Inculcation: ODEs §

AKA “non-linear dynamics”, “smooth dynamical systems”, “systems” even (as in most of the content in “systems biology”).

interpret an ODE as a vector field §

This is a standard geometric interpretation.

Did you ever think an ordinary differential equations book will have this picture:

This is Perko’s book Differential Equations and Dynamical Systems, Third Edition (2006), a good read, if youre familiar with Analysis in ${\mathbb{R}}^n$. Otherwise any physics text on “non-linear dynamics” works, for example Steven Strogatz’s Non-linear dynamics lecture videos and book.

vector fields and ODE dictionary §

What we do is, write a differential equation like

$${\dot{x}}_i=f_i(x_1,x_2,\dots ,x_n)$$

for all $1\le i\le n$ so we have $n$ equations for $n$ variables $x_i$ and make it even more compact by

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$$

where $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector field on the open domain $\mathcal{U}$.

This gives us a geometric pov on ODEs in ${\mathbb{R}}^n$, and we have a

solving differential equations	analysis and geometry of vector fields
an equation $\dot{x}=\mathbf{f}(x)$	a vector field $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$
solutions of the equation $\begin{array}{rl}\dot{x}(t)& =\mathbf{f}(x(t))\\ x(0)& =x_0\end{array}$	integral curves of the vector field $t\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
how solutions depend on initial conditions $x_0\mapsto x(t)$	flows of the vector field $x_0\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
conserved quantities	integrals of the vector field
(linearly) decoupling the differential equation $\dot{z_i}=f_i(z_i)$	(linear) coordinate transformation such that $\mathbf{f}={\sum }_i{f}_i(z_i){\hat{z}}_i$

We may convert ordinary differential equation of any order to first order by taking enough independent variables and defining them to be higher derivatives.

much elementary introduction §

A short introduction with examples from models in population dynamics:

A full semester course:

Steven Strogatz's Non-linear dynamics

and his book Nonlinear dynamics and chaos : with applications to physics, biology, chemistry, and engineering

This playlist (and the channel) consists of shorter videos:

more formal lectures §

ICTP's Dynamical systems

This is more a “dynamical systems” course, but has ODEs too. Lecture notes: SLI.pdf (bris.ac.uk)

And as I said above, this book has a pre-requisite of some analysis, but still one must cover the topics in the contents:

Tip Differential Equations and Dynamical Systems, Third Edition (2006)

Arnold’s Ordinary Differential Equations is also a good resource.

vector fields on smooth manifolds §

Should be motivated from Perko, classical mechanics, etc. We, very naturally, want to write ODEs on surfaces, toruses or other “surfaces” of even higher dimensions AKA manifolds.

• Define the Lie bracket of vector fields, exponential of vector fields

$$\exp :\mathrm{Vec}(M)\to \mathrm{LDiff}(M)$$

- think of  the Lie algebra of vector fields as "the Lie algebra" of the "Lie group" local diffeomorphisms on the manifold $M$.
- The flows of two vector fields commute $\iff$ their Lie bracket is $0$

• Symplectic geometry helps study Hamiltonian systems in a general setting.

For more on smooth manifolds: inculcation-smooth-manifolds.

Use topology in your study of ODEs

• On compact manifolds, every vector field is complete, that is, the solutions of ODEs exist globally in time.
• On the sphere, any vector field is zero atleast at one point, has atleast one fixed point (Hairy ball theorem!).
• In general, use index theorems to know about vector fields on manifolds.
• and more!

In general, study smooth Lie group action on smooth manifolds, whose special case is an $\mathbb{R}$-action - that is a complete vector field.

complete resource on the study of ODEs §

This series by Arnold exists:

Encyclopaedia of Mathematical Sciences Dynamical Systems vol 1-10

1. Dynamical Systems I: ODEs and smooth dynamical systems
2. Dynamical Systems II: Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics
3. Dynamical Systems III: Mathematical Aspects of Classical and Celestial Mechanics
4. Dynamical Systems IV: Symplectic Geometry and its Applications
5. Dynamical Systems V: Bifurcation Theory and Catastrophe Theory
6. Dynamical Systems VI: Singularity Theory I
7. Dynamical Systems VII: Integrable Systems, Nonholonomic Dynamical Systems
8. Dynamical Systems VIII: Singularity Theory II - Applications
9. Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour
10. Dynamical Systems X: General Theory of Vortices

go beyond finite dimension §

Interpret heat equation, fluid flows, Schrodinger equation as infinite dimensional ODE and watch the consequences:

interpret an ODE as an operator on a function space §

interpret as a plane distribution using a contact structure §

generalize to dynamical systems §

Should be motivated from ODEs, or just plain playing with functions.