Hopf Oscillator Research Articles

Mode dynamics of interacting neural populations by bi-orthogonal spectral decomposition.

A system of coupled bistable Hopf oscillators with an external periodic input source was used to model the ability of interacting neural populations to synchronize and desynchronize in response to variations of the input signal. We propose that, in biological systems, the settings of internal and external coupling strengths will affect the behaviour of the system to a greater degree than the input frequency. While input frequency and coupling strength were varied, the spatio-temporal dynamics of the network was examined by the bi-orthogonal decomposition technique. Within this method, effects of variation of input frequency and coupling strength were analyzed in terms of global, spatial and temporal mode entropy and energy, using the spatio-temporal data of the system. We observed a discontinuous evolution of spatio-temporal patterns depending sensitively on both the input frequency and the internal and external coupling strengths of the network.

Re-Entrant Hexagons and Locked Turing–Hopf Fronts in the CIMA Reaction

Aspects of the mode-interaction and pattern-selection processes in far-from-equilibrium chemical reaction–diffusion systems are studied through numerical simulation of the Lengyel–Epstein model. By varying the feed concentrations, a transition is observed in which hexagons are replaced by stripes and these again by inverted hexagons. The competition between Hopf oscillations and Turing stripes is investigated by following the propagation of a front connecting the two modes. In certain parameter regimes, mode-locking is found to occur. The front then moves an integer number of Turing stripes during an integer number of Hopf oscillations. This phenomenon can be seen as arising from depinning of the Turing front under influence of the Hopf mode.

Computer simulation of Turing structures in the chlorite-iodide-malonic acid system

The emergence, growth and stabilization of stationary concentration patterns in a continuously fed chemical reaction-diffusion system are studied through numerical simulation of the Lengyel-Epstein model. This model represents a key to understanding the recently obtained Turing structures in the chlorite-iodide-malonic acid system. Using the supply of iodine as a control parameter, the regularity of the hexagonal patterns that develop from the noise inflicted homogeneous steady state is examined. In the region where they are both stable, the competition between Hopf oscillations and Turing stripes is studied by following the propagation of a front connecting the two modes. Finally, examples are given for the types of structures that can develop when a gradient in feed concentration is applied to the system.

Stability of self-pulsing due to Hopf bifurcation in semiconductor lasers

A formulation is given of the stability criterion for oscillations arising at Hopf bifurcations in semiconductor lasers. A second-order perturbation expansion is utilized to provide nonlinear rate equations for the laser. The rate equations contain terms whose second-order partial derivatives are required in the definition of stability parameters. Using these parameters, a simple statement of the stability criterion is possible. Numerical calculations are included, using typical device parameters, and show the existence of both stable and unstable Hopf oscillations.