Vered Rom-Kedar

Department of computer science and applied mathematics
Weizmann Institute
Rehovot, Israel 76100



Research interests and publications:


To chronological publication list



Steep billiard-like potentials

Abstract       Publications

Transport theory

Abstract       Publications

Chaotic fluid mixing

Abstract       Publications

Exponentially small splitting

Abstract       Publications

Forced NLS & Parabolic resonances

Abstract       Publications

Modeling & medicine

Abstract       Publications


Steep Billiard-like potentials and steep Impacts

with  M. Kloc, L. Lerman, V. Gelfreich, A. Rapoport, K. Shah, D. Turaev


Main motivation and results:    One of the underlying assumptions of statistical mechanics is the Boltzmann ergodic hypothesis which roughly states that the dynamics of a sufficiently large number of small molecules in a box is ergodic. The mathematical investigation of this hypothesis had focused mainly on studying the dynamics of hard spheres, and lead to fundamental developments in the theory of dynamical systems. Studying the motion of n "real" particles, having smooth steep repelling potentials, is a mathematical challenge with important implications to Physics.


    To this aim we develop methodologies for analyzing Hamiltonian systems with steep repelling potentials by using the singular billiard limit. Mathematically - they allow to study far from integrable systems in arbitrary dimension. Physically - they allow to study when the widely used idealized billiard model works, and, more surprisingly, when it fails. These methodologies are being extended to steep impact systems - here, in the singular limit, particles move under non-trivial potential in the domain interior and reflect from the domain boundary as in the billiards.


    In [R15,R17,R21] we establish that in the two dimensional case the smooth steep potentials can indeed ruin ergodicity. These findings lead to an experimental realization of this phenomenon in an atom-optic system (Kaplan et al, PRL, 2001). We recently showed that similar approach may be utilized to predict and control the form of the chaotic scattering function of various scattering-by-steep-potentials problems [R29].


    In [R28], we made the multi-dimensional leap: we proved that elliptic orbits appear in systems with steep smooth potentials that limit to Sinai billiards for arbitrarily large dimension (see figure for effective stability islands in a 20-dimensional phase space). Admittedly, the considered special example cannot be directly linked to the smooth many particles case. Nonetheless, this construction, of stabilizing mechanism in multi-dimensional highly unstable systems, suggests that analogous stabilizing mechanisms may exist in the smooth Boltzmann problem. 

More recently [R34,R38, R44], we study classical particles accelerators: billiards with oscillating boundaries. While it has been already established that 2D accelerators can lead to unbounded growth of the averaged energy of ensembles of particles (i.e. to heating), we are the first to construct accelerators in which this growth is exponential in time. 


We have demonstrated that the exponential in time energy growth mechanism may be induced by a billiard with mixed phase space (the Oscillating Mushroom). Moreover, we  derived adiabatic theory for non-ergodic systems from which the exponential growth rate is predicted and is then corroborated numerically [R46].





 K. Shah, V.Gelfreich, V. Rom-Kedar
D. Turaev
Leaky Fermi Accelerators, PRE 91, 062920 (2015)

 V. Gelfreich, V. Rom-Kedar, D. Turaev 

 Oscillating mushrooms: adiabatic theory for a non-ergodic system Journal of Physics A : Mathematical and Theoretical, Volume 47 (Number 39). Article number 395101 . ISSN 1751-8113
   [R44] V.Gelfreich, V. Rom-Kedar and D. Turaev Fermi acceleration and adiabatic invariants for non-autonomous billiards Chaos 22, 033116 (2012);
21 pages).
  [R43] V. Rom-Kedar and D.Turaev, Billiards: a singular perturbation limit of smooth Hamiltonian flows"  Chaos 22, 026102 (2012);
21 pages)
  [R38] V. Gelfreich, V. Rom-Kedar, K. Shah, D. Turaev, Robust exponential accelerators
         PRL  106, 074101, 2011.
   [R34]  K. Shah, D. Turaev and V. Rom-Kedar, Exponential energy growth in a Fermi accelerator  Phys. Rev. E 81, 056205, 2010


A. Rapoport and V. Rom-Kedar

Chaotic scattering by steep potentials.

Phys. Rev E., 77, 016207 (2008).


A. Rapoport, V. Rom-Kedar and D. Turaev

Stability in high dimensional steep repelling potentials.

