Marko Budišić

Geometry of the ergodic quotient reveals coherent structures in flows

Abstract Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, and (b) to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold–Beltrami–Childress flow, which was the test-bed for alternative approaches: the Ulam’s approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincare map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill’s vortex flow, discovering unknown coherent structures in its state space. Finally, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.

Dynamic window based force reflection for safe teleoperation of a mobile robot via internet

The paper presents an Internet-based teleoperation system that enables a human operator to safely control a mobile robot in unknown and dynamic environments. The operator controls the robot using a joystick and a graphical user interface which displays images forwarded from the camera mounted on the robot. A sonar ring on the robot circumference is used to measure obstacle range information, and a dynamic window algorithm is used to convert that information into a force, which is than reflected to the operators hand via joystick, providing additional haptic information about obstacles in the robot vicinity. To overcome the instability caused by the unknown and varying time delay an event-based teleoperation system is employed to synchronize actions of the operator and teleoperated mobile robot. The experiments with the Pioneer 2DX mobile robot verified effectiveness of the developed system.

Mesochronic classification of trajectories in incompressible 3D vector fields over finite times

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we provide calculations that can be used to determine the number of expanding directions and the presence of rotation from the characteristic polynomial of the Jacobian matrix of mesochronic velocity. In doing so, we show that (a) the mesochronic velocity can be used to characterize dynamical deformation of three-dimensional volumes, (b) the resulting mesochronic analysis is a finite-time extension of the Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the two-dimensional mesochronic analysis from Mezic et al. \emph{A New Mixing Diagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extends to three-dimensional state spaces. Theoretical considerations are further supported by numerical computations performed for a dynamical system arising in fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.

Conditioning moments of singular measures for entropy optimization. I

Abstract In order to process a potential moment sequence by the entropy optimization method one has to be assured that the original measure is absolutely continuous with respect to Lebesgue measure. We propose a non-linear exponential transform of the moment sequence of any measure, including singular ones, so that the entropy optimization method can still be used in the reconstruction or approximation of the original. The Cauchy transform in one variable, used for this very purpose in a classical context by A.A. Markov and followers, is replaced in higher dimensions by the Fantappie transform. Several algorithms for reconstruction from moments are sketched, while we intend to provide the numerical experiments and computational aspects in a subsequent article. The essentials of complex analysis, harmonic analysis, and entropy optimization are recalled in some detail, with the goal of making the main results more accessible to non-expert readers.

An approximate parametrization of the ergodic partition using time averaged observables

An ergodic set in the state space of a measure-preserving dynamical system is an invariant set on which the system is ergodic. Moreover, it comprises points on statistically identical trajectories, i.e., time averages of any function along any two trajectories in the set are equal. The collection of such sets partitions the state space and is called the ergodic partition. We present a computational algorithm that retrieves a set of coordinates for ergodic sets. Those coordinates can be thought of as generalization of action coordinates from theory of Liouville-integrable systems. Dynamics of the system is embedded into the space of time averages of observables along the trajectories. In this space, the problem is formulated as a dimension-reduction problem, which is handled by the Diffusion Maps algorithm. The algorithm is demonstrated on a 2D map with a mixed state space.

Conditioning moments of singular measures for entropy maximization. II: numerical examples

If moments of singular measures are passed as inputs to the entropy maximization procedure, the optimization algorithm might not terminate. The framework developed in our previous paper demonstrated how input moments of measures, on a broad range of domains, can be conditioned to ensure convergence of the entropy maximization. Here we numerically illustrate the developed framework on simplest possible examples: measures with one-dimensional, bounded supports. Three examples of measures are used to numerically compare approximations obtained through entropy maximization with and without the conditioning step.