Linear and non linear dynamical systems. Fixed points and stability in 2D. Bifurcation. Numerical solution of complex problems. Discrete maps. Chaos. Pattern formation in reaction diffusion models.
Theory of stochastic systems. Perturbative techniques (van Kampen, Kramers-Moyal). Applications.
Learning Objectives
Knowledge acquired:
Basic knowledge in non linear physics and stochastic systems. Study of specific applications in physics, biology and ecology.
Competence acquired : How to construct a model, using the deterministic and stochastic viewpoints. Solve the model with appropriate algorithm.
Skills acquired (at the end of the course): Solve simple models, both deterministic and stochastic, in one or several dimensions, with analytical means and/or computer simulations.
Teaching Methods
6 CFU
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 150
Contact hours for: Lectures (hours): 48
Further information
Office hours
Dipartimento di Energetica S. Marta, Thursday ore 15:00-18:00
Type of Assessment
Oral discussion
Course program
1d systems. Fixed points. Stability. Bifurcation theory. Saddle node bifurcation. Transcritical bifurcation. Pitchfork bifurcation. 2D systems. Fixed points and linear stability. Limit cycles. Hopf bifurcation. Maps. Logistic maps. Bifurcation diagram and chaos. Lorenz model. Diffusion equation. Reaction diffusion systems. Pattern formation. Turing instability and waves. Introduction to stochastic systems. The concept of master equation. From the master equation to the underlying deterministic model. The van Kampen expansion. Kramers Moyal expansion. The Gillespie algorithm. The role of finite size fluctuations. Applications.