Border-collision bifurcation route to strange nonchaotic attractors in the piecewise linear normal form map

Abstract

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs.