A Student Seminar in Nonlinear dynamics and chaos



Instructors: Einat Aharonov, Vered Rom-Kedar, Uzi Smilansky and Eli Tziperman.


Day, time & location of seminar: Thursdays at 16:00-18:00 hrs in Sem. Rm A, Weissman Building. First meeting: April 11, 2002

Announcements Last updated: Apr 4.
Feel free to write or call me with any questions:
Vered, x3170, vered@wisdom.weizmann.ac.il


Introduction

Nonlinear models arise in a variety of fields of science, including fluid dynamics, geophysics, particle physics and biological models. The analysis of these models results in many cases in complicated bifurcation scenarios and sometimes chaotic dynamics. Interpretation of these results leads to interesting insights regarding the behavior of the systems. Students will be offered a variety of topics and papers to study and present in front of the class. The study of the papers will be guided by the corresponding researchers. Students are expected to have basic familiarity in nonlinear dynamics.

List of suggested topics and papers for student seminars

  1. Controlling chaos; El Nino's chaos; Controlling El Nino's chaos (Eli):
    1. Ott, E., and Grebogi, C., and Yorke, J., Controlling chaos, Phys. Rev. Lett., 1990, 64, 1196-1199.
    2. Tziperman, E., M. A. Cane and S. Zebiak, 1995: Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos. Journal of the Atmospheric Sciences, 52, 293-306. here
    3. E. Tziperman, H. Scher, S. Zebiak and M. A. Cane, 1997: Controlling spatiotemporal chaos in a realistic El Niņo prediction model. Physical Review Letters, 79, 6, 1034-1037. here

  2. Glacial dynamics: stochastic resonance, frequency locking and entrainment, chaos and relaxation oscillations (Eli):
    1. R. Benzi and G. Parisi and A. Sutera and A. Vulpiani, Stochastic resonance in climatic change, Tellus, 1982, 34, 10-16
    2. (?) R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A 14, L453-L457 (1981).
    3. a coherence resonance paper
    4. Ghil, M. Cryothermodynamics: the chaotic dynamics of paleoclimate, Physica D 1994, 77, 130-159
    5. Gildor, H. and E. Tziperman, 2000. Sea ice as the glacial cycles' climate switch: role of seasonal and orbital forcing. Paleoceanography, 15, 605-615. here

  3. Chaos synchronization, chaos synchronization in climate dynamics (Eli):
    1. Pecora LM, Carroll TL Synchronization in chaotic systems Phys Rev Lett 64 (8): 821-824 FEB 19 1990
    2. Duane GS, Tribbia JJ Synchronized chaos in geophysical fluid dynamics Phys Rev Lett 86 (19): 4298-4301 MAY 7 2001
    3. Duane GS, Webster PJ, Weiss JB Go-occurrence of Northern and Southern Hemisphere blocks as partially synchronized chaos J Atmos Sci 56 (24): 4183-4205 DEC 15 1999

  4. Geodynamics: Friction as it relates to Earthquakes (Einat):
    1. Creep, stick-slip and dry friciton dynamics Heslot et al. Phys Rev E, 49, pg. 4973, (1994)
    2. Non-linear dynamics of friction F Elmer J Phys A: Math Gen. 30, pg 6057 (1997)
    3. Transition from stick-slip to smooth sliding: An earthquake-like model Braun OM, Roder J PHYSICAL REVIEW LETTERS 88 (9): art. no. 096102 MAR 4 2002
    4. "slider-block" chapter from fractals and chaos by Turcotte.

  5. Geodynamics: The Earth's magnetic field (Einat):
    1. "Rikitake dynamo" chapter from: fractals and chaos by turcotte book
    2. Cook and Roberts (1970).

  6. Fluid Mixing and Transport (Vered)
    1. V. Rom-Kedar, A. Leonard and S. Wiggins [1990] An Analytical Study of Transport, Mixing and Chaos in an Unsteady Vortical Flow , Journal of Fluid Mechanics, volume 214, pp. 347-394; Rom-Kedar, V.; Poje, A. C. [1999] Universal properties of chaotic transport in the presence of diffusion. Phys. Fluids 11, no. 8, 2044-2057.
    2. G. Haller [2001], Lagrangian structures and the rate of strain in a partition of two dimensional turbulence, Physics of Fluids 13(11)

  7. Hamiltonian chaos (Vered):
    1. Bessi, Ugo; Chierchia, Luigi; Valdinoci, Enrico Upper bounds on Arnold diffusion times via Mather theory. J. Math. Pures Appl. (9) 80 (2001), no. 1, 105-129
    2. Haller, G. Diffusion at intersecting resonances in Hamiltonian systems. Phys. Lett. A 200 (1995), no. 1, 34-42.
    3. Anna Litvak-Hinenzon and Vered Rom-Kedar [2002] Parabolic resonances in 3 d.o.f. near integrable Hamiltonian systems, Physica D, to appear.

  8. Dynamical systems with small noise (Vered)
    1. Nils Burglund and Barbara Gentz -selected preprints [2001-2]; e.g. "Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations", or "The effect of additive noise on dynamical hysteresis", etc.
    2. Hamiltonian systems with noise: Mark. I. Freidlin [1998] "Random and Deterministic perturbatopms of nonlinear oscillators", Doc. Math. J. DMV, 223-235.

  9. Mathematical models for cancer (Vered)
    1. Bellomo, N.; Preziosi, L. Modeling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modelling 32 (2000), no. 3-4, 413-452 and references therein.

  10. Chaotic scattering of particles and waves (Uzy)
    1. U Smilansky in Chaos and Quantum Physics ed: M J Giannoni A Voros and J Zinn-Justin (North Holland, Amsterdam, 1992)
    2. "Chaos" Focus issue on "Chaotic Scattering" Chaos, vol 3 No. 4. 1993. Quantum chaos
    3. Rom-Kedar, Vered and Turaev, Dmitry [1999] Big Islands in dispersing billiard-like potentials. Phys. D 130, no. 3-4, 187-210.

 back to ESP home page