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Title: Unstable attractors and irregular transients in networks of pulse-coupled oscillators
Keywords: Pulse-coupled oscillators, Unstable attractors, Stable irregular transients, Chaotic irregular transients, Inhibitory, Excitatory
Issue Date: 5-Aug-2011
Source: ZOU HAILIN (2011-08-05). Unstable attractors and irregular transients in networks of pulse-coupled oscillators. ScholarBank@NUS Repository.
Abstract: We explore the mechanisms of some novel collective dynamical behaviors in networks of pulse-coupled oscillators. The model possesses three main characteristics: individual threshold dynamics to generate pulses which mediate the interactions, a delay time in the pulse transmission, and random disorder in the coupling structures. Specifically, we investigate unstable attractors and long irregular transients, whose mechanisms are unknown. We mainly use the event approach focusing on the microscopic events such as firing and receiving of pulses to study these collective behaviors. We first investigate the source of instability for unstable attractors in networks of excitatory pulse-coupled oscillators. Unstable attractors are a type of attractors whose nearby points within a neighborhood will almost leave this neighborhood. An oscillator fires and sends out a pulse when reaching the threshold. In terms of these firing events, we find that the unstable attractors have a simple property hidden in the event sequences. They coexist with active simultaneous firing events. That is, at least two oscillators reach the threshold simultaneously, which is not directly caused by the receiving pulses. We show that the split of the active simultaneous firing events by general perturbations can make the nearby points leave the unstable attractors. Furthermore, this structure can be applied to study the bifurcation of unstable attractors. Unstable attractors can bifurcate due to the failure of establishing active simultaneous firing events. We then study the dynamical mechanism of long chaotic irregular transients in networks of excitatory pulse-coupled oscillators by the event approach. We introduce a type of attractors with certain event structure: sequential active firing (SAF). By using the fraction of SAF attractors in phase space as an order parameter, a phase boundary between SAF and non-SAF attractors is located. Interestingly, the long chaotic transients occur near the phase boundary. The bifurcations of SAF attractors tend to induce irregular transients because passive firings are easier to be converted into active firings near the phase boundary. In addition, many SAF attractors bifurcate near the phase boundary. The above two facts can greatly enhance the average transient time near the phase boundary. Lastly, we investigate the long irregular transients in networks of inhibitory pulse-coupled oscillators, which are insensitive to infinitesimal perturbations. We focus on the dynamical formation of these irregular transients. Interestingly, it is found that the transient dynamics has a hidden pattern in phase space: it repeatedly approaches a basin boundary and then jumps from the boundary to a remote region in phase space. This pattern can be clearly visualized by measuring the distance sequences between the trajectory and the basin boundaries. The dynamical formation of these stable irregular transients originates from the intersection points of the discontinuous boundaries and their images. We carry out numerical experiments to verify this mechanism.
Appears in Collections:Ph.D Theses (Open)

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