If in a quasiperiodically driven system we have a smooth torus attractor at one value of control parameter and an SNA at some other, what are the transitions (bifurcations) observable on a road from one situation to another under a very slow parameter variation? One scenario consists in the so-called blowout bifurcation and occurs in the pitchfork bifurcation model. An interesting approach to a study of details of the onset of SNA was developed on a basis of renormalization group method.
Bifurcation scenarios in a quadratic map
It is natural to start examination of bifurcation scenarios in the model system
with the torus-doubling bifurcations (Anishchenko, 1982, Kaneko, 1983). As known, at e=0 we have a sequence of the period-doubling bifurcations with Feigenbaum's transition to chaos. The first bifurcation consists in stability lose of a fixed point and a birth of a stable period-two cycle, i.e. two points visited one after another. In presence of the quasiperiodic driving, instead of a stable fixed point we get an attractive closed invariant curve (the torus-attractor) T1, and the bifurcation consists in an event that two closed curves separate from the former one, which becomes unstable. They are visited one after one and compose a new attractor, the doubled torus T2 (see figure). At small e, increase of l, that is, a motion along the line TD on the parameter plane, leads to several subsequent torus bifurcations (T1-->T2-->T4-->:). In contrast to the Feigenbaum cascade, this sequence contains a finite number of bifurcations (the less e, the larger number of the bifurcations).
Heagy and Hammel (1994) have considered a mechanism responsible for restriction of the number ob torus doublings and associated a scenario of birth of SNA with it. This corresponds to a motion on the parameter plane on a road marked as HH and is illustrated by phase portraits in the following figure. After the birth of the doubled torus (two green curves), a parent torus continues to exist as an unstable invariant set (red curve). With increase of l the components of the doubled torus accept more and more wrinkled form, grow in width, and finally touch the parent invariant curve by their 'protuberances'. After the bifurcation, the attractive set does not contains disjoint components and is a unified single SNA.
Another scenario called torus fractalization has been described by Nishikawa and Kaneko (1996). Under variation of the control parameter the attractive invariant curve gradually develops wrinkling and becomes a fractal curve, which is an SNA. This takes place on a road marked FT and is illustrated by phase portraits in the following figure.
Transition to SNA on a road CI, consists in a sharp expansion of the domain occupied by the attractor, although the system still spends overwhelming part of time in a neighborhood of the former attractive invariant curve. It looks like intermittency. Hypothetically, Prasad et al. (1997) interpreted this transition with bifurcation of tori collision analogous to the collision of a stable and an unstable fixed point in a course of tangent (saddle-node) bifurcation. In fact, the stable torus collides not with an unstable torus-partner, but with a specific unstable invariant set identified by Kim, Lim and Ott (2003) in approach based on rational approximations as a 'ring-shape invariant set'. Apparently, the phenomenon has to be interpreted as a kind of crisis of the torus-attractor leading to appearance of SNA.
Transition to SNA in a model with tangent bifurcation
A real analog of the tangent bifurcation and of the Pomeau-Manneville intermittency is a transition in a driven circle map (Feudel et al, 1995; Osinga et al., 2001), and in the model based on the fractional-linear mapping with a modification introducing the re-injection mechanism (Kuznetsov, 2002):
It appears that there exists a critical value of the driving amplitude, ec, and the transition observed under increase of parameter b is of essentially different nature at e<ec and at e>ec.
Remark. In the model based on the fractional-linear map, the critical value of e exactly equals 2. An explanation is found with reformulation of the problem in terms of the Harper equation, for which the critical value ec=2 corresponds to the transition localization - delocalization.
In the first case (see the picture) two smooth invariant curves, a stable (green) and an unstable (red), approach each other, coincide, and disappear. The result is the birth of chaos via intermittency.
In the second case approach to the bifurcation is accompanied by formation of 'spikes' on the invariant curves. At the bifurcation they touch each other with these spikes (see the diagram below). Because of ergodicity in dynamics of the phase variable, the points of touch occupy a dense set, so at the bifurcation the formation is a fractal. This bifurcation is referred to as the fractal tori collision, and the result is the birth of SNA.
