Study-unit Code
Francesca Di Patti
  • Francesca Di Patti
  • 42 ore - Francesca Di Patti
Course Regulation
Coorte 2023
Learning activities
Formazione modellistico-applicativa
Academic discipline
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
One dimensional flows: flows on a line, bifurcations, flows on the circle. Two-dimensional flows: linear systems, phase plane, limite cycles. Chaos: Lorenz equations, one-dimensional maps. Introduction to Turing patterns.
Reference texts
S. H. Strogatz, Nonlinear dynamics and chaos, Westview Press.

E. Ott, Chaos in Dynamical Systems, Cambridge University Press.
Educational objectives
This course is aimed at newcomers to nonlinear dynamics, chaos and pattern formation. The principal themes of dynamical systems are introduced, both through examples and by explaining and proving fundamental and accessible results. The main goal is to understand behavior of states in a system, given a rule for how the state evolves and to predict the emergence of unexpected dynamics. The course is characterized by a very intuitive approach that allows the student to master a methodology that can be used both in the field of applications and in the didactic field.
The essential prerequisite is single-variable calculus and linear algebra, including curve-sketching, Taylor series, differential equations and eigenvalues and eigenvectors. Basic knowledge of multivariable calculus like partial derivatives, Jacobian matrix and divergence theorems is also recommended.
Teaching methods
Lectures on all subjects of the course and related exercises. Some models will be implemented in Matlab. Supporting material and a detailed program will be posted on unistudium.
Other information
It is recommended to attend the lectures.
Learning verification modality
The oral examination consists on an interview about 2/3 arguments treated during the course. This allows to verify the ability of knowledge and understanding, the ability to apply the acquired skills, the ability to display and learn. Operating time up to 30/45 minutes.
For information about services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa.
Extended program
Flows on a line: a geometric way of thinking, fixed points and stability, population growth, linear stability analysis, existence and uniqueness, impossibility of oscillations, potentials.
Bifurcations: saddle-node bifurcation, transcritical bifurcation, laser threshold, Pitchfork bifurcation, overdamped bead on a rotating hoop, imperfect bifurcations and catastrophes, insect outbreak.
Two-dimensional flows: definition and examples, classification of linear systems.
Phase plane: phase portrait, existence and uniqueness, fixed points and linearization, rabbit versus sheep, conservative systems, reversible systems, pendulum.
Limit cycles: examples, ruling out closed orbits, Poincarè-Bendixson theorem, Lienard systems, relaxation oscillators, weakly nonlinear oscillators.
Averaged equations, Hopf bifurcatuion, Brusselator model.
Lorenz equations: simple properties, chaos and strange attractor, Lorenz map.
One-dimensional maps: fixed points and cobwebs, logistic map, Lyapunov exponent.
Fractals: countable and uncountable sets, Cantor set, dimension od Self-Similar Fractals, box dimension, pointwise and correlation dimensions.
Turing patterns.
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