Ring Of Oscillators Research Articles

Observation of a Fast Rotating Wave in Rings of Coupled Chaotic Oscillators

We report on the discovery of a transition in rings of coupled electronic circuits in the chaotic regime to a collective periodic state characterized by a time scale that is between two and three orders faster than that corresponding to an isolated circuit. This transition arises from a linear instability in the uniform synchronized state of the ring through a symmetric Hopf bifurcation. The same type of transition is also observed for other coupled chaotic systems, e.g., a ring of Lorenz attractors.

The dynamics ofn weakly coupled identical oscillators

We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.

Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators

The results of an investigation of the structure and stability of various oscillatory modes of an inductance coupled ring of the van der Pol oscillators, using a nonlinear mode analysis, are presented. It has been demonstrated that all single modes, both standing and traveling waves, are stable, but that simultaneous two and three modes are not stable. Experiments have been performed using four- and five-oscillator rings and the results obtained agree well with the theory.

Mathematical modeling of the colorectal myoelectrical activity in humans.

Measurement of the colorectal myoelectrical activity in humans has revealed three basic patterns of behavior, comprising a lower frequency oscillation of about 0.05 Hz, a higher frequency of about 0.12 Hz and periods of zero activity. To simulate these myoelectrical patterns, the concept of linked relaxation oscillators is extended in this paper using three different mathematical model structures. In the three models two of the activity patterns are obtained as two stable limit cycle solutions produced by symmetrically coupling together two relaxation oscillators. The first model comprises a ring of interconnected oscillators which produces a third stable solution representing the higher frequency of oscillation. Summation of adjacent oscillator outputs reproduces the zero activity when the oscillators are in antiphase. By addition of an extra coefficient into the basic relaxation oscillator equation for the second model, the zero state becomes a stable condition and the three patterns are obtained without the necessity of a ring structure. The third model requires an in-situ change in a parameter value for the lower and higher frequencies to decay away to the zero activity condition.