S Wiggins University of Bristol Ana M Mancho

S. Wiggins, University of Bristol Ana M. Mancho, ICMAT (CSIC-UAM-UC 3 M-UCM) Barriers to Transport in Aperiodically Time-Dependent Two. Dimensional Velocity Fields: Nekhoroshev's Theorem and ''Almost Invariant'' Tori Funded by the Office of Naval Research: Grant No. N 00014 -01 -1 -0769. Dr. Reza Malek-Madani

Plan for the talk • The Dynamical Systems approach to Lagrangian transport: Motivation and Background Mathematical issues (and some history) associated with general time dependence, and finite time dependence • Issues associated with the application of the KAM theorem and Nekhoroshev’s theorem. • A Nekhoroshev theorem for two-dimensional, aperiodically time dependent velocity fields • Some examples

Original Connection with Dynamical Systems Theory: 2 -D, Incompressible, Time-Periodic Flows Phase Space Reduction to a 2 -D, Area Preserving, Poincare Map Physical Space

Dynamical Systems Structure Implications and Uses for Fluid Transport Geometrical Template Governing Invariant Manifold (material curve, surface) Transport. Basis for Analytical and Computational Methods for Computing Transport Quantities Probably implies rapid stirring Chaos (at least “somewhere”) Trapping of Fluid. Barriers to Transport KAM Tori

5 Early dynamical systems analysis of Lagrangian transport was applied to kinematic models, but do these mathematical results and techniques work for “real problems”? (Some do, and some don’t) What is a “real problem”? (It should be related to “data”) What are the issues and obstacles? Aperiodicity in Time “Finite-Time Velocity Field”

6 For finite time, aperiodically time-dependent velocity fields what about………. • Poincaré maps? (More generally, “how is dynamics generated and described by the velocity field? ”) • Hyperbolic trajectories? • Stable and unstable manifolds of hyperbolic trajectories? • Chaos? • Lyapunov exponents? • KAM tori? and many other “dynamical systems” concepts and quantities? ?

Some relevant mathematical results: nonautonomous systems Generating the dynamics (no flow, or single map) Dafermos, C. M. (1971). An invariance principle for compact processes. J. Di�. Eq. , 9, 239– 252. Miller, R. K. (1965). Almost periodic di�erential equations as dynamical systems with applications to the existence of almost periodic solutions. J. Di�. Eq. , 1, 337– 395. Sell, G. R. (1967 a). Nonautonomous di�erential equationa and topological dynamics I. The basic theory. Trans. Amer. Math. Soc. , 127(2), 241– 262. Sell, G. R. (1967 b). Nonautonomous di�erential equationa and topological dynamics II. Limiting equations. Trans. Amer. Math. Soc. , 127(2), 263– 283. Stable and unstable manifolds of hyperbolic trajectories Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Di�erential Equations. Mc. Graw-Hill, New York. de Blasi, F. S. and Schinas, J. (1973). On the stable manifold theorem for discrete time dependent processes in banach spaces. Bull. London Math. Soc. , 5, 275– 282. Irwin, M. C. (1973). Hyperbolic time dependent processes. Bull. London Math. Soc. , 5, 209– 217.

The spectrum of linear, nonautonomous systems • Lyapunov exponents • exponential dichotomies • Sacker-Sell spectrum Stability and attraction Kloeden, P. and Schmalfuss, B. (1997). Nonautonomous systems, cocycle attractors, and variable time-step discretization. Numerical Algorithms , 141– 152. Langa, J. A. , Robinson, J. C. , and Suarez, A. (2002). Stability, instability, and bifurcation phenomena in non-autonomous di�erential equations. Nonlinearity , 15, 887– 903. Meyer, K. R. and Zhang, X. (1996). Stability of skew dynamical systems. J. Di�. Eq. , 132, 66– 86. Sell, G. R. (1971). Topological Dynamics and Di�erential Equations. Van Nostrand-Reinhold, London. Shadowing Chow, S. N. , Lin, X. B. , and Palmer, K. (1989). A shadowing lemma with applications to semilinear parabolic equations. SIAM J. Math. Anal. , 20, 547– 557.

Chaos Lerman, L. and Silnikov, L. (1992). Homoclinical structures in nonautonomous systems: Nonautonomous chaos. Chaos , 2, 447– 454. Sto�er, D. (1988 a). Transversal homoclinic points and hyperbolic sets for non-autonomous maps i. J. Appl. Math. and Phys. (ZAMP) , 39, 518– 549. Sto�er, D. (1988 b). Transversal homoclinic points and hyperbolic sets for non-autonomous maps ii. J. Appl. Math. and Phys. (ZAMP) , 39, 783– 812. Wiggins, S. (1999). Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence. Z. angew. Math. Phys. , 50, 585– 616. Lu, K. and Wang, Q. (2010). Chaos in di�erential equations driven by a nonautonomous force. Nonlinearity , 23, 2935– 2973. Bifurcation Poetzsche, C. (2010 b). Nonautonomous bifurcation of bounded solutions I. A Lyapunov-Schmidt approach. Discrete and continuous dynamical systems-series B , 14(2), 739– 776. Poetzsche, C. (2011 a). Nonautonomous bifurcation of bounded solutions II. A shovel bifurcation pattern. Discrete Contin. Dyn. Syst. , 31(3), 941– 973. Poetzsche, C. (2011 b). Persistence and imperfection of nonautonomous bifurcation patterns. J. Di�. Eq. , 250(10), 3874– 3906. Rasmussen, M. (2006). Towards a bifurcation theory for nonautonomous di�erence equations. J. Di�erence Eq. Appl. , 12(3 -4), 297– 312.

Some relevant mathematical results: finite time dynamics Finite time stability Weiss, L. and Infante, E. F. (1965). On the stability of systems defined over a finite time interval. Proc. Nat. Acad. Sci. , 54(1), 44– 48. Dorato, P. (2006). An overview of finite-time stability. In L. Menini, L. Zaccarian, and C. T. Abdallah, editors, Current Trends in Nonlinear Systems and Control: In Honor of Petar Kokotovic and Turi Nicosia , Systems and Control-Foundations and Applications, pages 185– 194. Birkhauser, Boston. Finite time hyperbolicity and invariant manifolds Duc, L. H. and Siegmund, S. (2008). Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals. Int. J. Bif. Chaos , 18(3), 641– 674. Berger, A. , Son, D. T. , and Siegmund, S. (2008). Nonautonomous finite-time dynamics. Discrete and continuous dynamical systems-series B , 9(3 -4), 463– 492.

More finite time hyperbolicity Duc, L. H. and Siegmund, S. (2011). Existence of finite-time hyperbolic trajectories for planar hamiltonian flows. J. Dyn. Di�. Eq. , 23(3), 475– 494. Berger, A. (2011). On finite time hyperbolicity. Comm. Pure App. Anal. , 10(2), 963– 981. Berger, A. , Doan, T. S. , and Siegmund, S. (2009). A definition of spectrum for di �erential equations on finite time. J. Di�. Eq. , 246(3), 1098– 1118. Doan, T. S. , Palmer, K. , and Siegmund, S. (2011). Transient spectral theory, stable and unstable cones and Gershgorin’s theorem for finite-time di�erential equations. J. Di�. Eq. , 250(11), 4177– 4199. Recommended review paper Balibrea, F. , Caraballo, T. , Kloeden, P. E. , and Valero, J. (2010). Recent developments in dynamical systems: Three perspectives. Int. J. Bif. Chaos, 20(9), 2591– 2636.

“The Hyperbolic-Elliptic Dichotomy” All of the results above are concerned with hyperbolic phenomena In general, “hyperbolicity results” do not depend on the nature of the time dependence or whether or not the system is Hamiltonian Two fundamental perturbation theorems of Hamiltonian dynamics: the KAM theorem and the Nekhoroshev theorem--are there versions for aperiodic time dependence and finite time dependence (and can they really be applied to the study of transport in fluids? ).