Comm. Math. Phys, 279, 497-534, 2008.



V. Rom-Kedar and D.Turaev

Big islands in dispersing billiard-like potentials.

Physica D, 130:187-210, 1999.


A. Rapoport, V. Rom-Kedar and D. Turaev.

Approximating multi-dimensional Hamiltonian flows by billiards.

Comm. Math. Phys., 272(3), 567-600, 2007.


D. Turaev and V. Rom-Kedar

On smooth Hamiltonian flows limiting to ergodic billiards.

In Chaos, kinetics and nonlinear dynamics in fluids and Plasmas (Carry-Le Rouet, 1997), volume 511 of Lecture Notes in Phys., pages 17-50. Springer, Berlin, 1998



D. Turaev and V. Rom-Kedar

Islands appearing in near-ergodic flows.

Nonlinearity, 11(3):575-600, 1998.



D. Turaev and V. Rom-Kedar

Soft billiards with corners


J. Stat. Phys., 112(3-4):765-813, 2003


A. Rapoport, V. Rom-Kedar

Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials.


Chaos 16, 043108, 2006.




Multi-dimensional systems, forced NLS and Parabolic Resonances

with N. Paldor, Y. Dvorkin, A. Litvak Hinenzon and E. Shlizerman.


Main motivation and results: Near-integrable systems had been extensively studied since the days of Poincaré, yet, the structure of their chaotic solutions is still mostly unknown. A decade ago we discovered a new kind of instability in such systems, the parabolic resonance instability. It was discovered while studying a geophysical application [R9,R11,R12], and then we studied it in the multi-dimensional setting [R19,R20,R22]. More recently, we proposed that the hierarchy of bifurcation framework [RP11,R23] is convenient for identifying the dominant instability mechanisms that appear in specific multi-dimensional models.


    Notably, applying this framework to the conservatively forced Non-Linear Schrödinger (NLS) equation, conditions under which a truncated form of this equation has a parabolic resonance instability were derived [R23]. It was then demonstrated that such instabilities do appear in the original PDE model - the forced NLS - and correspond to a new type of solutions of this extensively studied equation [R24]. As a blind search for such solutions is a formidable numerical task, our analysis is pivotal for identifying what are the interesting parameter values and initial conditions for which new types of solutions - like the parabolic resonant ones - appear.


1. In [R32], we showed that the parabolic resonance mechanism can even cause spatial decoherence of small amplitude nearly flat initial data.

2. We have classified the structure of the unpertubed energy surfaces of a "generic" class of near-integrable two-degrees of freedom systems [R31] . This classification shows that there are some mechanisms of instability in two-d.o.f. systems that have not been studied yet!

3. In [R36] we developed a framework to study the near integrable PDE structure. This paper  was selected to be in the Nonlinearity Highly downloaded Collection for 2010. 

4. In [R35] we provide a framework for studyng analytically the parabolic resonance instability by using the adiabatic chaos framework. Most importantly, we show that the volume of the chaotic zone depends on the perturbation parameter as a power law, thus it is numerically observable.





 [R35]  V. Rom-Kedar and D. Turaev,  The symmetric Parabolic Resonance Instability,    Nonlinearity 23 1325-1351, 2010.
 [R36] E. Shlizermann and V Rom-Kedar, Classification of  solutions of the forced periodic nonlinear Schr\"{o}dinger equation   Nonlinearity 23 (9), 2183-2218, 2010.
[R32] E. Shlizermann and
V. Rom-Kedar;
Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos Physical Review Letters, 102, 033901, 2009.
 [R31]   M. Radnovic and
V. Rom-Kedar
Foliations of Isonergy Surfaces and Singularities of Curves Regular and Chaotic Dynamics,  Vol. 13, No. 6, pp. 645–668, J. Moser-80 memorial issue, 2008.
[R24] E. Shlizermann and
V Rom-Kedar
Three types of chaos in the forced nonlinear Schrodinger equation. Physical Review Letters, 96, 024104, 2006.

E. Shlizermann and
V. Rom-Kedar
Hierarchy of bifurcations in the truncated and forced NLS model Chaos 15(1) 013107, 22 pp, 2005.


V. Rom-Kedar

Parabolic resonances and instabilities

Chaos, 7(1):148-158, 1997.


