Applied Mathematics

DYNAMICAL SYSTEMS

Course objectives

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.

Channel 1

PAOLO BUTTA' Lecturers' profile

Program - Frequency - Exams

Course program

- Discrete and continuous time deterministic evolution processes. Reversible and irreversible processes. Phase space, time laws, and associated orbits. Stationary points and periodic points. Vector fields and phase flows. - Discrete-time dynamical systems. Maps and recursive laws. Fixed points, periodic and pre-periodic points. Invariant sets. Topological dynamical systems (continuous maps on metric spaces): limit sets, transitivity and topological mixing, chaos. Linear maps, contractions, hyperbolic linear maps. Conditionally periodic motion and translations on the torus. Theorem on the averages. Expanding endomorphisms of the circle, symbolic dynamics, Arnold's cat map. - Introduction to ergodic theory. Mean sojourn times and empirical measures, random initial data, time averages and space 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, Krylov-Bogolyubov theorem. - Review of systems of ordinary differential equations. Existence and local uniqueness of solutions, maximal solutions and complete systems. Lie derivative and first integrals. Transformation laws of vector fields. The flow box theorem. Differential equations on varieties: completeness of smooth vector fields on compact varieties. - Linear flows, contractions and hyperbolic flows. Hyperbolic points. Stable and unstable manifold for flows and diffeomorphisms. - Limit sets of a phase curve and their properties. Planar systems: conservative systems, isolated orbits and limit cycles. Poincaré sections. The van der Pol oscillator: an example of an attractive limit cycle. The Poincaré-Bendixson theorem. - Introduction to hyperbolic theory. Anosov's theorem on the structural stability of the Arnold map. Transverse intersections and homoclinic orbits. Hyperbolic sets. Shadowing lemma and its consequences. Periodically perturbed systems. Stroboscopic map. Melnikov formula and application to the dynamics of the forced pendulum.

Prerequisites

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.

Books

P. Buttà, Appunti per il corso di Sistemi Dinamici (in italian), available online at https://sites.google.com/uniroma1.it/paolobutta/didattica. Additional bibliographic material is indicated in these handouts or is made available at the aforementioned network address.

Teaching mode

Lectures with examples and exercises.

Frequency

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

Exam mode

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.

Bibliography

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.

Lesson mode