Codimension two bifurcations of nonlinear systems driven by fuzzy noise

By means of a fuzzy generalized cell mapping method, a Duffing–Van der Pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period-one attractors grow and merge as the intensity of the fuzzy noise is increased. By introducing a small symmetry-breaking parameter to break the symmetry, the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations. Considering both the intensity of the fuzzy noise and the symmetry-breaking parameter together as controls, a codimension two bifurcation of fuzzy attractors is defined, and examples of additive and multiplicative fuzzy noise are given. Such a codimension two bifurcation is fuzzy noise-induced effects which cannot be seen in the deterministic systems.