A. Litvak-Hinenzon and V. Rom-Kedar

On energy surfaces and the resonance web

SIAM J. Appl. Dyn. Syst. 3(4), 525—573, 2004.


V. Rom-Kedar and N. Paldor

From the tropic to the poles in forty days.

Bull. Amer. Mete. Soc., 78(12):2779-2784, 1997.


V.Rom-Kedar, Y. Dvorkin, and N. Paldor;

Chaotic Hamiltonian dynamics of particle's horizontal motion in the atmosphere.

Physica D, 106(3-4):389-431, 1997.



V. Rom-Kedar; Y. Dvorkin, and N.Paldor

Chaotic motion on a rotating sphere.

In M.F. Shlesinger, Zaslavsky, G.M.thinspace and U.Frisch, editors, Proc. on Le'vy Flights and Related Phenomena, Lecture Notes in Physics, pages 72-87. Springer-Verlag, 1995.


V. Rom-Kedar

Some characteristics of two-degrees-of-freedom Hamiltonian flows.

In Doveil Benkadda and Elskens, editors, Transport, Chaos and Plasma Physics, II, Proc., Marseille, pages 164-178, Singapore, 1996. World Scientific.

[RP7] pdf


V. Rom-Kedar

Transport in a class of n-d.o.f. systems.

In Hamiltonian systems with three or more degrees of freedom (S'Agar'o, 1995) , pages 538-543. Kluwer Acad. Publ., Dordrecht, 1999

[RP9] pdf


A. Litvak-Hinenzon and V. Rom-Kedar

Parabolic resonances in near integrable Hamiltonian systems.

In D.S. Broomhead, E.A. Luchinskaya, P.V.E. McClintock, and T.Mullin, editors, emStochaos: Stochastic and Chaotic Dynamics in the Lakes, pages 358-368. American Institute of Physics, Melville, NY, USA, 2000.

[R19] pdf


A. Litvak-Hinenzon and V. Rom-Kedar

Resonant tori and instabilities in Hamiltonian systems.

Nonlinearity, 15(4):1149-1177, 2002.

[R20] pdf


A. Litvak-Hinenzon and V. Rom-Kedar

Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems

Physica D, 164(3-4):213-250, 2002.

[RP12] pdf


E. Shlizermann and V. Rom-Kedar

Characterization of Orbits in the Truncated and Forced Nonlinear Shrodinger Model.

ENOC, 2005

[RP11] pdf


E. Shlizerman and V. Rom-Kedar

Energy surfaces and hierarchies of bifurcations - instabilities in the forced truncated NLS.


In P. Collet et al. editors, Chaotic Dynamics and Transport in Classical and Quantum Systems. Kluwer Academic Press in NATO Science Series C, 2004.




Modeling and Medicine: models of the hematopoesis system:


with R. Malka, E. Shochat MD/PhD, and Prof. L. Segel (deceased, 2005).


 Main motivation and results: Why do G-CSF (Granulocyte Colony Stimulating Factor) injections help some neutropenic patients (patients with very low counts of this kind of white blood cells) and not others?


    In [R27] we employ dynamical systems approach to develop a simple model (not Hamiltonian) that is adequate for describing the natural dynamics of the neutrophils and the G-CSF levels in the blood on a time scale of several days. In [R30] we use a variant of this model to study the response of patients that received chemotherapy and thus became neutropenic to various protocols of G-CSF support.


More precisely, in [R30] we built axiomatically a mathematical model that describes the neutrophils’ dynamics. We showed that patients fall into three groups according to their blood counts, and that these groups respond differently to the growth factor therapy. The first group responds well. The current treatment is ineffective for the second group, but our results imply that a change in the administration protocol may help. In the third group, treatment with growth factors alone is insufficient at any dose, and the patients need an alternative therapy. These conclusions stem directly from the mathematical model, which makes it possible to predict outcomes and improve treatment. While the immune system is exceedingly complex, the model is simple, and that’s precisely its strength. We are proposing to test these recommendations in clinical trials. If these succeed, treatment efficiency would increase, and perhaps some lives could be saved.


We study the development of bacterial infections under neutropenic conditions by studying the neutrophils-bacteria interaction in-vitro [R33] and by combining this study with the blood neutrophils dynamics [R39].




