202 Progress of Theoretical Physics, Vol. 72, No.2, August 1984 Oscillation and Doubling of Torus Kunihiko KANEKO Department of Physics, University of Tokyo, Tokyo 113 (Received February 20, 1984) Two features of the instability of torus along the amplitude direction are investigated, using various two·dimensional mappings.· First, the oscillation of a torus in a delayed logistic map is studied, which is related with the oscillation of an unstable manifold of a periodic saddle. The oscillation is also analyzed from another point of view, i.e., the synergetic effect of the rotation, stretching and folding, which is typically seen in the delayed piecewise· linear mapping. Critical phenomena after the onset of chaos are also discussed. Second, the doubling of a torus is reinvestigated. A torus doubling occurs only a finite number of times. The mechanism of interruption of the doubling is discussed from three points of view, i.e., relevant perturbation in the RG framework, fractalization and the intermittent·like bursts between the valleys of the multi· humped mapping. § 1. Introduction The onset of chaos from a torus motion is an important problem which· has been intensively and extensively studied quite recently.1l- 3l ) The instability due to the phase motion of a torus has been investigated by the use of a one-dimensional circle map.2)_1l) In general problems of the transition from torus to chaos, however, the instability along the amplitude direction is also important, which exhibits novel and interesting features in nonlinear physics. In the present paper we study two problems of the instability of a torus along the amplitude direction by resorting to various kinds of two-dimensional mappings. An important feature of the amplitude behavior of a torus is the oscillation, which 1· S been observed in a large class of two-dimensional mappings. In experiments on LIe- Benard convection, the oscillation of a torus was observed by Berge 23 ) and Sano. 29 ) In §2, we investigate the two-point delayed logistic map, which shows typically the oscillation of torus (and sometimes of chaos). The oscillation of a torus is understood in connection with the damped oscillation of an unstable manifold of a periodic saddle. Since the unstable manifold' is along the amplitude direction, the oscillation of a torus can be regarded as the representation of the instability along the amplitude direction. The experiment by Berge seems to be well explained by the results in § 2. In §3, we consider the oscillation from another point of view, i.e., a synergetic effect of the rotation, stretching and folding. In order to elucidate this effect we introduce a delayed piecewise-linear mapping, which is a simplified version of the map in §2. In usual two-dimensional mappings (and the flow with three variables), however, the oscillation is masked by a locking, from which chaos appears via period-doubling bifurca- tions. In three-dimensional mappings, the doubling of a torus itself is also possible, which has been found by Arneodo et al.,13) Franceschini (for a flow system),14) and by the author. 15) In these examples the doubling occurs only a finite number of times before chaos appears as is shown in Ref.15). In §4, we consider the mechanism of the interrup- tion of the doubling cascade of tori using a coupled circle and logistic map. Its mecha- nism is understood from three points of view, i.e., first, from a renormalization group viewpoint, second, from the {ractalization of torus/ 9 ) and lastly by the intermittent- Downloaded from https://academic.oup.com/ptp/article/72/2/202/1829127 by guest on 31 July 2022