


Politecnico di Torino  
Academic Year 2017/18  
01QNMNG Nonlinear systems for engineering 

Master of sciencelevel of the Bologna process in Mathematical Engineering  Torino 





Subject fundamentals
Beyond its intrinsic beauty and its geometric elegance, the development of a mathematical theory of dynamical systems is motivated by its fundamental role in analysis and control of nonlinear phenomena, complexity and deterministic chaos. The first part of the course addresses the study of Lyapunov stability, in its topological and differential version. This is one of the simplest, but basic concepts of dynamical systems theory. We focus essentially on the continuous time case. However, in order to introduce the Poincare' map approach to the stability problem of limit cycles, we treat also the discrete time case and Floquet theory. In addition, we present the method of center manifold and DulacPoincare' normal forms. The second part of the course deals in a more direct way with the asymptotic behavior of oscillatory nonlinear systems. We analyze systems which exhibit complex dynamics, strange attractors and chaotic attractors. Bifurcation theory is extended from the elementary case of equilibria to limit cycles and tori. One of the main goals of the course is the study and application of the descriptive function method and of the harmonic balance method. Some classical examples are presented in details: Chua's oscillator, Bernoulli's map and the logistic map. 
Expected learning outcomes
Ability of studying the stability of an equilibrium of a system of
ordinary differential equation in continuous time, using linearization, the second Lyapunov method, the center manifold theory. Ability of discussing simple types of bifurcations. Ability of determining and analyzing both from a qualitative and numerical point of view the oscillatory motions arising in dynamical systems of interest for application, especially for electrical and electronic engineering. 
Prerequisites / Assumed knowledge
Mathematical Analysis, Geometry, Fundamentals of electric circuits thought in a course of Physics

Contents
First part Linear systems. Dynamical systems in metric space (topological dynamics) Orbits of special type: equilibria, cycles, homoclinic, heteroclinic. The limit set and its properties. Poincare'Bendixson Theory. Stability and attraction of compact sets. Region of attraction. Lyapunov theorems and invariance principle. Linearization, topological equivalence. HartmanGrobman theorem. Center manifold theory and application. Classification of some elementary bifurcation. Normal form (DulacPoincare' theory). Discrete dynamical systems, Poincare' map, Floquet theory. Second part. Asymptotic behavior of nonlinear systems around a limit cycle. Limit cycles and Poincare' map. Oscillatory systems with complex behavior: the Chua oscillator. Strange attractors and chaotic attractors. Bifurcations of equilibria, cycles, tori. Method of the descriptive function. Lur'e systems. Method of the harmonic balance. Loeb criterion. Bernoulli map, logistic map. 
Assessment and grading criteria
The exam will consist in anora examination on the thoretical aspects of the course. The student must also solve and discuss some exercises and simulation on the tpics related to the second part of the course.

