S. Wiggins, University of Bristol Ana M. Mancho, ICMAT (CSIC-UAM-UC3M-UCM)

1. S. Wiggins, University of Bristol Ana M. Mancho, ICMAT (CSIC-UAM-UC3M-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. N00014-01-1-0769. Dr. Reza Malek-Madani

2. Plan for the talk 2 • 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

3. Original Connection with Dynamical Systems Theory: 2-D, Incompressible, Time-Periodic Flows 3

4. 4

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) 5 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………. 6 • 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????

7. Some relevant mathematical results: nonautonomous systems 7 Generating the dynamics (no flow, or single map) Dafermos, C. M. (1971). An invariance principle for compact processes. J. Diff. Eq., 9, 239–252. Miller, R. K. (1965). Almost periodic differential equations as dynamical systems with applications to the existence of almost periodic solutions. J. Diff. Eq., 1, 337–395. Sell, G. R. (1967a). Nonautonomous differential equationa and topological dynamics I. The basic theory. Trans. Amer. Math. Soc., 127(2), 241–262. Sell, G. R. (1967b). Nonautonomous differential 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 Differential Equations . McGraw-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.

8. The spectrum of linear, nonautonomous systems 8 • 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 , 14, 141–152. Langa, J. A., Robinson, J. C., and Suarez, A. (2002). Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity , 15, 887–903. Meyer, K. R. and Zhang, X. (1996). Stability of skew dynamical systems. J. Diff. Eq., 132, 66–86. Sell, G. R. (1971). Topological Dynamics and Differential 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.

9. Chaos 9 Lerman, L. and Silnikov, L. (1992). Homoclinical structures in nonautonomous systems: Nonautonomous chaos. Chaos , 2, 447–454. Stoffer, D. (1988a). Transversal homoclinic points and hyperbolic sets for non-autonomous maps i. J. Appl. Math. and Phys. (ZAMP) , 39, 518–549. Stoffer, D. (1988b). 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 differential equations driven by a nonautonomous force. Nonlinearity , 23, 2935–2973. Bifurcation Poetzsche, C. (2010b). Nonautonomous bifurcation of bounded solutions I. A Lyapunov-Schmidt approach. Discrete and continuous dynamical systems-series B , 14(2), 739–776. Poetzsche, C. (2011a). Nonautonomous bifurcation of bounded solutions II. A shovel bifurcation pattern. Discrete Contin. Dyn. Syst., 31(3), 941–973. Poetzsche, C. (2011b). Persistence and imperfection of nonautonomous bifurcation patterns. J. Diff. Eq., 250(10), 3874–3906. Rasmussen, M. (2006). Towards a bifurcation theory for nonautonomous difference equations. J. Difference Eq. Appl., 12(3-4), 297–312.

10. Some relevant mathematical results: finite time dynamics 10 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.

11. More finite time hyperbolicity 11 Duc, L. H. and Siegmund, S. (2011). Existence of finite-time hyperbolic trajectories for planar hamiltonian flows. J. Dyn. Diff. 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 differential equations on finite time. J. Diff. 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 differential equations. J. Diff. 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. More finite time hyperbolicity

12. “The Hyperbolic-Elliptic Dichotomy” 12 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?).

13. KAM/Nekhoroshev Theorems-The Set-up (Traditional Version) 13 The Hamiltonian (no explicit time dependence--yet) =0 Unperturbed Hamilton’s equations Trajectories of unperturbed Hamilton’s equations Domain filled with invariant tori

14. KAM Theorem--”Sufficiently nonresonant tori are preserved if the perturbation is sufficiently small” 14 Sufficient conditions for application of the theorem Action-angle variables (formulae exist, but virtually impossible to compute in typical examples) Dealing with resonances Nondegeneracy condition

15. Nekhoroshev Theorem: “A Finite Time Result” 15 “...while not eternity, this is a considerable slice of it.” (Littlewood) Action-angle variables Dealing with resonances (“the geometric argument”) Nondegeneracy condition

16. Recommended Reading 16 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., González, A., Jorba, A., and Villanueva, J. (2005). KAM theory without action-angle variables. Nonlinearity, 18(2), 855–895.

17. The idea behind “exponential stability estimates” 17 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

18. 18 The problem Estimate ratio of terms in the normal form series (ignore many constants) Stirling’s formula “Optimal choice of r--exponentially small remainder

19. Explicit time dependence 19 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.

20. A Nekhoroshev Theorem for General Time Dependence 20 The set-up The usual “trick” Corresponding Hamilton’s equations

21. After “setting up” the problem--steps in the proof 21 • 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

22. Example: “Reverse Engineering” Kolmogorov’s Proof of KAM 22 is an invariant torus for all values of and any time dependent functions b(t) Take as an example time dependence

23. FTLEs 23 (Does the KAM theorem apply?)

24. 24 Can you see an invariant torus at ? Can you see that no particles can cross ?

25. What about the case ? 25

26. Integrate trajectories for a longer time... 26 You will always get “artifacts” when you compute FTLEs. How do you know if they are real?

27. Summary and Conclusions 27 • 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”.