Characterization of Hydrodynamic Strange

The onset of chaotic motion in hydrodynamic systems is still not well understood. However, significant progress has been made in the past ten years. The evidence indicates that certain fluid systems generate chaotic motion through the nonlinear interaction of a small number of degrees of freedom. Examples include Couette-Taylor flow, Rayleigh-B~nard convection in restricted geometries, and vertically forced surface waves. In these cases the dynamics can be described by introducing a state or phase space much like that of classical statistical mechanics. The coordinates of the phase space could be, for example, the projections of the motion of the system onto the various modes into which it can be decomposed. If the system is dissipative, volumes in phase space contract over time, and the trajectories converge to a limit set known as an attractor. Nonperiodic states of deterministic systems are called "chaot ic ," and they are represented by topologically complex attractors that have been dubbed "s t range" [1, 2]. Typically, a strange attractor is an infinitely folded sheet of infinite extent located in a bounded region of phase space. A line will typically cut the attractor an infinite number of times. In a sense that can be made quite precise, the dimension of the attractor is fractional. The trajectories of a strange attractor also exhibit "sensitive dependence on initial conditions," which means that nearby trajectories diverge from each other exponentially in time. Suppose that a trajectory comes close (at time t l ) t o its position at an earlier time t0, so that we have a near recurrence. The behavior after t~ is quite different from that between t o and tl, because the new and old segments of the trajectory diverge from each other exponentially in time. This property prevents near recurrences from leading to near periodicity; the next near recurrence will not be an equal time interval later. Therefore, the time dependence will be nonperiodic, and power spectra of observable quantities will be broadband. The rates at which the trajectories diverge (or converge) in various locally defined orthogonal directions are called Lyapunov exponents. At least one of these must be positive for a chaotic flow. Recent experiments have confirmed that some flows just beyond the onset of chaos can be described by low dimensional strange attractors. This is a rather remarkable result, since the Navier-