Many living and inanimate systems exhibit nonlinear dynamics expressed in various ways: loss of system state stability and its evolution to another by symmetry breaking, extreme response to small disturbances, universal behaviors and more. These systems are most often described by ordinary (non-linear) or partial (non-linear) differential equations. The course aims to provide basic, analytical and computational tools for analyzing nonlinear systems, demonstrating them with models of physical, environmental, chemical and biological systems.
Specific topic:
• Motivation and examples of nonlinear behavior in nature: resonance in forced oscillators, chemical and biological oscillations, solitons in fluids and optics, electrical discharge, neural signals and cardiac arrhythmia, and patterns in arid regions
• Basic concepts in nonlinear dynamics and their demonstration with the help of ordinary differential equations (ODEs): Stability and linear instability of stationary solutions (fixed points), phase space, and basins of attraction
• Bifurcation theory, classification and characterization of bifurcations (saddle-node, transcritical, pitchfork, etc.), supercritical and subcritical bifurcations and relation to second and first phase transitions.
• Invariance and center manifold theorem
• Hopf bifurcation and limit cycles, Poincaré – Bendixson theorem, Floquet theory, Poincaré sections and mapping, periodic and quasi-periodic solutions
• Global bifurcations, such as homoclinic and heteroclinic, introduction to Shil'nikov homoclinic theory, connection to localized waves, such as excitable pulses and fronts
• Multiplicity of unstable solutions and chaotic dynamics, strange attractors, Lyapunov exponents, and fractals
• Energy dissipation vs conservation (Leaponov functional), dynamics in the Hamiltonian vs gradient systems, i.e., chemical potential (Lagrange multipliers), meta-stable states
• Introduction to spatially extended systems described by nonlinear partial differential equations (PDEs), examples of spatial patterns such as: Wave fronts in ferromagnetism, stripes and hexagons in fluid convections, waves in the heart and cells, resonant patterns (mode-locking) in under periodic forcing
• Numerical methods for solving non-linear differential equations: Finding roots, Runge-Kutta methods for solving initial value problem in ODEs, finite difference methods for boundary value problems in ODEs and PDEs.
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