Phase Oscillator Networks: Chimera States



We show some examples of chimera states in phase oscillator networks. All of these examples were computed using prism software. In all cases, we used Runge-Kutta 4th order, a time step 0.01 and long double precision. Apart from regarding phases as defined on \([0,1]\), we generally follow the conventions of Abrams and Strogatz' paper (Chimera States in a ring of nonlocally coupled oscillators, Int. Jnr. of Bifurcation and Chaos, 16 (1) (2000), 21-37). In particular we use an exponential kernel (with \(\kappa = 4\)) and take \(\alpha = 1.45\). Unless stated to the contrary we assume all phase oscillators have frequency term equal 0. We initialize using Kuramoto's protocol (see Abrams and Strogatz for details).

In Figure 1, we show the collective dynamics of 1000 phase oscillators over a time period of 12.67 seconds after allowing the system to evolve for 10,000 seconds (about 2.77 hours)


Figure 1. Collective Dynamics over 12.67 seconds


In Figure 2, we show the Chimera state after 10,012.67 seconds of system evolution.


Figure 2. Chimera State


Next we show what happens if we allow the frequency of the individual phase oscillators to be set randomly and uniformly within the range \(\pm 0.001 \). For these examples we take 504 oscillators. In figures 3 and 4 we show the result after evolving the system for 10,000 seconds. In figure 5, we show the chimera state after evolving for an additional 40,500 seconds.

Figure 3. Collective Dynamics over 12.67 seconds


Figure 4. Chimera State


Figure 5. Chimera State after 50,512.67 seconds (about 14 hours)


Finally we show collective dynamics and chimera state for a system of 8,000 phase oscillators after evolving the system for 7576.02 seconds (just over two hours).

Figure 6. Collective Dynamics of 8,000 oscillators over 12.67 seconds


Figure 7. Chimera State


Video

  • Chimera state movie (animated gif format). The movie shows the evolution of a chimera state for a 5000 node phase oscillator system. Computation was Runge-Kutta 4th order with a time step of 0.005. A frame of the chimera state was taken every 0.125 seconds. (Best viewed using a viewer with *no* post processing or anti-aliasing features.)

Research supported by NSF Grants DMS-0806321 and DMS-1265253. All images and movies copyright Michael Field, 2013. Images and Movies may be not be used without my written permission.
Research supported in part by DMS-0806321 and DMS-1265253 and in collaboration with Christian Bick. Software using prism and including some components based on ATLAS and CBLAS libraries as well as custom long double precision software.