S. Wiggins, University of Bristol

Finite Time and Aperiodically Time Dependent Dynamics: The Research Landscape— Past, Present, and Future

Funded by the Office of Naval Research: Grant No. N00014-01-1-0769. Dr. Reza Malek-Madani

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Plan for the talk

•Explain the title !Personal perspective on the rise of applied dynamical systems theory (and what happened to the theory of ordinary differential equations in the process?) !•What is taught in a first course on applied dynamical systems theory (example topics)—and what of this is valid for aperiodic and finite time dynamics? !

•The “Hyperbolic-Elliptic Dichotomy”—The KAM and Nekhoroshev Theorems !

•Finally, are there any lessons related to “how you choose research problems”?

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The Rise of Applied Dynamical Systems Theory•When—Late 70s early 80s !•Why—applications (and two influential books) Guckenheimer and Holmes [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer. Lichtenberg and Lieberman [1982] Regular and Stochastic Motion, Springer.

!• What role did the well-developed Theory of Ordinary Differential Equations play? Essentially none—and 3 great books were mostly overlooked !E. A. Coddington and N. Levinson {1984] Theory of Ordinary Differential Equations, Krieger !J. K Hale [2009] Ordinary Differential Equations, Dover. !P. Hartman [2002] Ordinary Differential Equations, SIAM.

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Typical Topics in a First Course on Dynamical Systems (with a “geometrical flavor")

!•Some general remarks on “nonlinearity” (and some “captivating” examples” •Generating the dynamics—maps and flows

•Linearization about fixed points

•Stable, unstable, and center manifolds of fixed points

•Invariance, stability, and attraction

•Bifurcation theory (mostly local)

•Chaos !

How much of this goes over for nonautonmous systems? Or time-dependent systems that are defined only for a finite time? !!

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Autonomous systemsGenerating the dynamics (flow, or single map)

General Property of the solutions

“Abstract” Notation

Definition of a Flow

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Some relevant mathematical results: nonautonomous systems

Generating the dynamics (no flow, or single map)

Definition of a Process

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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.

Some relevant mathematical results: nonautonomous systems

Generating the dynamics (no flow, or single map)

1t

0t t 2

( )φ 1t 0t x0, ,t 2( )φ 0t x0, ,

1tt 2( )φ , , ( )φ 1t 0t x0, ,

=

x0( )φ 0t x0, , =0t

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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 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.

Stability and attraction

Linearization: The spectrum of linear, non autonomous systems

• Lyapunov exponents • exponential dichotomies • Sacker-Sell spectrum

9 Example from A. J. Szeri, S. Wiggins, L. G. Leal [1991] On the dynamics of suspended microstructure in Unsteady, spatially inhomogeneous, 2-dimensional Fluid flows. J. Fluid Mech., 228, 207-241.

What is a DHT?

One dimensional vector field

Instantaneous stagnation point (not a solution)

General solution of the equation

The distinguished hyperbolic trajectory (DHT) (Note that all trajectories are hyperbolic)

16 Frozen Time Structure

Time Evolving Structure

11 But how is this trajectory “distinguished”?

“Forward attraction” “Pullback attraction”

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Barkana, I.. (2014). Defending the Beauty of the Invariance Principle!International Journal of Control, 87(1), 186-206.

Invariance and Invariant Sets

The LaSalle Invariance Principle

Theorem (J. P. LaSalle)

13An Example

What about nonautonomous systems?

Barkana, I.. (2014). Defending the Beauty of the Invariance Principle.!International Journal of Control, 87(1), 186-206.

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

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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.

Stable and unstable manifolds of hyperbolic trajectories

Katok, A, Hasselblatt, B. (1997). Introduction to the Modern Theory of Dynamical Systems. Cambridge! University Press.

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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.

Chaos

The Smale Horseshoe

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Bifurcation—“Standard” Bifurcation of Equilibria (Autonomous Systems)

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Example of an Autonomous Bifurcation of Equilibria: A Pitchfork Bifurcation

x = µx� x

3

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Poetzsche, C. (2011). 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.

Bifurcation—Nonautonomous SystemsLanga, J. A., Robinson, J. C., Suarez, A. (2002). Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity, 15, 887-903.

Nonautonomous Pitchfork Bifurcation Theorem (Langa et al.)

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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

Current Mathematical Research Motivated by ONRs Program on “Lagrangian Transport in Geophysical Flows

But there is a “forgotten” paper that lays much of the foundations…

20Duc, 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.

More finite time hyperbolicity

Recommended review paperBalibrea, 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.

21 “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?).

22KAM/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

23KAM 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)

25Recommended Reading

H. Scott Dumas, [2014] The KAM Story. A Friendly Introduction to the History, Content, and Significance of the Classical Kolmogorov-Arnold-Moser Theory. (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.

!!

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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

A. Fortunati and S. Wiggins (2014). Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence, Regular and Chaotic Dynamics, 19(3), 363–373.

A. Fortunati and S. Wiggins (2014). Persistence of Diophantine Flows for Quadratic, Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time dependence, submitted to Regular and Chaotic Dynamics.

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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 “partial” normal form via a “canonical transformation method”

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

•“Geometric argument”—inherited from the “original” geometric argument (Choose constants “optimally”).

A. Fortunati and S. Wiggins (2014). Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence, Regular and Chaotic Dynamics, 19(3), 363–373.

31An Example to “Play With”: “Reverse Engineering” Kolmogorov’s Proof of KAM

is an invariant torus for all values of

and any time dependent functions b(t)

S. Wiggins and A. M. Mancho (2014). Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev’s theorem and “Nearly Invariant” tori, Nonlinear Processes in Geophysics, 21, 165–185.

32Summary and Conclusions

•Reviewed a number of mathematical results related to non autonomous and “finite time” systems (all deterministic), with many being “forgotten” or not well-known.

•Highlighted the “hyperbolic-elliptic dichotomy”

•KAM theorems and Nekhoroshev Theorems--The latter may be more relevant for studies of Lagrangian transport (noted recent aperiodic versions of these theorems)

•Presented an example (“reverse-engineered” from Kolmogorov’s proof of KAM) which is an ideal “laboratory” for a variety of the issues raised.

•Highlighted a wealth of “important and undeveloped research directions”

•EXAMPLES ARE IMPORTANT (and show “the way”)