Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.
Do Arnold tongues really constitute a fractal set?
