Fuzzy Noise-Induced Codimension-Two Bifurcations Captured by Fuzzy Generalized Cell Mapping with Adaptive Interpolation

In this paper, the Fuzzy Generalized Cell Mapping (FGCM) method is developed with the help of the Adaptive Interpolation (AI) in the space of fuzzy parameters. The adaptive interpolation on the set-valued fuzzy parameter is introduced in computing the one-step transition membership matrix to enhance the efficiency of the FGCM. For each of initial points in the state space, a coarse database is constructed at first, and then interpolation nodes are inserted into the database iteratively each time errors are examined with the explicit formula of interpolation error until the maximal errors are just under the error bound. With such an adaptively expanded database on hand, interpolating calculations assure the required accuracy with maximum efficiency gains. The new method is termed as Fuzzy Generalized Cell Mapping with Adaptive Interpolation (FGCM with AI), and is used to investigate codimension-two bifurcations in two-dimensional and three-dimensional nonlinear dynamical systems with fuzzy noise. It is found that global changes in fuzzy dynamics are dominated by the underlying deterministic counterparts, and the fuzzy attractor expands along the unstable manifold leading to a collision with a saddle when a bifurcation occurs. The examples show that the FGCM with AI has a thirtyfold to fiftyfold efficiency over the traditional FGCM to achieve the same analyzing accuracy.