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, "Chaos and Integrability in Nonlinear Dynamics", Wiley & Sons, 1989 M. Cencini; F Cecconi, A. Vulpiani., “Chaos”, World Scientific 2010
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
S. Lepri.
Office hours
R. Livi: Tuesday 11.30-12.30
S. Lepri : su appuntamento
Type of Assessment
Oral examination
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. Generalized dimensions. Tological and metrical entropies. 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.