Network structures resulting from varying parameter . The gray disks symbolize populations, inhabited by individual oscillators symbolized by black dots. Their bidirectional coupling is represented by black lines. Each population has a self-coupling of unit strength 1. The population in what becomes the center for is coupled to the neighboring populations with strength ; the populations to the left and right are coupled with strength . The case of a triangular network is obtained for ; the character of the network has chain-like character for .
Effect of breaking the rotational symmetry on the bifurcation diagram for the SDS and DSD symmetries. The triangular case (see Ref. 13 ) with is shown in the left column in (a) and (c) for comparison with the case of broken symmetry in the right column in (b) and (d). The displayed curves are the saddle-node curve (solid red), the Hopf curve (dashed blue), and the homoclinic curve (dotted black). Dots mark the bifurcation points obtained by inspection of the phase plane. The homoclinic curve is an interpolation based on these points, whereas the solid curves were obtained analytically.
Bifurcation diagram for the SDS chimera (above) and the two DSD chimeras (below) in the -plane for a range of values. Saddle node (solid red) and Hopf curves (dashed blue) are shown. The curves related to the SDS chimera collapse onto the -axis at ; conversely, the two DSD curves collapse on the axis at and , respectively. It is seen that the Bogdanov–Takens point (black dots) of the upper DSD chimera follows the associated saddle-node curve as we vary .
Boundaries for the occurrence of saddle-node transitions in the -plane (i.e., saddle-node curves at ) shown for the SDS symmetry (a) and the two DSD symmetries in (b) [ is the boundary for the second DSD state seen in the upper part in Fig. 3(b) ]. The regions shaded in gray either have negative coupling or have coupling without chain-like character, as explained in the text. Throughout the regions labeled with No chimeras, we find no chimeras. The black dots indicate points of special interest. (A) is where the saddle-node curves detach from the origin -plane. (B) are the points for which the BT points collide with the -axis leading to the annihilation of the chimera state; , , and . (C) Intersection with boundaries of positive coupling; and . (D) Intersection of the saddle-node boundary with ; . The regions where chimera exists for , which are stable within the symmetry manifolds SDS and DSD, are hatched green.
Article metrics loading...
Full text loading...