To know how to deal with dynamical systems.

KNOWLEDGE AND UNDERSTANDING. At the end of the course the
students will know the foundations of the theory of dynamical systems.
The student will be able to solve simple exercises.

APPLYING KNOWLEDGE AND UNDERSTANDING. At the end of the course
the student will know how to apply techniques from dynamical systems
and ODE to solve problems of medium difficulty, using rigorous and
qualitative analysis.

MAKING JUDGEMENTS. At the end of the course the student will be able to
recognize and apply techniques to solve ODEs and dynamical systems.

COMUNICATIVE SKILLS. At the end of the course the student will be able
to express him/herself appropriately on the themes of the theory of
dynamical systems, the know-how and its abstract reasoning.

LEARNING SKILLS. At the end of the course the student will be able to
consult a medium level text on the theory of dynamical systems.

Analysis-Geometry-Physics 1-Analytical Mechanics

Introduction to dynamical systems. Dynamical Systems 1-dim. Bifurcations. Discrete dynamical systems in 1-dim. Scalar linear maps and the logistic map. Linear continuous dynamical systems Non-linear dynamical systems Stability theory. Gradient dynamical systems. Hamiltonian dynamical systems. Liouville theorem. Recurrence theorem and ergodic theory. Invariant sets. Hartman-Grobman theorem and linearization. Invariant manifolds. Non-linear planar systems. Poincaré-Bendixon theorem. Elements of chaotic dynamics. Advanced topics.

Lecture notes [MJ] - James D. Meiss, "Differential Dynamical Systems", SIAM, 2007 [W] - Wiggins S. "Introduction to applied nonlinear dynamical systems and chaos" Springer 2003 [PS2]- C. D. Pagani, S. Salsa, “Analisi Matematica2, vol 1, 2, Masson, 1993 [M]- J. D. Murray, “Mathematical Biology”, vol 1, Springer, 2003 [H-K]- J. Hale, H. Koçac, “Dynamics and bifurcations”, Springer-Verlag, 1991 [H-S]- M. W. Hirsch, S. Smale, “Differential equations, dynamical systems and linear algebra”, Academic press N.Y. 1974 [H-S-D]- M. W. Hirsch, S. Smale, R. L. Devaney, “Differential equations, Dynamical systems and an introduction to chaos”, Academic Press N.Y., 2004 [P]- L. Perko, “Differential equations and dynamical systems”, SpringerVerlag, 1991 [S]- S. Strogatz, “Nonlinear dynamics and chaos”, Westview, 1994 Dispense in rete: [B]-G.Benettin , Introduzione ai sistemi dinamici per la Scuola Galileiana A.A. 2011-12 http://www.math.unipd.it/~benettin/ [GE]- G. Gentile, “Introduzione ai sistemi dinamici” http://www.mat.uniroma3.it/users/gentile/2010-2011/testo/ Divulgativi I. Peterson, “Newton’s clock, Chaos in the solar system”, W. H. Freeman and Company, 1993

Introduction to dynamical systems. Examples and qualitative analysis. Elements of differential geometry: manifolds, tangent space, vector fields and flows. Basics of ordinary differential equations.

Dynamical systems in one dimension. Equilibria and stability, phase portrait. Examples. Dynamical systems depending on one parameter and bifurcations.

Discrete dynamical systems. Equilibria, stability and bifurcations. Examples. The logistic map. Conjugation. Chaotic dymamics. Symbolic dynamics.

Linear dynamical systems. Basics of linear algebra. The planar case. Exponential of a matrix and general solution. Stability for linear systems.

Non-linear dynamical systems. Omega-limits. Stability theory. Gradient systems.

Hamiltonian systems. Poisson dynamics. Liouville theorem. Recurrence theorem and ergodic theory.

Topological conjugation. Hartman-Grobman theorem. Invariant manifolds.

Planar dynamical systems. Poincaré index. Poincaré-Bendixon theorem and applications.

Elements of chaos theory. Lorenz system, fractals and strange attractors.

Advanced topics.

Classroom lessons and exercises sessions. Homework exercises.

The course will be supported using Moodle and MSTeams.

Changes to the above modalities, if necessary in order to comply with the application of the security protocols due to the COVID19 emergency, will be communicated in the website of the Department and of the Course

an oral colloquium
covering all the arguments discussed in class.
Usually a simple exercise and a theory question.