On bifurcations and local stability in 1-D nonlinear discrete dynamical systems

In this paper, a theory of bifurcations and local stability of fixed-points (or period-1 solutions) in one-dimensional nonlinear discrete dynamical systems is presented. The linearized discrete dynamical systems are discussed first, and the higher-order singularity and monotonic and oscillatory stability of fixed-points for one-dimensional nonlinear discrete dynamical systems are presented. The monotonic and oscillatory bifurcations of fixed-points (period-1 solutions) are presented. A few special examples in 1-dimensional maps are presented for a better understanding of the general theory for the stability and bifurcation of nonlinear discrete dynamical systems. Global analysis of period-2 motions for the sampled nonlinear discrete dynamical systems are carried out, and global illustrations of period-1 to period-2 solutions in the sampled nonlinear discrete dynamical systems are given.