Course media1. E. Meron, Nonlinear Physics of Ecosystems, CRC Press, 2015 (QH 541.15.M3M476)
2. M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press 2009 (Q 172.5.C45C76).
3. S. H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, Addison-Wesley 1994; 2nd edition WestView Press, 2014 (Q 172.5.C45S767).
תצורות מרחב וזמן במערכות לא לינאריות (2022)
About course - General
Topic outline
Pattern forming systems
Thermal convection: Rayleigh-Benard system, hydrodynamic equations
Chemical reactions: Belousov-Zhabotinsky reaction. Oregonator model, FitzHugh-Nagumo model
Dryland ecosystems: Vegetation patterns, water-biomass feedback loops, models
Dimensional analysis
Dimensionally independent quantities, derivation of scaling laws, non-dimensional forms of dynamic equations, examples
Pattern-forming systems as low-dimensional dynamical systems
Dimension of a dynamical system, linear stability analysis, stationary and oscillatory instabilities, bifurcation diagrams, local vs. global bifurcations, variational and non-variational systems, excitable systems, codimension-2 bifurcations.
Reading material: Section 2.2 of book
Linear stability analysis of spatially extended systems
The Swift-Hohenberg equation. Linear stability analysis of uniform states in small systems. Center Manifiold. Linear stability analysis of uniform states in large systems using the Swift-Hohenberg equation and the FitzHugh-Nagumo model as examples. Classification of instabilities in spatially extended systems.
Amplitude equations
Derivation of amplitude equations. Amplitude equation for stripe patterns using the SH model (the NWS equation). Amplitude equation for uniform oscillations using the FHN model (the CGL equation). Phase dynamics. Phase instabilities of periodic patterns and traveling waves. Stationary patterns in 2d. Limitations of the amplitude equation approach.
Front dynamics
Fronts in bistable systems, singular perturbation analysis, front instabilities
Periodically forced systems
Uniform time-periodic forcing of oscillatory systems: frequency locking, multiplicity of stable phase states, fronts. Stationary space-periodic forcing of pattern forming systems: wavenumber locking, multiplicity of stable phase states, front.
Section 9
Section 10
Section 11
Section 12
Section 13