Mathematical tools for modeling [SMATB210]

Learning outcomes

Dynamical systems are characterized by a state that evolves in time and play an essential role in many fields of application of mathematics, such as exact sciences (physics, chemistry, biology), computer science or economics and management.

At the end of this course, the students will be able to model simple dynamical phenomena, simulate them on a computer, and use the main mathematical tools to study their behavior.

Objectives

Provide an introduction to mathematical modeling and nonlinear system analysis.

Content

The course is pluridisciplinary and will address the student's interests in his/her favourite subject or in general. Theoretical notions will be illustrated with many concrete examples (biological mechanisms, electricity, neuroscience, etc.).

A first part of the course will introduce the different types of dynamical systems and basic notions (state space, trajectory, attractor). It will also cover the main system modeling techniques (Lagrange fomalism, linearization).

The second part of the course will provide an introduction to basic notions and mathematical tools for studying dynamical systems: phase portrait, cobweb diagram, equilibrium point, stability, periodic orbits and limit cycles.

Table of contents

• Different types of systems; state space; a first example: finite automata
• Modeling a 1D discrete-time system ; cobweb diagram ; stable and unstable fixed point ; periodic orbit
• Modeling a continuous-time system : Newton - Lagrange formalisms ; examples: compartment systems ; linearization technique
• Planar systems: trajectories and phase portraits ; behaviors of linear systems
• Nonlinear systems : existence and uniqueness of solutions ; local stability of the fixed point (Hartman-Grobman theorem); limit cycle; introduction to bifurcation theory
• Stochastic systems: master equation; random process; Langevin equation; Fokker-Planck equation