VoLUME 70, NUMBER 3 PH YSICAL REVIEW LETTERS On-Off' Intermittency: A Mechanism for Bursting 18 JANUARY 1993 N. Platt IVaval Surface Warfare Center, Silver Spring, Maryland 20903 50-00 E. A. Spiegel Department of Astronomy, Columbia University, iVew York, JVew York l0027 C. Tresser IBM T, J. 8'atson Laboratories, Yorktown Heights, 1Vew York I 0598 (Received l9 May l992) On-off' intermittency is an aperiodic switching between static, or laminar, behavior and chaotic bursts of oscillation. It can be generated by systems having an unstable invariant (or quasi-invariant) mani- fold, within which is found a suitable attractor. We clarify the roles of such attractors in producing in- termittency, provide examples, and relate them to previous work. PACS numbers: 47. 20. Ky, 05. 45. +b, 47. 52.+j The simplest chaotic systems follow similar trajectories over and over again but they never exactly repeat. This behavior has long been recognized in celestial mechanics [I], but only in recent times have simple models for such aperiodic oscillatory behavior proliferated. More than this, the time dependence of a variable in a chaotic sys- tem can lead to signals with a variety of distinct forms. These range from weak aperiodic modulations of a periodic signal to apparently random switching amongst qualitatively diFerent kinds of oscillations. The latter behavior is called intermittency, probably after the usage of this word in IIuid dynamics [2]. In fluid turbulence, the term was introduced to describe sig- nals from probes in fluids that alternated between flat portions and bursty ones, interpreted as laminar and tur- bulent states of the fluid. We shall say that a process producing this form of intermittency, switching abruptly from extended periods of stasis to bursts of large varia- tion, manifests on-o+intermittency. A model of intermittency in terms of simple dynamical systems was given by Pomeau and Manneville (PM) [3]. In their discrete time model, a system spends a long time near a weakly unstable fixed point, or a quasif'txed point, whose image is not far from the point itself. With the in- troduction of an aperiodic recurrence mechanism that turns trajectories back toward this unstable fixed point, an intermittent signal is produced. Pomeau and Manne- ville also proposed a classification for various types of in- termittency corresponding to diFerent modes of instabili- ty of the fixed point. The fixed point in the PM model corresponds to a periodic orbit in the continuous time system that they im- plicitly describe. Hence their intermittency is generally not of the on-oF type, for continuous systems. For a sim- ple example of on-oF behavior, we can parallel their mod- el with a diFerential equation admitting a critical point that loses stability with the tuning of a parameter. More generally, we may use any other weakly unstable, invari- ant objects representing states near to which the system will tend to spend long times. In fact, the objects that or- ganize the behavior do not even have to be invariant. It is enough that they be quasi-invariant in the systems that enter their neighborhoods and remain there for a long time. Intermittent systems can be constructed around ei- ther invariant or quasi-invariant objects. We shall speak only of the former for brevity, though the latter could serve as well and have been used in previous models of in- termittency. An example can be constructed from the unstable invariant object devised by Grebogi, Ott, and Yorke [4], with the introduction of a reinjection mecha- nism into their model, though the signal produced in this way would again generally not be of the on-oF kind. On-oF intermittency occurs when the unstable object lies in the hyperplane x~ = =x~=0 of the phase space where the coordinates x i, . .. , x~ with K & Ã ~~ are suitably chosen. Though many systems are capable of producing on-oF signals, they may far less often be detected since the "suitable" variables may not arise nat- urally, nor be discovered easily. Good coordinates are likely to be natural in many real problems where a small set of variables is observable but a larger number of "hid- den variables" is believed to be implicated. The solar cy- cle involves turbulent convection; stock market prices are influenced by various economic factors; in the wild, popu- lations of certain species are environmentally influenced. In these cases, "crashes" are seen — sunspots rarely oc- curred in Newton's time and species may come close to extinction and yet survive. In these examples, the codirnension N — K is quite large and it is not clear whether the evolution occurring in the complementary space should be characterized as deterministic or random. It is a feature of the mechanism we propose that this distinction does not matter very much for the shape of the observed signals. This insensi- tivity may at first surprise those aware of Takens' theorem [51 ensuring the possibility of estimating the di- mension of phase space from a time signal. However, an essential assumption of this theorem is that the variable 1993 The American Physical Society 279