Volume 149, number 2,3 PHYSICS LETFERS A 17 September 1990 Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos Kunihiko Kaneko Instituteof Physics, College ofArts and Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan Received 17 May 1990, accepted for publication 9 July 1990 Communicatedby A.R. Bishop Structural stability of fully developed spatiotemporal chaos (FDSTC) is confirmed. The stability is sustained by the destruction of all windows through spatiotemporal intermittency and supertransients. Transition to intermittency is found to occur succes- sively, via type-I and then type-Il transients. In type-I transients, the transient time increases algebraically with system size, while it increases exponentially in type-I!. Decay of positive Lyapunov exponents is gradual in the former and abrupt in the latter. In FDSTC, spatial and temporal correlations are found to decay exponentially, as measured by mutual information. The existence of a finite correlation length assures the density of thermodynamic quantifiers. Spatiotemporal chaos is a complex dynamical are separated: the dynamics consists of phenomenon with many degrees of freedom, emerg- x~ (i) —i’x’ (i) =f(x~ ( i)) and then applying the dis- ing in spatially extended systems. It appears in a crete Laplacian operator x~÷ 1 (i) = (1 — )x’ (i) + broad area of natural phenomena. Qualitative, quan- ~[x’ (i+ l)+x’ (i—i)]. titative, and theoretical understanding of spatiotem- CML has originally been introduced to model tur- poral chaos remains one of the most important prob- bulent behavior as a synthesis of Landau’s picture on lems in nonlinear dynamics. turbulence [15] and Rössler’s hyperchaos [161. As a simple model for spatiotemporal chaos, cou- Landau has regarded turbulence as a direct product pled map lattices (CML) have been proposed [1—3]. of periodic states (quasiperiodic state with many in- A CML is a dynamical system with a discrete time, commensurate frequencies). This direct product discrete space, and continuous state [1—14]. Al- state, however, is not stable. It is easily locked to a though there are various types of CML, we restrict lower-dimensional torus [2], or attracted to a nearby ourselves here to the following diffusive coupling case strange attractor [17,2]. here: Not only a high-dimensional quasipenodic state x~~1(i) = (1 — ~ )f(x~ ( j)) but also a low-dimensional chaotic state is structur- ally unstable. In the logistic map, for example, chaos + ~ [f(x~ (1 + 1)) +f(x~ ( i— 1))] (1) cannot exist in an open set in the parameter space where n is a discrete time step and i is a lattice point [18]. Periodic windows are dense, although the cha- (i= 1, 2, ..., N= system size) with a periodic bound- otic state also has positive measure in the parameter ary condition. Here the mapping function f(x) is space. Quantifiers such as the Lyapunov exponent chosen to be the logistic map f(x) =1—ax 2 (cou- have infinitely many drops, if plotted as a function pled logistic lattice). Results to be presented here are of the bifurcation parameter (see fig. 1 a). The cha- applied to other maps and other couplings. otic state is structurally unstable, even if it observable. Separation of procedures is essential in the con- On the other hand, a direct product of chaos ((hy- struction of CML [1,4,11]. In the above model, lo- per)°°chaos [13]) may be structurally stable and may cal nonlinear transformation and diffusion processes provide a metaphorical model for turbulence. In the 0375-9601/90/S 03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland) 105