Continuously pleated torus or strange nonchaotic attractors seen as continuously fractalized quasiperiodic attractors

Toroidal chaos designates a chaotic solution that is structured in the neighborhood of a torus. For being chaotic, such a torus needs to be discontinuously fractalized to present a Cantor set, ensuring the great sensitivity to initial conditions required for chaos. A fractal is related to self-similarity and a non-integer dimension. In fact, although clearly introduced as such by Mandelbrot, it was never written in a paper related to strange nonchaotic attractors that an object can be fractal in two very different ways: under a continuous way, as a coastline, or in a discontinuous way, as a Cantor set. If the latter is necessarily related to chaos, the former can, for instance, be associated with a continuous torus fractalization leading to a strange nonchaotic attractor. Moreover, although being fractal, the torus is still regular and, consequently, a strange nonchaotic attractor is a particular type of quasiperiodic attractor. We will, therefore, focus on showing with the help of three different cases-produced by two coupled van der Pol oscillators or by a quasiperiodically driven Duffing system, that the fractalization is actually continuous when strange nonchaotic attractors are observed. In one of the cases, it will be shown that the largest Lyapunov exponent wrongly suggests chaos rather than a strange nonchaotic attractor. We also investigate how, in the presence of symmetry, an attractor merging crisis-induced intermittency can be observed, just after two symmetry-related strange nonchaotic attractors.