### Abstract

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant component with positive topological entropy. The FTBEs approach the value of the topological entropy from below as the length and number of trajectories is increased; we conjecture that this result holds for a general class of ergodic, mixing systems. Furthermore, FTBEs are computed robustly with respect to the numerical time step, details of braid representation, and choice of initial conditions. We find that, in the class of systems we describe, trajectories can be re-used to form different braids, which greatly reduces the amount of data needed to assess the complexity of the flow.

Received 07 February 2015
Accepted 13 July 2015
Published online 27 July 2015

Lead Paragraph:
In geophysical flows and many other applications, it is important to know where things can go, and where they come from. This is the study of transport and its cousin, mixing. Modern methods are most powerful when we know the flow perfectly through its velocity field. But, when the data comes from, say, ocean floats, it is very sparse and cannot completely characterize transport. We use tools from the mathematical branch of topological dynamics to tease out as much information as possible about mixing in the flow, in particular, the degree of “entanglement” that a set of trajectories achieves. We call this measure the Finite-Time Braiding Exponent (FTBE). In this paper, we show how parameters that describe the flow, or the manner in which the flow is measured, affect the value of the Finite-Time Braiding Exponent by studying a simulated example of a flow with well-developed mixing.

Acknowledgments:
The authors thank Michael Allshouse, Margaux Filippi, Tom Peacock, and Elaine Spiller for helpful discussions, and the anonymous reviewers for detailed comments. Much of this work was completed while the second author was a visiting fellow of Trinity College, Cambridge. This research was supported by the U.S. National Science Foundation, under Grant No. CMMI-1233935.

Article outline:

I. INTRODUCTION
II. MATHEMATICAL BACKGROUND
A. Representation of braids and loops
B. Finite-time braiding exponents
III. CONSTRUCTING BRAIDS AND COMPUTING FTBEs
A. Practical considerations
B. The Aref blinking vortex flow
C. Robustness of braid construction
1. Time step
2. Angle of the projection axis
D. Parameters for braid construction
1. Initial conditions and length of trajectories
2. Number of strands
IV. EXTRAPOLATING TOPOLOGICAL ENTROPY FROM FTBEs
V. DISCUSSION

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