Transport in Transitory Dynamical Systems

We introduce the concept of a “transitory” dynamical system—one whose time-dependence is confined to a compact interval—and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the “action” of relevant heteroclinic orbits at the intersection of invariant manifolds of “forward” and “backward” hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.

[1]  Anatoly Neishtadt,et al.  On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom , 1987 .

[2]  Steven L Brunton,et al.  Fast computation of finite-time Lyapunov exponent fields for unsteady flows. , 2010, Chaos.

[3]  D. Edwards,et al.  An Introduction to the Physics of High Energy Accelerators: Edwards/An Introduction , 2008 .

[4]  S. Wiggins,et al.  Transport in two-dimensional maps , 1990 .

[5]  S. Wiggins,et al.  Lobe area in adiabatic Hamiltonian systems , 1991 .

[6]  S. Wiggins,et al.  Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach , 2006 .

[7]  M. Rio,et al.  The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields , 2010, 1003.0377.

[8]  P. Schmelcher,et al.  Evolutionary phase space in driven elliptical billiards , 2009, 0904.3636.

[9]  Filip Sadlo,et al.  Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction , 2007, IEEE Transactions on Visualization and Computer Graphics.

[10]  M. Kruskal,et al.  Asymptotic Theory of Hamiltonian and other Systems with all Solutions Nearly Periodic , 1962 .

[11]  Knobloch,et al.  Mass transport and mixing by modulated traveling waves. , 1989, Physical review. A, General physics.

[12]  R. MacKay,et al.  Flux and differences in action for continuous time Hamiltonian systems , 1986 .

[13]  Gwm Gerrit Peters,et al.  Experimental/Numerical Analysis of Chaotic Advection in a Three-dimensional Cavity Flow , 2006 .

[14]  D. Escande,et al.  Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory , 1991 .

[15]  Anthony Leonard,et al.  Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems , 1991 .

[16]  George Haller,et al.  Finite time transport in aperiodic flows , 1998 .

[17]  Dana D. Hobson,et al.  An efficient method for computing invariant manifolds of planar maps , 1993 .

[18]  Kamran Mohseni,et al.  A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. , 2010, Chaos.

[19]  G. Haller,et al.  Eddy growth and mixing in mesoscale oceanographic flows , 1997 .

[20]  A M Mancho,et al.  Distinguished trajectories in time dependent vector fields. , 2008, Chaos.

[21]  Knobloch,et al.  Chaotic advection by modulated traveling waves. , 1987, Physical review. A, General physics.

[22]  George Haller,et al.  Geometry of Cross-Stream Mixing in a Double-Gyre Ocean Model , 1999 .

[23]  Stephen Wiggins,et al.  Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics , 2000 .

[24]  K. Yagasaki.  Invariant manifolds and control of hyperbolic trajectories on infinite- or finite-time intervals , 2008 .

[25]  Robert W. Easton,et al.  Transport through chaos , 1991 .

[26]  Sanjeeva Balasuriya,et al.  Melnikov theory for finite-time vector fields , 2000 .

[27]  Cary,et al.  Adiabatic-invariant change due to separatrix crossing. , 1986, Physical review. A, General physics.

[28]  Stephen Wiggins,et al.  Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow , 1998 .

[29]  G. Froyland,et al.  Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows , 2009 .

[30]  Meiss,et al.  Relation between quantum and classical thresholds for multiphoton ionization of excited atoms. , 1988, Physical review. A, General physics.

[31]  Julio M. Ottino,et al.  A case study of chaotic mixing in deterministic flows: The partitioned-pipe mixer , 1987 .

[32]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[33]  Hans Hagen,et al.  Efficient Computation and Visualization of Coherent Structures in Fluid Flow Applications , 2007, IEEE Transactions on Visualization and Computer Graphics.

[34]  Christopher K. R. T. Jones,et al.  Quantifying transport in numerically generated velocity fields , 1997 .

[35]  J. Meiss,et al.  Resonance zones and lobe volumes for exact volume-preserving maps , 2008, 0812.1810.

[36]  S. Wiggins,et al.  Finite-Time Lagrangian Transport Analysis: Stable and Unstable Manifolds of Hyperbolic Trajectories and Finite-Time Lyapunov Exponents , 2009, 0908.1129.

[37]  George Haller,et al.  Uncovering the Lagrangian skeleton of turbulence. , 2007, Physical review letters.

[38]  R. MacKay.  A variational principle for invariant odd-dimensional submanifolds of an energy surface for Hamiltonian systems , 1991 .

[39]  L. Markus,et al.  II. ASYMPTOTICALLY AUTONOMOUS DIFFERENTIAL SYSTEMS , 1956 .

[40]  Stefan Siegmund,et al.  Hyperbolicity and Invariant Manifolds for Planar nonautonomous Systems on Finite Time Intervals , 2008, Int. J. Bifurc. Chaos.

[41]  S. Wiggins,et al.  On the structure of separatrix-swept regions in singularly-perturbed Hamiltonian systems , 1992, Differential and Integral Equations.

[42]  Julio M. Ottino,et al.  Laminar mixing and chaotic mixing in several cavity flows , 1986, Journal of Fluid Mechanics.

[43]  Stephen Wiggins,et al.  Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets , 2002 .

[44]  Julio M. Ottino,et al.  Experiments on mixing due to chaotic advection in a cavity , 1989, Journal of Fluid Mechanics.

[45]  Hassan Aref.  The development of chaotic advection , 2006 .

[46]  James D. Meiss,et al.  Transport in Hamiltonian systems , 1984 .

[47]  G. Haller.  Distinguished material surfaces and coherent structures in three-dimensional fluid flows , 2001 .

[48]  Jerrold E. Marsden,et al.  Lagrangian coherent structures in n-dimensional systems , 2007 .

[49]  James D. Meiss,et al.  Resonances in area-preserving maps , 1987 .

[50]  Christopher K. R. T. Jones,et al.  Lagrangian Motion and Fluid Exchange in a Barotropic Meandering Jet , 1999 .

[51]  K. Mohseni,et al.  Vortex Shedding over a Two-Dimensional Airfoil: Where the Particles Come From , 2008 .

[52]  Stephen Wiggins,et al.  Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics , 2000 .

[53]  J. Marsden,et al.  Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows , 2005 .

[54]  G. Haller,et al.  Lagrangian coherent structures and mixing in two-dimensional turbulence , 2000 .

[55]  Sanjeeva Balasuriya,et al.  Cross-separatrix flux in time-aperiodic and time-impulsive flows , 2006 .