Critical points and dynamic systems with planar hexagonal symmetry

In this investigation, we detect and utilize critical points of functions with hexagonal symmetry in order to study their dynamics. The asymmetric unit in a parallelogram lattice is chosen as the initial searching region for a critical point set in a dynamic plane. The accelerated direct search algorithm is used within the parallelogram lattice to search for the critical points. Parameter space is separated into regions (chaotic, periodic or mixed) by the Ljapunov exponents of the critical points. Then the generalized Mandelbrot set (M-set), which is a cross-section of the parameter space, is constructed. Many chaotic attractors and filled-in Julia sets can be generated by using parameters from this kind of M-sets.