# Dynamical Systems

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.Â  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 importantÂ  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 oneÂ  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

Euler-Lagrange equation

Geodesic flows

Maupertuis principle

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.)

commuting flows

symmetry groups

reduction

Lie, Noether, PoincarÃ©, Marsden-Weinstein, …

Equilibrium (Russian mountains)

static and dynamical equilibrium

Nash equilibrium, equilibrium states in thermodynamics/statistical mechanics

general equilibria in economics

Stability of equilibrium

Lyapunov …

Quasi-Periodicity (The Sun rises every day)

intrinsic torus action

quasi-periodic motion

integrable system

KAM theory

exponential time stability

Liouville

Resonances

concentration

synchronization

Birfurcation (To B or not to B)

period doubling

way to chaos

hyperbolicity

(Re)normalization

Normal forms

Multi-scale normalizations

Torus actions in renormalization

PoincarÃ© Dulac Birkhoff Bogoliubov …

Path integrals ?

Perturbations

Deterministic, stochastic perturbations

Decomposition

spectral decomposition

phase space decomposition: Conley etc

five-state decomposition

time scales

Entropy

dynamical systems from point of view of information theory

second law of thermodynamics

useless energy

forgetting

information rate

topological vs measure-theoretic entropy

Bernoulli shift, Sinai-Ornstein theorem

Ergodicity

Boltzmann

Invariant measures

Birkhoff-Khinchin

….

Stability

Compensation rarity/stability

Structural stability

Anosov flows

Self-organization (Order out of chaos)

dissipative systems, Ramsey, starting anew

Stochastic systems

Stochastic processes

Different ways to view stochastic systems

Modeling of financial markets

Quantization (The world is non-commutative)

Geometric quantization

Semi-classical quantization

Deformation quantization

Quantum counterparts of classical phenomena

Duality

Particle/field

Duality in decompositions

evolutionary systems / dynamical systems

… (to be added)

References

Elementray introductions:

* 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

???

Ergodic theory

* Sinai

* Mane

Complex (million-component) systems:

* Gros, Complex and adaptive dynamical systems: a primer arxiv:0807/4838

Applications

* 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, Itâ€™s about time: an overwiew of dynamical approach to cognition