VOLUME 65, NUMBER 24 PHYSICAL REVIEW LETTERS 10 DECEMBER 1990 Transition to Chaos for Random Dynamical Systems Lei Yu, Edward Ott, "' and Qi Chen Laboratory for Plasma Research, University of Maryland, Co/lege Park, Maryland 20742 (Received 9 July 1990) %'e study the transition to chaos for random dynamical systems. Near the transition, on the chaotic side, the long-time particle distribution (which is fractal) that evolves from an initial smooth distribution exhibits an extreme form of temporally intermittent bursting whose scaling we investigate. As a physical example, the problem of the distribution of particles floating on the surface of a fluid whose flow velocity has a complicated time dependence is considered. PACS numbers: 05.45.+b, 03. 40. 6c The problem of the transition to chaos in deterministic systems has been the subject of much interest, and, for low-dimensional dynamics, it has been found that this transition most often occurs via a small number of often observed routes (e. g. , period doubling, crises, intermit- tency, and quasiperiodicity). In this paper we discuss the transition to chaos for random dynamical systems, and, in particular, random maps, ' r„i|=M(r„, n). Here the second argument of M arises because M is chosen randomly at each iterate n according to some rule. Our principal result concerns the fact that there can be a transition with variation of a parameter from a situation where an initial cloud of particles eventually permanent- ly clumps at a point to a situation where the particles are eventually distributed on a fractal. ' In particular, we find that, near the critical point, the time dependence of the spatial extent of the fractal particle cloud can exhibit an extreme form of temporally intermittent bursting. One possible physical motivation for studying this problem comes from consideration of the convection of particles in a flowing fluid. Recently there has been much interest in this subject. In particular, fluid flows with smooth large-scale spatial structure can lead to complicated chaotic motion of convected particles. The consequences of this for mixing have been emphasized, and it has also been pointed out that gradients of the convected particle density can concentrate on a fractal. For the most part, these studies have dealt with the case of flow velocities whose time dependence either is steady and three dimensional or else is time periodic and two di- mensional. Furthermore, until recently, investigations have concentrated on the case of chaotic particle motion in which the impurity particles move with the same ve- locity as the fluid. Here we emphasize two important eAects which lead to the consideration of non-area- preserving random maps: (1) Situations where the parti- cle motion is not the same as the fluid motion can arise. (In this case the particle motion will typically not be in- compressible even though the Auid motion is. ) (2) It is common for the time dependence of the Eulerian fluid velocity v(x, t ) to be more complicated than sim pie periodic or steady time behavior. As a simple example, we consider particles floating on the surface of a fluid located in z ~ 0. The flow velocity v=v&+i, zo is incompressible. The floating particles only move in the surface, however, and hence their veloc- ity, which is vi = v, x o+ U~y o, is typically not incompres- sible, V vi =— tlv. /BzWO. We now consider a specific Aow v v] +v2+v3 where vi =azzo — ayyo leads to a compressive component to v& (i.e. , V~ v& i =— a), v2 =pyxo is a steady shear flow, and v3 is a vortical flow component with complicated time dependence. To facili- tate the calculations we take v3=ksin[x+8(t)]Q„B(t nT)yo. In— tegrating the equation of motion for Aoating particles, dr/dt =v~, where r =(x, y), over one time unit T, we obtain x„+i =[x„+a '(1 — e ')y„]mod(2tr), y„+i =e 'y„+ksin(x„+i+8„), where (x„, y„) denotes the particle position just after time t =nT, 8„8(nT), and we have set T=p=1 (which can be done without loss of generality since T and P can be absorbed by suitable normalization). If we think of 8(t) as varying chaotically in time, then 8„can be considered a random variable. Thus we are led to the consideration of a random map, ' Eq. (I). We emphasize that a random-map problem is expected in general and is not specific to the particular functional form of the mod- el flow which results in (1). In particular, say we in- tegrate the equation dr/dt = vi (r, t ) for a general vi(r, t) from time t =nT to time t =(n+1)T. Then r, +i =M(r„, n), where M maps r(nT)= r„ forward one- time unit, but, because of the nonperiodic time depen- dence of vi, M depends on the time interval n. If the dependence of vi(r, t ) on t is chaotic, then the variation of M(r, n ) with n is effectively random and this random- ness is large if T is greater than some suitable correlation time of the flow. %e expect that such random-map problems will arise in a variety of physical contexts, in addition to the problem of particles floating on a fluid surface. In what follows we first numerically examine Eq. (1) to discover possible universal behavior for ran- dom dynamical systems and then present analyses pro- viding an understanding of that behavior. The two Lyapunov exponents, Xl and X2 (we take Xi ) k2), for the system (1) characterize the exponential rate of separation or approach of two nearby points both 1990 The American Physical Society 2935