Mechanism and Verification of Strange Nonchaotic Attractors in a Quasi-Periodically Forced Nonsmooth System

The generation mechanisms of Strange Nonchaotic Attractors (SNAs) in smooth dynamical systems have been extensively investigated, yielding substantial insights, yet nonsmooth systems remain relatively understudied due to increased verification complexity. This paper investigates multiple routes to SNAs in a quasi-periodically forced nonsmooth Logistic map, including the torus-doubling, intermittency, bubbling, and incremental routes. The incremental route exhibits distinct features: during the transition from quasi-periodic oscillation to SNA, attractor doubling appears in phase trajectories without torus-doubling bifurcation, and the 3-torus quasi-periodic attractor evolves directly into an SNA without full separation. For verification, the Lyapunov exponent is employed for checking the nonchaotic property, while phase sensitivity, power spectrum, and spectral distribution function are used for verifying the strange property. Significantly, this paper employs a quantitative method (quantifying strange property) to directly validate the strange property of the SNA. This method can directly determine whether the attractor possesses the strange property.