Module page
Sistemi Dinamici
| academic year: | 2013/2014 |
| instructors: | Piero Negrini, Paolo Butta' |
| degree courses: | Mathematics (magistrale) Mathematics for applications (magistrale) |
| type of training activity: | caratterizzante |
| credits: | 6 (48 class hours) |
| scientific sector: | MAT/07 Fisica matematica |
| teaching language: | italiano |
| period: | II sem (03/03/2014 - 13/06/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- General aspects. Deterministic evolution processes. Vector fields and phase flows. Structure of phase curves: singular points and periodic orbits. Transformation law of vector fields. Flow-box theorem. Discrete dynamical systems (maps).
- Introduction to bifurcation theory. Bifurcation of singular points, necessary condition for the existence of bifurcation critical values. Examples of bifurcations for autonomous systems in one dimension depending on a real parameter.
- Elements of linear theory and applications. Matrix exponential, properties and strategies for the computation. Linearization around singular points. Adapted norms and exponential rate of convergence. Stability and instability detected by the linear part.
- Qualitative study of differential systems in the plane. Limit sets of phase curves and their properties. Attractive equilibria and basins of attraction. Study of the plane pendulum with linear friction. Bendixson criterium for the non-existence of periodic orbits. Isolated orbits and limit cycles. Poincaré sections and first return maps. An example of attractive limit cycle: the van der Pol oscillator. The Poincare-Bendixson theorem.
- Elements of hyperbolic dynamics. Hyperbolic points. Stable and unstable manifolds for flows and diffeomorphisms. Global structure of the invariant manifolds. Homoclinic intersections and orbits. Hyperbolic sets. Shadowing lemma and its consequences. Periodically perturbed systems and Poincarés theorem on periodic solutions. Melnikov formula. Application to the dynamics of the forced pendulum: existence of chaotic motions.
- Hamiltonian mechanics.
Elementary facts on Hamiltonian systems. Symplectic transformations. Action-angle variables. Conditionally periodic motion. The Arnold-Liouville theorem on the integrability of Hamiltonian systems. Hamilton-Jacobi equation and examples of separation of variables. Classical perturbation theory.
Suggested reading: P. Buttà, P. Negrini, Note del corso di Sistemi Dinamici, available at http://www.mat.uniroma1.it/people/butta/didattica/. Supplementary bibliographic suggestions can be found in these notes or is available at the same web page.
Type of course: standard
Knowledge and understanding: Successful students will learn rigorous and advanced concepts in the theory of nonlinear dynamical systems, with focus on Hamiltonian and hyperbolic systems.
Skills and attributes: Successful students will be able to apply the theoretical acquired knowledges to the analysis of nonlinear evolutionary models arising in Applied Sciences.
Personal study: the percentage of personal study required by this course is the 65% of the total.