[R42] Roy Malka, Baruch Wolach, Ronit Gavrieli, Eliezer Shochat and Vered Rom-Kedar Evidence for bistable bacteria-neutrophil interaction and its clinical implications   J. Clin Invest. doi:10.1172/JCI59832, 2012.  See also commentary .
 [R39] R. Malka and V. Rom-Kedar Bacteria--Phagocytes Dynamics, Axiomatic Modelling and Mass-Action Kinetics, Mathematical Biosciences and Engineering, 8(2), 475-502, 2011.
 [R33] R. Malka R, E. Shochat, V. Rom-Kedar;    Bistability and Bacterial Infections. PLoS ONE 5(5): e10010. doi:10.1371/journal.pone.0010010, 2010.

[R27] pdf Abstract

E. Shochat, V. Rom-Kedar and L. Segel

G-CSF control of neutrophils dynamics in the blood

Bull. Math. Biology 69(7), 2299-2338, 2007.

[R30] pdf Abstract

E. Shochat and V. Rom-Kedar

Novel strategies for G-CSF treatment of high-risk severe neutropenia suggested by mathematical modeling

Clinical Cancer Research  14, 6354-6363, October 15, 2008 2008.


Chaotic fluid mixing :

with R. Aharon, D. F. Carlson, E. Fredj, H. Gildor, R. Mundel

         A. Poje, A. Leonard and S. Wiggins.


Main motivation and results: The field of chaotic advection is concerned with the study of mixing of passive scalars in fluids with specified velocity fields that are time-dependent yet non-turbulent. Since the important paper by Aref in 1984, it attracts a growing community of researchers coming from diverse areas of science and engineering and involves both practical problems and beautiful mathematical questions.


    In my Ph.D. studies we considered time periodic planar flows (the OVP model) and introduced several dynamical systems concepts and tools to this field; We identified the unstable manifold and KAM tori as the observed objects in flow visualizations and as controllers of transport, discovered lobe dynamics, and established the direct relation between lobe dynamics and fluid transport [R3,RP1]. These ideas influence theoretical and experimental studies of fluid mixing problems arising in geophysical applications, traditional lab scale experiments and microfluidics (e.g. these findings became the main topic of S. Wiggins popular book "Chaotic Transport in Dynamical systems").


    In [R16] we proved that for a large class of two-dimensional time-periodic fluid flows some mixing characteristics have a universal dependence on the flow frequency even in the non-perturbative regime. These results were realized experimentally (see e.g. Horner et al. JFM 02).  In [R36] we show that upper bounds on eddy diffusivity may be found by combining LCS computations from RF data and airial photos of separating lines.


      In [41] we show that minute 3d convection can untangle the chaotic tangles of the surface mixing.

      In [47] we show that extreme values of observables provide simple, intuitive,
and computationally cheap diagnostics for fluid mixing; Moreover, they also enable a significant data reduction, since it is possible to extract from their cumulative distribution functions much of the relevant information regarding the existence, location, size and motion of the coherent structures.






 [R47] R. Mundel, E. Fredj, H. Gildor and V. Rom-Kedar, "New Lagrangian diagnostics for characterizing fluid flow mixing" Physics of Fluids, 26, no. 12 (2014): 126602
[R41] Rotem Aharon, Vered Rom-Kedar, and Hezi Gildor, "When complexity leads to simplicity: ocean surface mixing simplified by vertical convection" Phys. Fluids 24, 056603 (2012); \
 [R36] D. F. Carlson, E. Fredj, H. Gildor, V. Rom-Kedar   Deducing an upper bound to the horizontal eddy diffusivity using a stochastic Lagrangian model  Environmental Fluid Mechanics, 1567-7419 (Print) 1573-1510 (Online),  2010.


V. Rom-Kedar, A. Leonard, and S. Wiggins

An analytical study of transport, mixing and chaos in an unsteady vortical flow

J. Fluid Mech., 214:347-394, 1990.


A. Leonard, V. Rom-Kedar, and S.Wiggins

Fluid mixing and dynamical systems.

proc. intl. conf. on the physics of chaos and systems far from equilibrium. In Nuclear Physics B (Proc. Suppl.), volume2, pages 179-190, 1987


V. Rom-Kedar and

A.C. Poje

Universal properties of chaotic transport in the presence of diffusion

Phys. of Fluids, 11(8):2044-2057, 1999.




