Study On The Chimera States Of Coupled Oscillators

The collective behavior in coupled oscillators has been researched for years, especially in the aspect of synchronization. It has very wide applications in the biology and neuroscience. One of the most famous models is the Kuramoto model. In recent years,some research groups have found the"Chimera state"behavior in the identical Kuramoto model, where some of the oscillators are synchronized, and other oscillators are asynchronous. This collective behavior has been named"Chimera state". Many creatures sleep with only half of their brain at a time[1]. Such unihemispheric sleep was first reported in dolphins and some other sea mammals, and has now been seen in birds and referred in lizards [2]. When brain wave is recorded in the experiment, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neuros oscillating out of phase, whereas the sleeping side is highly synchronized. This phenomenon is called half-brain sleep in biology. Many scholars are studying this phenomenon. Abrams et al. use the simplest system of Kuramoto model to explain this phenomenon[3]. Our work is to observe the collective behavior from the aspect of time delay in coupling term and size asymmetry based on the Abrams model. After that we propose a new model to explain the alternating chimera states in the most mammals.Fisrtly, considering that the finite velocity of transmission between the neurons will result a time delay, we introduce time delay in the Abrams model to study the influence to the three kinds of chimera state. We find that the fixed time delay can result in phase transition, but the Chimera states of system have robustness to the random time delay.Secondly, we consider the influence of the size asymmetry in the Abrams model. Because it has two subgroups, and the size of every subgroup is equal, the first subgroup is always synchronized. So we try to change the size of the first subgroup, and observe the collective behavior. We find that the Chimera state of system has robustness to the size asymmetry.Thirdly, the sleep mode is one half brain working and another half brain sleeping in some mammals. After some time it turn over. That is, the two half brain are always alternating sleep. So we propose a new model to understand this mechanism and explain the phenomenon. It is useful for us to recognize and understand the real biological phenomenon.From the research of the above, we find that Abrams model has robustness to the time delay and the size asymmetry. We propose the new model to explain the alternating chimera state in many sea mammls. We find the system has robustness to the some parameters of the system and the external signal.