Conservative dynamics in phase oscillator networks

Abstract

The interaction between phase oscillators is conservative if the phase volume is conserved throughout the dynamics. We derive a general condition, based on the notion of a pair-Hamiltonian, for the pairwise couplings to be conservative. The conservative networks with Winfree-type and Kuramoto-Daido-type couplings are also discussed. It is demonstrated that although, in contradistinction to genuine Hamiltonian dynamics, there is no exact pairwise symmetry of the Lyapunov exponents, the Lyapunov spectrum for a large network is nearly symmetric. The concept is also generalized to triplet and quadruplet couplings.

Figures (5)

• Figure 1: Trajectories at the conservative Winfree-type coupling of two oscillators. Parameters are given in text.
• Figure 2: Poincaré map at the conservative Winfree-type coupling of three oscillators. Parameters are given in text. For some initial conditions, the trajectory forms a closed curve (torus in the phase space), while at other initial conditions, chaotic sets are produced.
• Figure 3: Cumulative distributions of the largest Lyapunov exponent for $N=3,4$ and different coupling strengths $\varepsilon$ (lines of different colors), calculated from $1000$ independent runs as described in the text.
• Figure 4: The sums of the largest and of the smallest Lyapunov exponents vs the largest exponent for conservative networks of 5 phase oscillators.
• Figure 5: Spectra of Lyapunov exponents for several values of $N$, for random networks with Kuramoto-type coupling \ref{['eq:12']}. The spectrum appears nearly anti-symmetric with respect to the midpoint $i=N/2$.