KAM/Nekhoroshev Theorems-The Set-up (Traditional Version) The Hamiltonian (no explicit time dependence--yet) =0 Unperturbed Hamilton’s equations Trajectories of unperturbed Hamilton’s equations Domain filled with invariant tori

KAM Theorem--”Sufficiently nonresonant tori are preserved if the perturbation is sufficiently small” Sufficient conditions for application of theorem Action-angle variables (formulae exist, but virtually impossible to compute in typical examples) Dealing with resonances Nondegeneracy condition

Nekhoroshev Theorem: “A Finite Time Result” “. . . while not eternity, this is a considerable slice of it. ” (Littlewood) Action-angle variables Dealing with resonances (“the geometric argument”) Nondegeneracy condition

Recommended Reading H. Scott Dumas, The KAM Story. A Friendly Introduction to the History, Content, and Significance of the Classical Kolmogorov-Arnold-Moser Theory. to be published soon (World Scientific). See also de la Llave, R. , Gonza lez, A. , Jorba, A. , and Villanueva, J. (2005). KAM theory without action-angle variables. Nonlinearity, 18(2), 855– 895.

The idea behind “exponential stability estimates” Transform to a “normal form” (ignoring resonances, and other things) Evolution of the action variables of the normal form The “standard estimate” Estimate holds on an interval [0, T], where

The problem Estimate ratio of terms in the normal form series (ignore many constants) Stirling’s formula “Optimal choice of r--exponentially small remainder

Explicit time dependence KAM Jorba, A. and Simo, C. (1996). On quasiperiodic perturbations of elliptic equilibrium points. SIAM J. Math. Anal. , 27(6), 1704– 1737. Sevryuk, M. B. (2007). Invariant tori in quasiperiodic non-autonomous dynamical systems via Herman’s method. Discrete Contin. Dyn. Syst. , 18(2 & 3), 569– 595. Broer, H. W. , Huitema, G. B. , and Sevryuk, M. B. (1996). Quasi-Periodic Motions in Families of Dynamical Systems, volume 1645 of Lecture Notes in Mathematics. Springer-Verlag, New York, Heidelberg, Berlin. Nekhoroshev Giorgilli, A. and Zehnder, E. (1992). Exponential stability for time dependent potentials. Z. angew. Math. Phys. (ZAMP), 43, 827– 855. Background Giorgilli, A. (2002). Notes on exponential stability of Hamiltonian systems. In Dynamical Systems. Part I. Hamiltonian Systems and Celestial Mechanics, Pisa. Centro di Recerca Matematica Ennio De Giorgi, Scuola Normale Superiore.

A Nekhoroshev Theorem for General Time Dependence The set-up The usual “trick” Corresponding Hamilton’s equations

After “setting up” the problem--steps in the proof • Construct a normal form via a “canonical transformation method” • No need for a “geometric argument” (problem is too simple). Choose constants “optimally”. “Model Statement” of a Theorem

Example: “Reverse Engineering” Kolmogorov’s Proof of KAM is an invariant torus for all values of and any time dependent functions b(t) Take as an example time dependence

FTLEs (Does the KAM theorem apply? )

Can you see an invariant torus at ? Can you see that no particles can cross ?

What about the case ?

Integrate trajectories for a longer time. . . You will always get “artifacts” when you compute FTLEs. How do you know if they are real?

Summary and Conclusions • Reviewed a number of mathematical results relevant to studies of Lagrangian transport from the dynamical systems point of view (all deterministic) • Highlighted the “hyperbolic-elliptic dichotomy” • KAM theorems and Nekhoroshev Theorems--The latter may be more relevant for studies of Lagrangian tramsport (there is an aperiodic version) • Presented an example (“reverse-engineered” from Kolmogorov’s proof of KAM) showing that FTLEs do not always reveal significant flow structures (in fact, they are “invisible to FTLEs) and they can give rise to “artifacts”.