Learning objectives
Aim of the course is to present some mathematical models arising from Mechanics or more generally from Applied Sciences. Methods for the qualitative analysis of systems of ODEs are presented, with particular attention to equilibrium solutions and stability.
Prerequisites
Knowledge of theory of ordinary differential equations is suggested.
Course unit content
Dynamical systems: definitions, properties. Stability. Liapunov's methods for stability of equilibrium solutions. Linear models: from harmonic oscillator to resonance phenomena. Non linear model in population dynamics: the Lotka-Volterra system, prey-predator models, mathematical models in epidemiology. Non linear oscillators: Van der Pol equation, Duffing equation. An introduction to bifurcation theory: stationary bifurcations, Hopf bifurcations and limit cycles. Poincarè-Bendixson Theorem for planar systems. Chaotic systems: Lorenz model. Discrete dynamical systems: Feigenbaum map, period-doubling bifurcations.
Bibliography
G.L. CARAFFINI, M. IORI, G. SPIGA, Proprietà elementari dei sistemi dinamici, Appunti per il corso di Meccanica Razionale, UNIVERSITA' DEGLI STUDI DI PARMA, a.a 1998-99;
G. BORGIOLI, Modelli Matematici di evoluzione ed equazioni differenziali, Quaderni di Matematica per le Scienze Applicate/2, CELID, TORINO, 1996;
R. RIGANTI, Biforcazioni e Caos nei modelli matematici delle Scienze applicate, LEVROTTO & BELLA TORINO, 2000;
M.W HIRSCH, S. SMALE, Differential Equations, Dynamical Systems and Linear Algebra, ACADEMIC PRESS, NEW YORK, 1974; J.D. MURRAY, Mathematical Biology, SPRINGER-VERLAG, NEW YORK, 1989;
J. GUCKENHEIMER, P. HOLMES, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectors Fields, SPRINGER-VERLAG, NEW YORK, 1983.
Teaching methods
The course consists mainly in oral lectures; a fundamental integration is represented by the numerical simulations in Matlab environment of the mathematical models presented and discussed during lectures.
Assessment methods and criteria
Oral exam, supported by a qualitative and numerical study of a dynamical system coming from applications
