Mathematics

PAOLO BUTTA'
PAOLO BUTTA'

General targets: To acquire advanced knowledge in the theory of dynamical systems.

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on hyperbolic systems and applications in mechanics, like stability theory. Moreover, they will learn part of the general theory of hyperbolic invariant sets, with applications to homoclinic intersections, chaotic motion, and ergodic theory, in the framework of concrete mechanical systems.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study equilibrium stability problems both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

General targets:
To acquire advanced knowledge in the theory of dynamical systems.

Specific targets:

Knowledge and understanding: Students who have passed the exam will have acquired rigorous and advanced theoretical knowledge in the field of dynamical systems theory, with focus on applications in mechanics and applied sciences in general. They will learn elements of stability theory and hyperbolic theory (such as homoclinic intersections and existence of chaotic motions). They will also learn elements of the theory of topological dynamical systems and of ergodic theory.

Applying knowledge and understanding: Students who have passed the exam will be able to: i) study stability problems of equilibria and cycles, both when this is recognized by the linear part and by the methods of Liapunov's theory; iii) analyze planar systems that exhibit self-oscillation phenomena; iv) formalize in concrete problems the concepts of intersection of stable and unstable manifolds and the related chaotic phenomena; v) apply the basic techniques of ergodic theory to concrete problems.

Making judgements: Students who have passed the exam will be able to use the acquired knowledge in the analysis of nonlinear evolutionary models arising in Applied Sciences.

Communication skills: Students who have passed the exam will have gained the ability to communicate and expose concepts, ideas and methodologies of the theory of dynamic systems.

Learning skills: The acquired knowledge will allow students who have passed the exam to deepen, in an individual and autonomous way, techniques and methodologies of the theory of dynamical systems.

The course requires the knowledge of the topics and basic tools that are introduced in the courses of analysis, linear algebra, geometry and rational mechanics of the three-year degree in mathematics.

- Discrete and continuous time deterministic evolution processes. Vector fields and phase flows. Transformation laws of vector fields. The flow box theorem. Stroboscopic maps and Poincaré sections.

- Linear flows and maps, adapted norms, and decomposition in invariant subspaces. Central subspace, conditionally periodic motion and translations on the torus. Theorem on the coincidence of means.

- Hyperbolic points. Stable and unstable manifold for flows and diffeomorphisms.

- Limit sets of phase curves and their properties. Attractive equilibria, basins of attraction and Liapunov’s functions. Stability of cycles: Poincaré sections and first return maps.

- Planar systems. Conservative systems. Isolated orbits and limit cycles. The van der Pol oscillator: an example of an attractive limit cycle. The Poincaré-Bendixson theorem.

- An outline of local bifurcation theory. Examples of bifurcations for autonomous systems in one dimension depending on a real parameter. An example of Hopf bifurcation.

- Topological dynamic systems: transitivity and topological mixing, chaoticity. Arnold's cat map. Anosov's theorem on the structural stability of the Arnold's map. Transversal intersections and homoclinic orbits. Hyperbolic sets. Shadowing lemma and its consequences. Periodically perturbed systems. Melnikov formula and pplication to the dynamics of the forced pendulum.

- Introduction to ergodic theory: random distributions of initial data, temporal and spatial averages. Invariant measures and recurrence properties. The ergodic theorems of Birkhoff-Khinchin and Von Neumann. Ergodicity, mixing and convergence to equilibrium. Examples: Baker's map, Bernoulli schemes, translations and algebraic automorphisms of the torus. Ergodic and mixing measures of continuous maps on metric spaces.

P. Buttà, Appunti per il corso di Sistemi Dinamici (in italian), available online at http://www.mat.uniroma1.it/people/butta/didattica. Additional bibliographic material is indicated in these handouts or is made available at the aforementioned network address.

V.I. Arnold: Metodi geometrici della teoria delle equazioni differenziali ordinarie. Editori riuniti 1989. (Oppure la versione inglese V.I. Arnold: Geometrical methods in the theory of ordinary differential equations, New York, etc. Springer, 1988.)
V.I. Arnold and A. Avez: Ergodic problems of classical mechanics. W.A. Benjamin, 1968.
M.W. Hirsch, S. Smale, R.L. Devaney: Differential Equations, Dynamical Systems and an Introduction to Chaos. Second/Third Edition. Academic Press/Elsevier 2003/2013.
A. Katok, B. Hasselblatt: Introduction to the modern theory of dynamical systems. Cambridge university press, 1995.
A. Katok, B. Hasselblatt: A first course in dynamics: with a panorama of recent developments. Cambridge university press, 2003.

Lectures with examples and exercises.

Attendance at lessons is recommended for a good understanding of the course.

The exam aims to evaluate learning through an oral test, in which the student must demonstrate that he has acquired sufficient knowledge of the covered topics.

(1) The asymptotic behavior of planar systems using the Poincaré-Bendixson theory.
(2) The theorem on invariant manifolds and examples.
(3) Linear flow on the torus with particular emphasis on the case of quasi-periodic motion.
(4) The van der Pol oscillator, as an example of attractive limit cycle and as a tool for constructing an example of a Hopf bifurcation.
(5) The notions of topological transitivity and mixing, and chaoticity, illustrated by examples.
(6) Instability and sensitive dependence on the initial data for the dynamics in the prossimity of an hyperbolic set.
(7) Consequences of the existence of transverse intersections of stable and unstable manifold of a hyperbolic point (emergence of chaos).
(8) The structural stability of the Arnold cat map (Anosov's theorem).
(9) The notions of ergodicity and mixing, illustrating their meaning and differences also through specific examples.