Courses ¨

NONLINEAR OSCILLATIONS AND DISSIPATIVE STRUCTURES
(A.A.Polezhaev)

  • Basic concepts
    • Open systems, non-equilibrium states, thermodynamics of open systems
    • Subordination principle
    • Active media, autovawes, dissipative structures
  • Elements of qualitative theory of dynamical systems
    • Phase portrait, trajectory, basic types of bifurcations in a plane
    • Characteristic Lyapunov exponents
    • Concept of attractor, strange attractors
    • Puancare map
    • Bifurcations in multidimensional systems, chaos in dynamical systems and scenarios (ways) of its appearance
    • Fractal structures and their dimension
  • Autowave structures
    • Models of excitable media
    • Systems with multiple stationary states
    • Autowave regimes in media with diffusion, displacement waves, traveling pulses, spiral waves.
    • Target patterns
  • Theory of stationary dissipative structures
    • Dissipative structures, turing bifurcation
    • Sharp and quasi-harmonic DS
    • Construction of DS solutions near the Turing bifurcation
    • Method of construction of sharp DS
  • Spatial-temporal patterns in physical and chemical systems
    • The Prigogine-Lefevr-Nicolis model (Brusselator)
    • Belousov-Zhabotinsky reaction, Fild-Noice model (Oregonator)
    • Hydrodynamic instabilities, the Benard effect
    • Autowave media in solid matter.
    • Heat waves and non-uniform stationary states
  • Self-organization in biological systems
    • The pronlem of biological self-organization
    • The concept of biological (morphological) field, positional information
    • The role of DS in morphogenesis, chemical basis of morphogenesis
    • Morphogenesis of hydra, the Gierer-Mainhardt model
    • Mechanism of pattern formation on animal skins, morphogenesis in insects
    • The role of currents in processes of self-organization
    • Models of mechanisms of pattern formation in bacterial colonies and Dictyostellium Discoideum amoebae
    • "Mechanical" models of morphogenesis

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