Winding number instability in the phase-turbulence regime of the Complex Ginzburg-Landau Equation

We give a statistical characterization of states with nonzero winding number in the Phase Turbulence (PT) regime of the one-dimensional Complex Ginzburg-Landau equation. We find that states with winding number larger than a critical one are unstable, in the sense that they decay to states with smaller winding number. The transition from Phase to Defect Turbulence is interpreted as an ergodicity breaking transition which occurs when the range of stable winding numbers vanishes. Asymptotically stable states which are not spatio-temporally chaotic are described within the PT regime of nonzero winding number.Comment: 4 pages,REVTeX, including 4 Figures. Latex (or postscript) version with figures available at http://formentor.uib.es/~montagne/textos/nupt

The Application of Lagrangian Descriptors to 3D Vector Fields

Since the 1980s, the application of concepts and ideas from Dynamical Systems Theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian Descriptors (LDs), a tool that has been successfully applied to time dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of Normally Hyperbolic Invariant Manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.S. Wiggins acknowledges the support of ONR Grant No. N00014-01-1-0769 and EPSRC Grant no. EP/P021123/1. A. M. Mancho acknowledges the support of ONR grant N00014-17-1-3003. V. J. Garc�?a-Garrido, J. Curbelo and A. M. Mancho thankfully acknowledge the computer resources provided by ICMAT. C.R. Mechoso was supported by the U.S. NSF grant AGS-1245069