Spatio-temporal Patterns in Nonlinear Systems

Semester A,  2009/10

Instructor: Ehud Meron

Office: Building 54, room 333

Tel: 6477556, 6596926,il



Wednesday: 15-17,  90/234

Thursday: 9-10, 32/113


List of topics:

  1. Instabilities in physical systems: Rayleigh-Benard convection. Self focusing and optical solitons. Chemical oscillations. Vegetation patterns.
  2. Mathematical formulation of instabilities: The transcritical, saddle node, pitchfork and Hopf bifurcations. Imperfect bifurcations. Local and global bifurcations. Excitable systems.
  3. Spatio-temporal patterns in small systems. Normal forms and the Center Manifold theorem.
  4. Amplitude equations. Applications to electromagnetic waves in dispersive media, convection, and chemical oscillations. The Non-Linear Shrodinger (NLS) equation, the Newell-Whitehead-Segei (NWS) equation, and the Complex Ginzburg-Landau (CGL) equation.
  5. Phase dynamics. The Eckhaus, zigzag and Benjamin-Feir instabilities. Phase and amplitude turbulence.
  6. Front propagation in bistable systems. Gradient vs. non-gradient systems. The non-equilibrium Ising-Bloch (NIB) bifurcation. Transverse instabilities.
  7. Singular perturbation theory of stationary and traveling patterns in bistable and excitable media.
  8. Spiral waves. Spiral turbulence. Kinematic theory.
  9. Forced oscillations. Frequency locking in spatially extended systems.


pattern formation in nature


  1. S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, (Addison-Wesley, Reading Mass., 1994). Call #:Q 172.5.C45S767.
  2. M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, (Cambridge University Press, 2009).
  3. P. Manneville, Dissipative Structures and Weak Turbulence, (perspectives in Physics, New York, 1990). Call #: QA 871 .M33.
  4. A.C. Newell and J.V. Moloney, Nonlinear Optics, (Addison-Wesley, New York, 1992). Call #: QC446.2.N48.
  5. J.D. Murray, Mathematical Biology, (Springer-Verlag, New York, 1989). Call #: QH 323.5.M88.
  6. J. Guckenheimer and P. Holmes Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, (springer-Verlag, New York, 1983). Call #: OA 1.A647.
  7. M. Cross and P .C. Hohenberg, Pattern Formation outside of Equilibrium, Rev. Mod. Phys. 65 (1993) 2.
  8. J.J. Tyson and J.P. Keener Singular Perturbation Theory of 8. Traveling Waves in Excitable Media, Physica D 32 (1988) 327.
  9. E. Meron, Pattern Formation in Excitable Media, Phys. Rep. 218 (1992) 1.


Grading policy:  

Grades will be determined by evaluating personal projects to be submitted in a limited period of time.

A prerequisite for having a grade is the submission of all homework assignments.

Homework assignments

Lecture notes