This page is devoted to the doctoral course
Basic Principles of Dynamical Systems
that I will teach in Toulouse in 2012-2013. I’ll try to write lecture notes for everylecture, and then compile them into a small book.
The idea of this course is to give a survey of the most basic principles of dynamical systems, which can be understood by non-specialists. The subject of dynamical systems is very vast, because everything which can move is a dynamical system, that is to say that everything is a dynamical system if we want to view it so. Dynamical systems is not simple a mathematical subject, but rather a point of view.
In this course, we want to look at dynamical systems from many different points of view, and try to see what are the most important questions to ask about dynamical systems, what are the mathematical tools which can be used or which will be needed to model and study dynamical systems coming from the real world, and also what are the possible applications of the ideas of dynamical systems to “pure” mathematical subjects, e.g. algebraic geometry, etc. I will try my best to explain the most important principles of dynamical systems in the most intuitive way possible, with many concrete examples., and allusion to applications in many different fields, from geometry, physics, chemistry, biology to economics, politics, psychology, etc.
Officially, the course will last only 20 hours, say 2 hours a week during 10 week. But I think that 20 hours is not enough even to cover the most basic principles. So I will probabaly lecture more than 20 hours, as long as there are still people who want to listen. Each lecture will be devoted to one principle of dynamical systems, together with its implications, and relations with the other principles. I will avoid as much as possible complicated mathematical formulas in the lectures (but will nevertheless give references, write details with some rigorous proofs in the appendices to my lecture notes). Instead, I will try to give many visual illustrations.
What are the most fundamental principles of dynamical systems?
This is an open-ended question, up for debate. There are many principles, and it is not easy (or possible) to decide which of them is more important. Here I will try to write down a list of what I consider as some of the most important principles (accroding to my own tastes). This list itself is “dynamic”, i.e. it can change with the time:
Variational principle (Everyone is trying to minimize something)
Modeling (using variational principles to get dynamical equations from observations):
probllems&phenomena -> Observations -> Equations -> Solutions&properties -> Interpretations -> back to problems&phenomena
Least action, path of least resistance
Lagrangian and Hamiltonian formalizm
Constraints: holonomic and non-holonomic
Determinism (God doesn’t play dices)
Existence and uniqueness of solutions.
Methods from functional analysis for proving existence and uniqueness.
The notion of weak solutions, viscosity solutions.
The phenomenon of “something coming out from nothing” in the non-unique case.
Predictability and forcasting.
Does free will exist ?!
Cauchy, Kovalevskii, Leray, …
Conservation laws (Money moves from one pocket to another)
conservation of energy and momentum
invariant tensors (volume form, symplectic structure, etc.)
Lie, Noether, Poincaré, Marsden-Weinstein, …
Equilibrium (Russian mountains)
static and dynamical equilibrium
Nash equilibrium, equilibrium states in thermodynamics/statistical mechanics
general equilibria in economics
Stability of equilibrium
Quasi-Periodicity (The Sun rises every day)
intrinsic torus action
exponential time stability
Birfurcation (To B or not to B)
way to chaos
Torus actions in renormalization
Poincaré Dulac Birkhoff Bogoliubov …
Path integrals ?
Deterministic, stochastic perturbations
phase space decomposition: Conley etc
dynamical systems from point of view of information theory
second law of thermodynamics
topological vs measure-theoretic entropy
Bernoulli shift, Sinai-Ornstein theorem
Self-organization (Order out of chaos)
dissipative systems, Ramsey, starting anew
Different ways to view stochastic systems
Modeling of financial markets
Quantization (The world is non-commutative)
Quantum counterparts of classical phenomena
Duality in decompositions
evolutionary systems / dynamical systems
… (to be added)
(to be added)
* Marsden and Ratiu: Introduction to mechanics and symmetry.
* Katok and Hasselblatt, Introduction to the modern theory of dynamical systems
* Roussarie ?
* Sternberg ?
Varioational principles in dynamics
* Kuperschmidt ?
Systems as fields (hydrodynamics etc)
* Kozlov, Dynamical systems X: Genral theory of vortices
Complex (million-component) systems:
* Gros, Complex and adaptive dynamical systems: a primer arxiv:0807/4838
* Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE Transactions on automatic control, Vol 36 (1991), No 3. (Electronics and automatic control)
* Gelder & Port, It’s about time: an overwiew of dynamical approach to cognition