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 phi 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-Neimark-Sacker bifurcation which gives rise to the alternation of periodic and quasi-periodic dynamics organized by Arnold tongues in the (phi, a)-plane. Inside a tongue a periodic attractor typically either undergoes a period doubling cascade, which leads to chaotic dynamics, or a Hopf-Neimark-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 Henon-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. (C) 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved