, where all oscillators are rotating at the same rate, and that in the synchronised state the system has a regular structure related to the distribution of the frequencies of the individual oscillators. Using a geometric description we show how changes in the non-linear coupling function can cause pitchfork and saddle-node bifurcations which create or destroy stable and unstable synchronised solutions. We apply these results to show how in-phase and anti-phase solutions are created in a system with a bi-modal distribution of frequencies.
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