(Color online) Real-valued units in a permutation-symmetric smooth network dynamical system [Eq. (5)] cannot pass each other because they cannot cross the invariant manifolds (dashed) which are fixed points of the permutation symmetries [Eq. (3)]. A projection of two trajectories (black, solid) onto the - plane is sketched, one on the invariant set and one that stays on one side of the invariant manifold.
(Color) Breakdown of order preservation in permutation symmetric pulse-coupled networks: (a) dynamics of pulse-coupled units [Eq. (8)] with for and otherwise, , , , coupling strength , and initial condition near the synchronous state to which the dynamics converges. In the process of synchronization, the units pass each other; (b) magnification.
(Color online) Model. Time evolution [Eq. (8)] of the potential and the corresponding phase of one oscillator [Eqs. (10) and (11)]: (a) the time evolution of the oscillator’s potential, which evolves freely for one period and then receives an inhibitory pulse of strength at time causing a potential jump from to ; (b) the time evolution of the corresponding phase that jumps according to the transfer function (12) from to .
(Color) Order preservation and its breakdown in the time evolution of a network of pulse-coupled oscillators [Eqs. (10) and (11)] with , [Eq. (21)]. The phases of all oscillators (color-coded) are shown vs. time . Insets show magnifications as indicated: (a) time evolution for and with approach to stable synchronous state where the ordering of the oscillators is conserved; (b) similar dynamics that synchronize more slowly for and ; (c) time evolution for and shows that oscillators pass each other; (d) similar dynamics with slower synchronization for and . Notice that in (d) all oscillators exchange their ordering, while in (c) only the blue oscillator overtakes the red and yellow oscillator.
(Color online) Parameter dependence of the transition between order conservation and its breakdown in - -parameter space of the pulse-coupled network used in Figure 4. For each parameter set ( in steps of , in steps of ), the system was initialized 250 times with uniform distributed phases . Shown is the percentage of these runs where the ordering is not conserved. The solid red line, given by Eq. (22), indicates the theoretically predicted transition between the regimes where the order is conserved or units pass each other. The parameter values used for Figure 4 for the cases of order conservation (•, ▴) and overtaking (+, ▾) are indicated. Because eigenvalues are close to zero near the transition line, the dynamics synchronize faster for ▴ and ▾.
(Color online) Dynamics of the ordering of the oscillator phases (, , ). Without loss of generality, we labeled all possible orderings of the oscillators from to . We start the system close to the synchronous state and plot the ordering index at discrete times just after the reception of all pulses of one cycle (return map) for (), (), and (▪).
(Color online) (a) Phase response curve [Eq. (25)] and (b) interaction function [Eq. (26)] used in the simulations shown in Figure 8.
(Color) Symmetrically pulse-coupled identical oscillators with different average frequencies: (a) dynamics of phase oscillators [Eqs. (24)–(26)] with and numerically integrated using Euler’s method with step size . Dashed lines indicate the beginning of one period of a frequency locked state; (b) pulse generation times of the two oscillators in (a) indicated by vertical bars showing the convergence of the dynamics to the periodic frequency locked state from initial condition , . By permutation symmetry of the network, the exchange of the initial phases leads to frequency locked state.
The phases of the N units directly after the events of the sequence
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