Nonlinear time series analysis: Power spectra, embedding techniques, reconstruction of attractors, calculation of dimensions. Prediction techniques, surrogate analysis. Control of chaos, synchronization.
Stability, Lyapunov functions. Center manifolds, normal forms for flows and maps. Local and global bifurcations. Melnikov's method. Hyperbolic systems and strong chaos. Symbolic dynamics, Markov and generating partitions. Elements of the thermodynamic formalism.
Hamiltonian dynamics: canonical transformations, symplectic group, action-angle variables, Hamilton-Jacobi equation. Liouville-Arnold integrability and dynamical symmetry. Non-integrable systems, canonical perturbation theory, small-divisor problem. KAM theorem. Area-preserving maps, fixed points, Poincare-Birkhoff theorem. Homoclinic and heteroclinic cycles. Local and global chaos.
Quantum Chaos: WKB and beyond. Semiclassical quantization. Periodic orbit theory. Elements of random matrix theory. Simple models of quantum chaos. Dynamical localization and experimental realizations in atomic, mesoscopic and optical systems.
Integrability in dynamical systems: Different kinds of integrability: Liouville-Arnold, algebraic, analytic, C-integrability. Painleve analysis, extension to the PDEs encountered in physical applications: KdV, modified KdV, sine-Gordon, nonlinear Schrodinger and Burgers equations. Nonlinear waves, solitary waves and solitons. Infinite number of conservation laws. Elementary ideas on quantum integrability: Bethe ansatz.
Dynamics of networks. Regular, small-world, growing, scale-free and random networks. Comparison with percolation. Evolution on networks. Physical applications.
Stochastic dynamical systems: Discrete and continuous random variables. Shot noise, Poisson noise, white noise. Gaussian processes. Markov chains, Markov processes, birth-and-death processes. Brownian motion, Langevin equation, master equation, Fokker-Planck equation. Correlation functions, power spectra, Wiener-Khinchin theorem. Diffusion processes, stochastic differential equations. Reaction-diffusion equations with noise.
Turbulence: Navier-Stokes equation, Boussinesq approximation, symmetries, conservation laws. Phenomenology of fully-developed turbulence. Kolmogorov's 1941 theory. Intermittency. Multiscaling and RG methods. 2D turbulence.