Scheda programma d'esame
ANDREA MILANI COMPARETTI
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KnowledgeBasic knowledge on matrix algebra and linear dynamical systems, on the concepts of stability, equilibrium, first integral, on finite difference equations. Specific knowledge on qualitative theory of dynamical systems, especially in the plane, on conservative systems with Newton. Lagrange and Hamilton formalism, on discretization of ordinary differential euqations, on non-integrable and chaotic dynamical systems. Significant examples of applications to problems in mechanics, in economics, and in model problems.
Assessment criteria of knowledgeIn the written exam (3 hours, 3 exercices) the student must demonstrate his/her knowledge of the course material and to solve computationally simple problems. During the oral exam the student must be able to demonstrate his/her knowledge of the course content and logical structure, i.e. give proofs.
- Final oral exam
- Final written exam
- Periodic written tests
Delivery: face to face
- attending lectures
- individual study
- Task-based learning/problem-based learning/inquiry-based learning
SyllabusDiscrete and continous dynamical systems. Linear dynamical systems and their solutions, resonance. Stability, instability, sinks, Lyapounove exponents and functions, Newtonian systems, conservative and with dissipation. Gradient systems, separatrix, limit sets, Poincare'-Bendixon theory. Hamiltonian systems with one degree of freedom, area preserving flow, integrability and time law, Legendre transform and Lagrange equation, constrained motion, canonical transformations, action-angle variables. Discrete dynamical systems and finte difference equations, examples in economics, discretization with Euler method, with standard map. The standard map of the pendulum: chaotic regions, the separatrices, homoclinic points, hyperbolic sets, Smale's horsehoe, ordered regions, Lyapounov exponents, definitions of chaos.
BibliographyA. Milani, Introduzione ai sistemi dinamici, Seconda edizione riveduta e corretta, Edizioni Plus, Pisa, 2009; 256 pagine + CD-ROM
Ultimo aggiornamento 14/11/2016 17:27