• Lecture 1. Introduction to nonsmooth maps. Poincare section. Impact section. Nonsmooth bifurcations (transition, annihilation, period adding). General classification of behaviours in nonsmooth linear maps of arbitrary dimension. Specialisation to 1D and 2D. Higher order solutions. Non-existance of solutions.
• Lecture 2. Nonsmooth nonlinear maps (1/2 & 3/2 power law nonlinearities). Maps with a gap (positive and negative)
• Lecture 3. Examples of codimension 2 nonsmooth bifurcations in maps (smooth & non- smooth and two non-smooth).
• Lecture 4. Nonsmooth nonlinear map taken from a high speed milling example: flip bifurcations, Neimark-Sacker bifurcations, period 2 solutions with flyovers, explanation of chaos.
• Lecture 5. Enhanced secure communication using nonsmooth maps. Explanation of observed behaviour in DC/DC converters. Effects of noise & smoothing on nonsmooth bifurcations. Unsolved

• Lecture 1.Analytical formalisms to describe nonsmooth continuous-time systems
• Lecture 2. Concept of solution, existence and uniqueness
• Lecture 3. Sliding and Zeno phenomena
• Lecture 4. Discontinuity-induced bifurcations (DIBs) in continuous-time systems:
• 4.1. Boundary equlibrium bifurcations
• 4.2. Grazing bifurcations
• 4.3. Bifurcations in Filippov systems with sliding
• Lecture 5. Classification of DIBs: the discontinuity mapping 6. Examples and applications 6) Using nonsmooth bifurcations for control system design

• Lecture 1: General principle of Dynamical System Theory

Topics: Preliminars and Examples of Dynamical systems. Poincarè recurrence theorem. Periodic orbits and invariant measures. Time and spatial average of classical observables: Birkhoff Ergodic Theorem. Ergodicity, mixing and decay of correlations. Hyperbolicity and statistical properties of orbits (SRB and maximal entropy measures). Oseledec Theorem and Lyapunov exponents. Spectral characterization and properties of hyperbolic systems: Koopman and Perron-Frobenious operators.

• Lecture 2: Quantum Mechanics on compact phase space

Topics: The space of quantum states and quantization of observables. The discrete Heisenberg grouo and the Metaplectic representation. Quantum dynamics over the torus. Quantized cat maps, baker map and sawtooth maps. The Wigner function over the torus: phase space representation of quantum mechanics.

• Lecture 3: Statistical Properties of quantum chaotic systems

Topics: Eigenfunctions and eigenvalues in quantum chaotic systems. Random Matrix Theory and the BGS conjecture. Random waves conjecture. Quantum Unique Ergodicity and Scars. Values distribution of eigenfunctions

• Lecture 4: Evolution of coherent states, Eherenfest time and correspondence

Topics: Brief introduction to coherent states, coherent states on the torus, Wigner transformation of coherent states. Evolution of coherent states under quantized linear automorhism. Egorov Theorems and breakdown of correspondance. Eherenfest time and Quantum Fidelity.

• Lecture 5: Quantum Chaos & Number Theory

  Topics: Number Theoretic Backgrounds: divisibility, diophantine equations, prime numbers, modular arithmetic, the Legendre symbols and quadratic reciprocity.
  
  Arithmetic eigenfunctions of the quantum cat maps. Quantum degeneracy and the Little Fermat Theorem. Hecke operators and equidistribution of eigenfunctions. Quantum chaos and the distribution of zeros in the Riemann Zeta function.

• Lecture 1. Travelling fronts in dimension 1
  • solutions of ordinary differential equation u''-cu'+f(u)=0
  • sign of the speed c
  • existence, monotonicity and/or uniquness properties for bistable or monostable KPP nonlinearities f
• Lecture 2. Dynamical issues
  • global stability of one-dimensional fronts
  • asymptotic spreading speed in R^N
• Lecture 3. Conical bistable fronts in R^N (1)
  • maximum principles in unbounded domains
  • monotonicity and further qualitative properties
• Lecture 4. Conical bistable fronts in R^N (2)
  • formula for the speed of propagation
  • existence of bistable conical fronts in R^N by sub- and super-solution method
• Lecture 5. Curved KPP-type fronts in R^N
  • interaction of two planar fronts
  • interaction of a measurable sum of planar fronts
  • qualitative properties of curved KPP fronts

• Lecture 1. Basic concepts
  • Ordered metric spaces
  • Order-preserving semiflows
  • Limit sets
  • Various examples
  • Convergence near a manifold of equilibria
• Lecture 2. Generic convergence results
  • Limit set dichotomy theorem
  • Almost everywhere convergence
  • Instability of periodic orbits
  • Unstable manifolds of equilibria
  • Connecting orbits
• Lecture 3. Symmetry and stability
  • Order preserving systems in the presence of symmetry
  • Does stability imply symmetry?
  • Applications to elliptic and parabolic PDE's

• Lecture 4. Criteria for blow-up
  • Examples of blow-up
  • Kaplan estimates
  • Fujita exponent
  • Generalized Fujita results
• Lecture 5. Blow-up profiles
  • Self-similar solutions
  • Local blow-up profile
  • Type I and type II blow-up
  • Stability of blow-up profile

• Lecture 2
  • Invariant manifolds theorems for infinite dimensional dynamical systems
    • Special features of infinite dimensional systems.
    • Types of partial dierential equations that can be treated.
  • The theorem of Chen, Hale and Tan.
    • Statement of the theorem.
    • Sketch of the proof.
    • Some basic applications.
• Lecture 3
  • The Navier-Stokes equations.
    • The physical origin of these equations.
    • The vorticity formulation and the Biot-Savart law.
    • Strong solutions vs. weak solutions.
    • The existence and uniqueness of solutions of the two-dimensional Navier-Stokes and vorticity equations.
• Lecture 4
  • Behavior of “small” solutions of the Navier-Stokes equation.
    • Scaling variables.
    • Invariant manifolds near the origin.
    • Optimal decay rates and invariant manifolds.
  • Oseen vortices and the global center manifold for the two-dimensional Navier-Stokes equation.
  • Local stability of Oseen vortices for arbitrary Reynolds number.
• Lecture 5
  • Lyapunov functionals and global stability of Oseen vortices.
  • Vortex solutions for the three-dimensional Navier-Stokes equations.
    • Burgers vortices and related solutions.
    • Three-dimensional stability of Burgers vortices.