The existence of strange nonchaotic attractors in the quasiperiodically forced Ricker family

In this paper, the Ricker family (a population model) with quasiperiodic excitation is considered. The existence of strange nonchaotic attractors (SNAs) is analyzed in a co-dimension-2 parameter space by both theoretical and numerical methods. We prove that SNAs exist in a positive measure parameter set. The SNAs are nowhere differentiable (i.e., strange). We use numerical methods to identify the existence of SNAs in a larger parameter set. The nonchaotic property of SNAs is verified by evaluating the Lyapunov exponents, while the strange property is characterized by phase sensitivity and rational approximations. We also find that there is a transition region in a parameter plane in which SNAs alternate with chaotic attractors.
Strange nonchaotic attractor (SNA) is one of the most important topics in nonlinear dynamics. An SNA has a geometric structure that is fractal (strange property), with the largest Lyapunov exponent being negative, ensuring the nonchaotic property in the dynamical sense. However, there are few studies on the existence of SNAs by both theory and numerical methods. In this paper, we consider the Ricker family population model with quasiperiodic excitation, and the existence interval of SNAs is obtained by combining theoretical and numerical methods in a parameter space. Our results show that the SNAs are discontinuous almost everywhere in a positive measure parameter space. Numerically, we show that SNAs exist in a larger parameter region.