Ordinary differential equations gave rise to different branches of mathematics: they stimulated the development of calculus by Newton, the invention of topology by Poincaré, and motivated the study of Lie groups. On the other hand, the theory of differential equations is a field of application of all the disciplines mentioned above, with complex analysis and algebraic geometry added to this list. The present course follows the modern exposition given by Arnold in his 1985 textbook. This exposition is enriched by elements of the analytic theory of ODEs and very first steps in the general theory of dynamical systems.
Prerequisites
Real Analysis ( Foundations of Analysis, Analysis I, Analysis II: Differentiation and Integration from the list of our courses), Linear Algebra (Advanced Linear Algebra from the list of our courses).
Curriculum
- Processes described by ODEs.
- Examples from ecology, mechanics, and electricity.
- Elementary methods of integration and symmetries.
- General theory: existence and uniqueness of solutions in real and complex domains.
- Small parameter method.
- Linear theory.
- Phase flows of linear vector fields and exponentials of linear operators.
- Linear equations of higher order.
- Resonance.
- Small-amplitude oscillations.
- Stability of singular and fixed points.
- Limit cycles and their stability.
- Poincaré–Bendixson theorem.
- The Smale horseshoe.
- Linear equations in the complex domain.
- Monodromy group.
- Elements of Frobenius theory.
Textbooks
- V. I. Arnold, Ordinary Differential Equations.