Transport theory of periodically forced systems :

with A. Litvak Hinenzon and G.M. Zaslavsky


 Main motivation and results: The transport theory of periodically forced Hamiltonian systems is concerned with characterizing the motion of ensembles of initial conditions from one region in phase space to another.


    In [R2] we observed that the geometry of the unstable and stable manifolds determines exactly all the transport rates between certain phase space regions for a general class of forced planar flows. This discovery of the popular lobe dynamics lead to a burst of theoretical and even some experimental studies in diverse fields of Physics ([R2] was generalized to various settings and was followed by numerous publications on its role in fluid mixing, chemical reactions, optics, astrophysics, ship stability, electroconvection etc.). In subsequent works I developed perturbational methods for calculating topological characteristics of these manifolds' complex geometry (called homoclinic tangles). These studies lead to analytical estimates of the transport rates for non-trivial time scales, to a lower bound on the topological entropy of homoclinic tangles in open flows [R1,R7,R8], to analytical methods for detecting multi-pulse solutions [R8] and to insights regarding the structure of strange attractors in slightly dissipative system [R10].





V. Rom-Kedar

Transport rates of a family of two-dimensional maps and flows

Physica D, 43:229-268, 1990.


V. Rom-Kedar and S. Wiggins

Transport in two-dimensional maps

Archive for Rational Mech. and Anal., 109(3):239-298, 1990.



Homoclinic tangles - classification and applications.

Nonlinearity, 7:441-473, 1994.



Secondary homoclinic bifurcation theorems

Chaos, 5(2):385-401, 1995.


A. Litvak Hinenzon and V. Rom-Kedar

Symmetry breaking perturbations and strange attractors.

Phys. Rev. E, 55(5):4964-4978, 1997.


V. Rom-Kedar and G.M. Zaslavsky

Islands of accelerator modes and homoclinic tangles.



Chaos, 9 (3):697-705, 1999.


V. Rom-Kedar

Homoclinic structures in open flows.

In J.-D. Fournier and P.-L. Sulem, editors, emLarge Scale Structures in Nonlinear Physics, Proc., Lecture Notes in Physics, pages 50-72, Villefranche-sur-Mer, 1991. Springer-Verlag.


Y. Ponty, A. Pouquet, V. Rom-Kedar, and P.L. Sulem

Dynamo in a nearly integrable chaotic flow.

In M.R.C. Proctor, editor, Proc. NATO ASI: Theory of Solar and Planetary Dynamos, pages 241-248, Cambridge, MA, 1992. Cambridge University Press.


V. Rom-Kedar

The topological approximation method.

In Doveil Benkadda and Elskens, editors, Transport, Chaos and Plasma Physics, Proc. , pages 39-57, Marseille, 1994. World Scientific


V. Rom-Kedar and G.M. Zaslavsky

Chaotic kinetics and transport (overview).

Chaos, 10(1):1-2, 2000. Focus Issue: Chaotic kinetics and Transport, eds V. Rom-Kedar and G. Zaslavsky. Including papers from the Workshop on Nonlinear Dynamics: Chaos, Transport and Transition to Turbulence held at the New York University, New York, November 1998.


V. Rom-Kedar

Frequency spanning homoclinic families

Commun. Nonlinear Sci. Numer. Simul., 8(3-4):149-169, 2003. Chaotic transport and complexity in classical and quantum dynamics.



Exponentially small splitting :




C.Amick, S.C.E. Ching, L.P. Kadanoff, and V.Rom-Kedar

Beyond all orders: Singular perturbations in a mapping

J. Nonlinear Sci., 2:9-67, 1992.



V.Rom-Kedar, L.P. Kadanoff, S.C.E. Ching, and C.Amick

The breakup of a heteroclinic connection in a volume preserving mapping.

Physica D, 62:51-65, 1993


S. Pekarsky and V. Rom-Kedar

Uniform stochastic web in low-dimensional Hamiltonian systems.

Phys. Lett. A, 225:274-286, 1997.


S. Pekarsky and V. Rom-Kedar

Degeneracies and instabilities of the motion of a charged particle - the 4d web-map model.

Nonlinearity, 10(4):949-963, 1997.