Basic Course I
Dynamical variables; phase space; flows and maps; deterministic chaos; linear autonomous systems; phase plane analysis; classification of singular points; stability and asymptotic stability; stable, unstable, and centre manifolds; Hartman-Grobman theorem; limitations of linear stability analysis; centre manifold theorem, limit cycles, Poincaré-Bendixon theorem.
Dissipative systems; one-dimensional maps; Bernoulli, tent, logistic maps; period-doubling route to chaos; intermittency; Frobenius-Perron equation; invariant density; two-dimensional maps; Baker transformation, cat map, standard map, and evolution of phase space density; Markov partitions, Kolmogorov-Sinai entropy, fractal sets; Multifractals, generalized dimension; cellular automata – a brief introduction.
Basic Course II
Hamiltonian systems, canonical transformations, Liouville-Arnold integrability, action-angle variables; Differential geometric aspects of Hamiltonian mechanics; Symplectic geometry and manifolds; Poincaré sections, area-preserving transformations; canonical perturbation theory,;Poincaré-Birkhoff theorem; small-divisor problem, KAM theorem, Green residue theorem; applications to problems in celestial mechanics
Quantum chaos; semi-classical chaos; WKB and EBK quantization; elements of random matrix theory; dynamical localization
Basic Course III
Linear and nonlinear wave equations; dispersive waves; solitary waves and solitons; nonlinear Schrödinger equation, sine-Gordon equation, Gross-Pitaevskii equation, Korteweg de Vries equation; integrability, bilinear methods; AKNS method, Painleve analysis, Lax pairs; Bäcklund transformations, Inverse scattering method; chaotic solitons, rogue waves and applications; Applications to plasma physics.
Basic Course IV
Time series – analysis and characterization; stochastic versus deterministic signals, stochastic resonance; noise reduction techniques; Lyapunov exponents; correlation and information dimensions; attractor reconstruction; empirical mode decomposition; strange non-chaotic attractors; applications to geophysics and climate.
Experimental and Laboratory Course
Random matrix theory and its applications:
Origin of RMT in Mathematics (Multivariate statistics), In Nuclear Physics, Extension to other branches of Physics/Mathematics
Classical Random Matrix Ensembles: Gaussian Ensembles, Wishart-Laguerre Ensembles, Circular ensembles: Matrix models, Invariance properties, Eigenvalue statistics, Correlation functions using orthogonal-polynomials method, Universal properties (e.g., Nearest-neighbour spacing distribution, Ratio of consecutive spacings), Asymptotic results (e.g., Wigner semicircle and Marchenko-Pastur densities)
Symmetry-crossover ensembles Matrix models, Connection with log-gas and Calogero-Sutherland models, Eigenvalue statistics, Brief discussion of applications
Non-Hermitian ensembles: Ginibre ensembles, Non-symmetric matrices
Many-body interactions and Embedded ensembles
Applications: Quantum chaos, Quantum kicked rotors, Quantum transport in mesoscopic systems, Chaotic mesoscopic cavities, Heidelberg vs. Mexico approach for modelling scattering, Landauer conductance, Shot-Noise power, Universal conductance fluctuations; Multiple Input Multiple Output (MIMO) channel communication, Channel matrix with Rayleigh fading, Shannon channel capacity; Growth models
Spatiotemporal patterns and dynamics in Classical and Quantum Physics
Introduction to patterns and spatiotemporal dynamics
Patterns in chemical systems. Reaction diffusion equations, the Turing instability and its analysis
Patterns in fluid systems. Taylor Couette flow and patterns. Instabilities which lead to patterns
Patterns in networks/granular media
Spatiotemporal dynamics, spatio-temporal intermittency, chimeras, techniques of analysis.
Patterns in quantum systems
Synchronization and control
Control of chaos: Methods: OGY, Delayed, Adaptive, Thresholding, Oscillation Death.
Synchronization of chaos: Complete, Generalized, Phase, Lag & Anticipation, Explosive, intermittent, Amplitude Envelop, and Partial Synchronization; Characterization & Stability of Synchronization; Synchronization in Hamiltonian systems.
Experimental demonstration of synchronization and its applications in electronic circuits.