Inculcation: ODEs

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

main article: inculcation

interpret any general ODE as a vector field and study it visually §

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 you’re 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 $\mathbf{f}={\sum }_i{f}_i(z_i){\hat{z}}_i$

This is a standard geometric interpretation.

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:

formal theory of ODEs §

Real analysis on $\mathbb{R}$ and ${\mathbb{R}}^n$ gives a foundation to general theory of ODEs.

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.
• Gerald Teschl’s book on ODEs and dynamical systems is amazing!

hard analysis perspective on ODEs §

generalizing ODEs to manifolds: flow of vector fields on smooth manifolds §

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

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

• The Lie bracket of vector fields, exponential of vector fields

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

is defined to be the solution of the ODE, the “flow” map, defined by the vector field. - We may think of the Lie algebra of vector fields as “the Lie algebra” of the “Lie group” local diffeomorphisms on the manifold $M$.

Question

Show that the flows of two vector fields commute $⟺$ their Lie bracket is $0$.

Thus a smooth dynamical system AKA an ODE generalized to manifolds is a pair

$$(M,X)$$

where $M$ is a manifold and $X$ is a (smooth) vector field on $M$.

motivating Hamiltonian vector fields §

Given a smooth $H:{\mathbb{R}}^2\to \mathbb{R}$, how can we produce a vector field that “preserves” this function? Well Hamilton’s equations on ${\mathbb{R}}^2$

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}\frac{\partial H}{\partial x}\\ \frac{\partial H}{\partial y}\end{array}\right]$$

produces a vector field perpendicular to the gradient of $H$ which does the job! But in doing so, this “Hamiltonian vector field” of $H$ produces nice geometrical properties. For example it has zero divergence, implying its flow preserves area (in ${\mathbb{R}}^2$ )!

Symplectic geometry helps study the geometry of such vector fields in a general setting on “Symplectic manifolds” which are smooth manifolds with the minimum structure needed to define Hamiltonian vector fields.

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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 2-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

applications §

method using characteristic curves for first-order PDEs §

Sturm-Liouville equations §

generalizations §

go beyond finite dimension §

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

interpret a linear ODE as an operator on a function space §

interpret as a plane distribution using a contact structure §

to dynamical systems §

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

motivating dynamical systems §

What is “dynamics”? A dynamical system, in general, is a monoid action on a set.

Ignoring the generality, let’s consider the definition of a discrete dynamical system: a function

$$f:X\to X$$

on a set is considered to be a discrete dynamical system (autonomous) where the iterations of the map

$$\mathsf{Id},f,f^2,f^3,\dots$$

are considered the time map for $t=0,1,2,3,\dots$. Thus orbit of a point $x\in X$ is defined to be

$$x,f(x),f(f(x)),f(f(f(x))),\dots$$

Simply put, just “iterations of a map” produces the dynamics.

Using this simple, extremely general definition, we reinterpret a lot of math in terms of “dynamics”:

• Linear algebra is study of iterations of one linear map on a vector space. The canonical forms essentially decomposes the dynamics into indecomposable pieces.
• Finite group theory studies group action on a (finte) set are by definition “dynamics”
• A module of a ring is just a special group action by the additive group of the ring which also plays well with the multiplicative structure. Representation theory in most cases studies such modules.
• There are various applications of theorems that are “dynamical”
  • proof of existence of ODEs using contraction mapping theorem
• Solutions of an ODE (with unique solutions) on $M$ produces a flow map

$${\varphi }^t:M\to M$$

on the phase space $M$ that map initial conditions $x$ to solution curves ${\varphi }^t(x)$ (this is the definition). Thus this is a “continuous-time dynamics”.

• A common proof of existence of solutions to ODEs uses a fixed point theorem, which is in turn a “dynamical” theorem!
• Machines have a natural monoid action.

Link to original