Hopf-Hopf Bifurcation and Chaos in A Ring Neural Oscillator due to A Single Shortcut

Bifurcations and chaos in a ring of seven sigmoidal neurons with unidirectional inhibitory coupling caused by a single shortcut are examined. In the absence of shortcuts, the network has the form of a ring oscillator and has a stable and an unstable periodic solution, which are generated through the Hopf bifurcations from the origin. In the presence of an inhibitory shortcut from the third neuron to the first neuron, the loci of the two periodic solutions intersect at a codimension-two Hopf-Hopf bifurcation point in the parameter plane. The loci of the Neimark-Sacker bifurcations of the periodic solutions emanate from the Hopf-Hopf bifurcation point and stable quasiperiodic solutions are generated. Multiple chaotic attractors are generated through period-doubling cascades of periodic solutions in the Arnold's tongues emanating from the loci of the Neimark-Sacker bifurcations. The crises of the chaotic attractors occur so that transient chaos and crisis-induced intermittency are observed. A single shortcut brings about these rich dynamical properties in a simple ring neural network.