Geometry of the ergodic quotient reveals coherent structures in flows

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, (b) 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 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 Poincar\'e 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. In the end, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen

New Directions in Non-Relativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems

In non-relativistic mechanics the center of mass of an isolated system is easily separated out from the relative variables. For a N-body system these latter are usually described by a set of Jacobi normal coordinates, based on the clustering of the centers of mass of sub-clusters. The Jacobi variables are then the starting point for separating {\it orientational} variables, connected with the angular momentum constants of motion, from {\it shape} (or {\it vibrational}) variables. Jacobi variables, however, cannot be extended to special relativity. We show by group-theoretical methods that two new sets of relative variables can be defined in terms of a {\it clustering of the angular momenta of sub-clusters} and directly related to the so-called {\it dynamical body frames} and {\it canonical spin bases}. The underlying group-theoretical structure allows a direct extension of such notions from a non-relativistic to a special- relativistic context if one exploits the {\it rest-frame instant form of dynamics}. The various known definitions of relativistic center of mass are recovered. The separation of suitable relative variables from the so-called {\it canonical internal} center of mass leads to the correct kinematical framework for the relativistic theory of the orbits for a N-body system with action -at-a-distance interactions. The rest-frame instant form is also shown to be the correct kinematical framework for introducing the Dixon multi-poles for closed and open N-body systems, as well as for continuous systems, exemplified here by the configurations of the Klein-Gordon field that are compatible with the previous notions of center of mass.Comment: Latex, p.75, Invited contribution for the book {\it Atomic and Molecular Clusters: New Research} (Nova Science

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic R\"ossler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.Comment: 12 pages, 13 figure

Centers of Mass and Rotational Kinematics for the Relativistic N-Body Problem in the Rest-Frame Instant Form

In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem. The Wigner hyperplanes define the intrinsic rest frame and realize the separation of the center-of-mass. Three notions of {\it external} relativistic center of mass can be defined only in terms of the {\it external} Poincar\'e group realization. Inside the Wigner hyperplane, an {\it internal} unfaithful realization of the Poincar\'e group is defined. The three concepts of {\it internal} center of mass weakly {\it coincide} and are eliminated by the rest-frame conditions. An adapted canonical basis of relative variables is found. The invariant mass is the Hamiltonian for the relative motions. In this framework we can introduce the same {\it dynamical body frames}, {\it orientation-shape} variables, {\it spin frame} and {\it canonical spin bases} for the rotational kinematics developed for the non-relativistic N-body problem.Comment: 78 pages, revtex fil

Recurrence-based time series analysis by means of complex network methods

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos (2011

Orbital clustering in the distant solar system

The most distant Kuiper belt objects appear to be clustered in longitude of perihelion and in orbital pole position. To date, the only two suggestions for the cause of these apparent clusterings have been either the effects of observational bias or the existence of the distant giant planet in an eccentric inclined orbit known as Planet Nine. To determine if observational bias can be the cause of these apparent clusterings, we develop a rigorous method of quantifying the observational biases in the observations of longitude of perihelion and orbital pole position. From this now more complete understanding of the biases we calculate that the probability that these distant Kuiper belt objects would be clustered as strongly as observed in both longitude of perihelion and in orbital pole position is only 0.2%. While explanations other than Planet Nine may someday be found, the statistical significance of this clustering is now difficult to discount