The dynamics of a fold-and-twist map

Inspired by a discrete-time predator-prey model we introduce a planar, noninvertible map composed of a rigid rotation over an angle φ and a quadratic map depending on a parameter a. We study the dynamics of this map with a particular emphasis on the transitions from orderly to chaotic dynamics. For a=3 a stable fixed point bifurcates through a Hopf-NeÄ­mark-Sacker bifurcation which gives rise to the alternation of periodic and quasi-periodic dynamics organized by Arnold tongues in the (φ,a)-plane. Inside a tongue a periodic attractor typically either undergoes a period doubling cascade, which leads to chaotic dynamics, or a Hopf-NeÄ­mark-Sacker bifurcation, which leads in turn to a new family of Arnold tongues. Numerical evidence suggests the existence of strange attractors with both one and two positive Lyapunov exponents. The former attractors are conjectured to be Hénon-like, i.e., they are formed by the closure of the unstable manifold of a periodic point of saddle type. The folded nature of such attractors is the novel feature of this paper.