The mean fields [panels (a) and (b)] and their phases [panel (c)] for two interacting ensembles of van der Pol oscillators (12) with . The modulation function of the coupling are shown in (a) and (b) with dashed lines. To make picture (c) clearer, we subtract a linear phase growth and plot the value , where is chosen close to the empirical mean frequency , which slightly differs from due to nonlinearity. is shown with solid line and with dashed one.
Evolution of amplitudes and phases for the mean fields of coupled ensembles described by slow complex amplitudes [panels (a) and (b)] and in the phase approximation [panels (c) and (d)] with . Solid lines: variables ; dashed lines: variables .
Stroboscopic maps over the period of external modulation for (a) the ensembles of van der Pol oscillators (12), (b) for the ensembles described by slow complex amplitudes (5), and (c) for the ensembles in the phase approximation (8); in all cases, . In all cases, the dynamics seems to be well described by the Bernoulli map (16). The observed splitting of the “lines” [the most pronounced in panel (a)] appears because of the presence of transversal fractal structure of the attractors (see Fig. 4): distinct filaments of the attractor give rise to distinct filaments on the phase iteration diagram due to imperfection of the phase definition.
Projections of the stroboscopic maps on the plane of the order parameters: for the van der Pol oscillators (left column); for the ensembles described by slow complex amplitudes (center column); and for the ensembles in the phase approximation. The bottom row shows enlargements to make the fractal transversal structure evident. Here, .
Stroboscopic maps over the period of external modulation for individual oscillators. Left panel: oscillator at the center of the band with ; right panel: oscillator with .
Bifurcation diagrams for the ensembles in the complex amplitude formulation (5) showing dynamical regimes in dependence on the ensemble size . (a) Periods for periodic regimes. The values of for which no period is plotted correspond to the chaotic states. (b) Three largest Lyapunov exponents (first: filled circles; second: pluses; and third: diamonds). Only a few regimes with small number of oscillators have two positive exponents. In panel (c), we show only positive largest Lyapunov exponents, in a logarithmic scale, to demonstrate that it does not tend to decrease for large ensemble sizes .
Lyapunov exponents plotted vs index for the ensembles described by complex amplitudes (5) with . exponents corresponding to the phases are close to zero, while the rest exponents corresponding to the stable amplitudes are negative. The right panel shows the region with small ; in all cases, only the first exponent is positive.
Article metrics loading...
Full text loading...