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

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Plan for the talkPlan 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

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

Phase Space

Physical Space

Reduction to a 2-D,

Area Preserving,

Poincare Map

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Dynamical Systems

Structure

Implications and Uses for

Fluid Transport

Invariant Manifold

(material curve, surface)

Geometrical Template Governing

Transport. Basis for Analytical and

Computational Methods for Computing

Transport Quantities

Chaos

Probably implies rapid stirring

(at least “somewhere”)

KAM Tori

Trapping of Fluid. Barriers to Transport

55Early 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?

“Finite-Time Velocity

Field”

Aperiodicity in Time

66For 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????

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Dafermos, C. M. (1971). An invariance principle for compact processes. J. Di . Eq., 9, 239–252. ff

Miller, R. K. (1965). Almost periodic di erential equations as dynamical systems with ffapplications to the existence of almost periodic solutions. J. Di . Eq., 1, 337–395. ff

Sell, G. R. (1967a). Nonautonomous di erential equationa and topological dynamics I. The basic theory. ffTrans. Amer. Math. Soc., 127(2), 241–262. Sell, G. R. (1967b). Nonautonomous di erential equationa and topological dynamics II. Limiting equations. ffTrans. Amer. Math. Soc., 127(2), 263–283.

Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Di erential Equations . McGraw-Hill, New York.ff

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.

Some relevant mathematical results: nonautonomous systems

Generating the dynamics (no flow, or single map)

Stable and unstable manifolds of hyperbolic trajectories

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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 di erential equations. Nonlinearity , 15, 887–903.ff

Meyer, K. R. and Zhang, X. (1996). Stability of skew dynamical systems. J. Di . Eq., 132, 66–86. ff

Sell, G. R. (1971). Topological Dynamics and Di erential Equations . Van Nostrand-Reinhold, London. ff

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.

Stability and attraction

Shadowing

The spectrum of linear, nonautonomous systems

• Lyapunov exponents

• exponential dichotomies

• Sacker-Sell spectrum

9Lerman, L. and Silnikov, L. (1992). Homoclinical structures in nonautonomous systems: Nonautonomous chaos. Chaos , 2, 447–454.

Sto er, D. (1988a). Transversal homoclinic points and hyperbolic sets for non-autonomous maps i. J. Appl. ffMath. and Phys. (ZAMP) , 39, 518–549. Sto er, D. (1988b). Transversal homoclinic points and hyperbolic sets for non-autonomous maps ii. J. Appl. ffMath. 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 , ff23, 2935–2973.

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. Di . Eq., ff250(10), 3874–3906.

Rasmussen, M. (2006). Towards a bifurcation theory for nonautonomous di erence ffequations. J. Di erence Eq. Appl., 12(3-4), 297–312. ff

Chaos

Bifurcation

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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.

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.

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.

Finite time hyperbolicity and invariant manifolds

Some relevant mathematical results: finite time dynamics

Finite time stability

11 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. ff

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 ffequations on finite time. J. Di . Eq., 246(3), 1098–1118.ff

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. ff ff

More finite time hyperbolicity

Recommended review paper

More finite time hyperbolicity

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.

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“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?).

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KAM/Nekhoroshev Theorems-The Set-up (Traditional Version)

=0

The Hamiltonian (no explicit time dependence--yet)

Unperturbed Hamilton’s equations

Trajectories of unperturbed Hamilton’s equations

Domain filled with invariant tori

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KAM Theorem--”Sufficiently nonresonant tori are preserved if the perturbation is sufficiently small”

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

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Action-angle variables

Dealing with resonances (“the geometric argument”)

Nondegeneracy condition

Nekhoroshev Theorem: “A Finite Time Result”

“...while not eternity, this is a considerable slice of it.” (Littlewood)

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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., Gonzalez, A., Jorba, A., and Villanueva, J. (2005). KAM theory

without action-angle variables. Nonlinearity, 18(2), 855–895.

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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

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The problem

Estimate ratio of terms in the normal form series (ignore many constants)

Stirling’s formula

“Optimal choice of r--exponentially small remainder

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Explicit time dependence

Nekhoroshev

KAM

Giorgilli, A. and Zehnder, E. (1992). Exponential stability for time dependent potentials. Z. angew. Math. Phys. (ZAMP), 43, 827–855.

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.

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.

Background

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A Nekhoroshev Theorem for General Time Dependence

The set-up

The usual “trick”

Corresponding Hamilton’s equations

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“Model Statement” of a Theorem

•Construct a normal form via a “canonical transformation method”

After “setting up” the problem--steps in the proof

•No need for a “geometric argument” (problem is too simple). Choose constants “optimally”.

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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

FTLEs23

(Does the KAM theorem apply?)

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?

Can you see an invariant torus at

Can you see that no particles can cross

?

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What about the case ?

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Integrate trajectories for a longer time...

You will always get “artifacts” when you compute FTLEs. How do you know if they are real?

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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”.