Bifurcation and scaling at the aging transition boundary in globally coupled excitable and oscillatory units

Abstract

Following a previous paper [Phys. Rev. E 88, 052907 (2013)], we study in detail the mechanism of aging transition in globally and diffusively coupled excitable and oscillatory units. Here two of the three models taken up in the earlier work are used, each composed of a large number of units with their bifurcation parameters forming a uniform distribution. The control parameters are the coupling strength and the average of the bifurcation parameters. The present work is mostly devoted to a region of the phase diagram near the aging transition boundary with the coupling strength greater than its critical value for the onset of bistability and hysteresis. The bifurcation structure of each system at the aging transition boundary is investigated theoretically as well as numerically. Moreover, we show that critical scaling laws of order parameters S and M used in the previous paper are different depending on which region of the coupling strength to be chosen.

Boundary in the dynamic phase of globally coupled oscillatory and excitable units

“…As this ratio reaches a critical value, the system makes a transition from the dynamic phase to the static one, which is defined as an aging transition [6]. Two major scenarios for the onset of oscillation are Hopf bifurcation [6][7][8][9][10][11][12][13][14][15] and saddle-node bifurcation on an invariant curve (SNIC) [14,[16][17][18]. Initially, our studies focused on the former scenario [6][7][8][9], but in recent years, our work has been concerned with the latter.…”

“…Initially, our studies focused on the former scenario [6][7][8][9], but in recent years, our work has been concerned with the latter. To be more specific, our target is globally coupled oscillatory and excitable units in which bifurcation parameters are continuously distributed [17,18]. The phase diagram of such a system is constructed using the coupling strength as the abscissa and the average of bifurcation parameters as the ordinate.…”

“…The difference between DP1 and DP2 is that the mean field is asymptotically (i.e., for t → ∞) constant in DP1 and oscillates in DP2, where the system size is assumed to be infinite. Our previous papers [17,18] deal with the aging transition boundary which separates the SP and a DP (DP1 or DP2).…”

“…II. An important order parameter denoted by M [6,27,28], which appears in the previous papers [17,18], is a standard deviation of Z s temporal variations, i.e.,…”

“…Phase diagrams for the case of a uniform distribution of a j : (a) γ = 0.1 and (b) γ = 0.2. In each panel, DP2 and DP1 are, respectively, situated on the upper and lower sides of the boundary, and the aging transition boundary is not shown (see[17,18] for details). Moreover, the red pluses show the boundary between DP1 and DP2 based on simulation results (N = 10 000), in which the boundary was obtained as the mean of the two nearest values of a with M > 0.05 and M < 0.05, as a was decreased for each value of K, where the step size of a is 0.02 or 0.01.…”