Elements of Applied Mathematics

Timetable

Module aim (aims)

The main goal of this course is to offer an introduction to classical methods of appliedmathematics. We will focus on the qualitative theory of systems of ordinary differential equations (ODEs). Following a review of standard results concerning existence and uniqueness of solutions and their continuous dependence on parameters, we will study linear system, stability theory, invariant manifolds, ending with a survey of periodic and homoclinic solutions. In the optimistic variant, the specific topics to be discussed will include.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Real analysis and basic differential equations, Linear algebra.

Syllabus

1. Elements of the ODE TheoryWeek 1: (a) existence of solutions (2.1, 2.2), (b) uniqueness of solutions (2.2)Week 2: (c) dependence on parameters (2.3)Week 3: (d) flows defined by differential equations (2.5)2. Linear Systems and StabilityWeek 4: (a) properties of linear systems (1.3, 1.4)Week 5: (b) solutions with homogeneous systems with constant coefficients (1.6, 1.7, 1.8)Week 6: (c) critical points and linearized stability (1.9, 2.6)Week 7: (d) Lyapunov functions and nonlinear stability (2.9)3. Hyperbolic TheoryWeek 8: (a) stable and unstable manifolds of dynamical systems (2.7, 2.10)Week 9: (b) linearization of hyperbolic systems (2.8)Week 10: (c) center manifold and nonlinear stability (2.11, 2.12)Week 11: (d) normal forms (2.3)4. Periodic and Homoclinic OrbitsWeek 12: (a) Floquet theory and stability of periodic solutions (3.3)Week 13: (b) Poincar'e maps (3.4, 3.5)Week 14: (c) Poincar'e–Bendixon theory (3.6, 3.7, 3.8)Week 15: (d) index theory and separatrix orbits (3.12)

Reading list

Primary Reference:1. L. Perko, Differential Equations and Dynamical Systems, Third Edition, Springer, (2000), ISBN 0–387–95116–4. Supplemental Reference:2. S. Lynch, Dynamical Systems with Applications Using MAPLE, Second Edition, Birkhäuser, (2010). ISBN 978–0–8176–4389–8.3. R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, (1982). ISBN0–12–497280–2.4. J. Cronin, Ordinary Differential Equations — Introduction and Qualitative Theory, Third Edition,CRC Press, (2008). ISBN 987–0–8247–2337–8.