Hysteresis in the Lorenz Equations

1. Much of the recent literature on chaos in dynamtransitions can be intermittent. ical systems has been concerned with pointing out the All these phenomena, and particularly hysteresis, behavioural similarity of many different types of can be explained on the basis of a suitable non-monomodels. The transition to aperiodic motion via the tone difference equation [11], such as that relating (Feigenbaum) period-doubling cascade occurs in difsuccessive maxima of Z constructed by Lorenz [12], ference equations [1], differential equations [2] and which exhibits a pronounced cusp. In turn, this can fluid experiments [3]. The phenomenon of intermitbe readily understood as being due to the occurrence tency [4] has also been widely observed in the above of a homodinic orbit in the system [13]: as r passes types of system [5], which argues persuasively that it through the value at which the homoclinic “explosion” is a “universal” kind of behaviour. To a lesser extent, takes place, a strange invariant set of trajectories is hysteresis (representing the concurrent existence of produced, including an infinite number of periodic different stable trajectories in dynamical systems) is orbits [14,151. It may be better to think of periodalso commonly observed [6], and indeed might be doubling windows as being a result of the existence expected in any system which can behave intermitof homocinicity in a system (which “produces” the tently: for example, the Rössler equations exhibit orbits which are then “absorbed” by the period. hysteresis, period-doubling, and intermittency in doubling window) rather than as being primarily due various ranges of their parameter space [7]. to successive bifurcations of a parent periodic orbit. This view is derived from that of Sparrow, as expressed 2. The best-known dynamical system, the Lorenz in his forthcoming book [15]. equationsX= —oX+ aY, Y(r_Z)X_ Y,Z=XY — bZ, is known to exhibit period-doubling [8] and 3. The “Lorenz map” relating successive maxima intermittency [9], and one might reasonably expect ofZ on the strange attractor should be a curve crossed hysteresis as well; indeed, it is well-known [10] that with a Cantor set; that Lorenz observed a single-valued for the “standard” values a = 10, b = 8/3, the nonfunction is directly due to the strong contraction rate trivial fixed points (Z = r — 1, ...) coexist (stably) of phase volumes. In turn, this can be considered as with the strange attractor for 24.06 ~r ~ 24.74, and being due to the large value of a in his computation. hence hysteresis Occurs in this range: for higher r, We have taken advantage of this observationto con-