Symmetry-breaking and fractal dependence on initial conditions in dynamical systems: One-dimensional noninvertible mappings

It is shown that symmetry caused degeneracy leading to presence of two co-existing attractors may lead to fractal dependence on initial conditions in one-dimensional (1D) noninvertible maps. It is demonstrated that in the case of 1D dynamical systems with symmetry the basin boundary structure is determined by the mechanism of symmetry breaking.