E. Ott "Chaos in Dynamical Systems", Cambridge University press, 2002
S.H. Strogatz, "Nonlinear Dynamics and Chaos: With Applications to physics, biology, chemistry and enieneering, Perseus Books, Cambridge 1994
M. Tabor, "Choas and Integrability in Nonlinear Dynamics", Wiley \& Sons, 1989
Learning Objectives
Knowledge acquired:
Theoretical and computational aspects on dynamical systems
Competence acquired :
Rigorous and numerical mathematical methods for the study of dynamical models of interest for physics
Skills acquired (at the end of the course):
Modelling and analysis of physical problems in the language of dynamical systems
Prerequisites
Clssical Mechanics, analysis, geometry and mathematical methods for physics
Teaching Methods
6 CFU
Lectures hours: 48
Further information
The first part of the course is given by R. Livi; the second one (from chaos theory) is given by A. Torcini.
Office hours
R. Livi: Tuesday 11.30-12.30
A. Torcini: Friday 15.00-16.00
Website: --
Type of Assessment
Oral test
Course program
Nonlinear dynamical systems: the compound pendulum and the standard map, billiards, Rayleih-Benard instability, Lorenz-Saltzmann model. Models of population dynamics, chemical reactions (Beluzov-Zhabotinsky).
Differential equations and applications: existence and uniqueness of solutions, conservative and dissipative systems, singular points, linearization, Lyapunov functional, central manifold theorem, Floquet analysis, Poincare' section, Volterra model and Van der Pol oscilator. Bifurcations: unidimensional central manifold, Hopf bifurcation, subarmonic bifurcation, invariant tori. Deterministic chaos: map of the interval, relation with renormalization group, comparison with experiments, intermittency of type I, III and II, map of te circle, frequency crossing. Diagnostic of Chaos: power spectra, Lyapunov exponents, geometry of strange attractors (fractals), experimental approaches. Invariant measures: ergodic systems, introduction to the ergodic theory in dynamical systems, multifractals, the tent map, Smale application and transverse homoclinic points. Integrable systems, perturbed integrable systems. KAM theorem. Biliiards as examples of mixed systems.