Ordinary Differential Equations

Notes from Hirsch and Smale's Differential Equations, Dynamical Systems and Linear Algebra, 1st ed. The theory of linear operators on finite-dimensional vector spaces in $ℝ^{n}$ $(TeX formula: \mathbb{R}^n)$ and some of $ℂ^{n}$ $(TeX formula: \mathbb{C}^n)$ , as a starting point for the analysis of nonlinear systems and ODEs on manifolds.

• First examples

$\frac{dx}{dt} = ax$ $(TeX formula: \frac{dx}{dt} = ax )$

also written as

$x'(t) = ax(t)$ (TeX formula:
x'(t) = ax(t)
)

is the simplest differential equation. a denotes a constant. $f (t) = K e^{a t}$ $(TeX formula: f(t) = Ke^{at})$ is a solution, since

$f'(t) = aKe^{at} = af(t)$ $(TeX formula: f'(t) = aKe^{at} = af(t) )$

there are no other solutions: let $u (t)$ (TeX formula: u(t)) be any solution and compute the derivative of $u (t) e^{- a t}$ $(TeX formula: u(t)e^{-at})$ :

$\begin{aligned} \dfrac{d}{dt} \left( u\left( t\right) e^{-at}\right) = u'\left( t\right) e^{-at} + u\left( t\right) \left( -ae^{-at}\right) \ =au\left( t\right) e^{-at}-au\left( t\right) e^{-at} = 0\end{aligned}$ $(TeX formula: \begin{aligned} \dfrac{d}{dt} \left( u\left( t\right) e^{-at}\right) = u'\left( t\right) e^{-at} + u\left( t\right) \left( -ae^{-at}\right) \\ =au\left( t\right) e^{-at}-au\left( t\right) e^{-at} = 0\end{aligned} )$

therefore $u (t) e^{- a t}$ $(TeX formula: u(t)e^{-at})$ is a constant K. this can only be true if $u (t) = K e^{a t}$ $(TeX formula: u(t) = Ke^{at})$

the constant K in the solution is determined if the value of u₀ at any single point t₀ is specified. if $x (t_{0}) = u_{0} ⟹ K e^{a t_{0}} = u_{0} = K$ $(TeX formula: x(t_0) = u_0 \implies Ke^{at_0} = u_0 = K)$ . for simplicity we often take t₀=0, without loss of generality. it is common to restate the differential equation in the form of a complete initial value problem:

$x' = ax, \; x(0)=K.$ $(TeX formula: x' = ax, \; x(0)=K. )$

constant a is considered a parameter. if a changes, so do the solutions. the sign is crucial here:

$a > 0 ⟹ {lim}_{t \to \infty} K e^{a t} = \pm \infty$ $(TeX formula: a>0 \implies \lim_{t \to ∞} Ke^{at} = ±∞)$ , depending on sign of K.
$a = 0 ⟹ K e^{a t} = K$ $(TeX formula: a=0 \implies Ke^{at} = K)$
$a < 0 ⟹ {lim}_{t \to \infty} K e^{a t} = 0$ $(TeX formula: a<0 \implies \lim_{t \to ∞} Ke^{at} = 0)$

(Image "Figure 1") — Figure 1 - the equation is stable in a sense if *a≠0*. if a is replaced by a sufficiently close b, the qualitative behavior of the solutions does not change. but if *a=0* the slightest change in it leads to a radical change. we say *a=0* is a *bifurcation point* in the one-parameter family of equations x'(t) = ax, a∈ℝ

consider a system of two uncoupled equations in two unknown functions:

$\begin{aligned} x_{1}'=a_{1}x_{1} \ x_{2}'=a_{2}x_{2} \end{aligned}$ $(TeX formula: \begin{aligned} x_{1}'=a_{1}x_{1} \\ x_{2}'=a_{2}x_{2} \end{aligned} )$

or in vector notation:

$𝐱'=A𝐱=\left( a_{1},a_{2}\right) \begin{pmatrix} x_{1} \ x_{2} \end{pmatrix}$ $(TeX formula: 𝐱'=A𝐱=\left( a_{1},a_{2}\right) \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} )$

with solutions

$\begin{aligned} x_{1}(t)=K_1 e^{a_1 t} \ x_{2}(t)=K_2 e^{a_2 t} \end{aligned}$ $(TeX formula: \begin{aligned} x_{1}(t)=K_1 e^{a_1 t} \\ x_{2}(t)=K_2 e^{a_2 t} \end{aligned} )$

from a geometric point of view, $𝐱 (t) = {(x_{1} (t), x_{2} (t))}^{T}$ $(TeX formula: 𝐱\left( t\right) =\left( x_{1}\left( t\right) ,x_{2}\left( t\right) \right)^T)$ specifies a curve $x (t) : ℝ \to ℝ^{2}$ $(TeX formula: x\left( t\right) :\mathbb{R} \rightarrow \mathbb{R} ^{2})$ , and the system of derivatives expresses a field of tangent/gradient vectors $\nabla 𝐱 (t) = 𝐱' (t) = {(x'_{1} (t), x'_{2} (t))}^{T}$ $(TeX formula: ∇𝐱(t) = 𝐱'\left( t\right) =\left( x'_{1}\left( t\right) ,x'_{2}\left( t\right) \right)^T)$ to that curve. for instance, this is the gradient vector for $𝐱' = (2, - 1 / 2) (\begin{matrix} x_{1} \\ x_{2} \end{matrix})$ $(TeX formula: 𝐱'=\left( 2, -1/2 \right) \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix})$ at $𝐱 = {(x_{1}, x_{2})}^{T} = {(1, 1)}^{T}$ (TeX formula: 𝐱 = (x_1, x_2)^T = (1, 1)^T)

more generally:

solving the system with initial conditions (u₁, u₂)^T at t=0 means finding in the plane a curve 𝐱(t) that satisfies the system and passes through point u. the trivial solution (x₁(t), x₂(t)) = (0,0) is also considered a curve. the family of all solution curves is called the phase portrait of the dynamical system.

some solutions to $𝐱' = A 𝐱 = (\begin{matrix} 2 & 0 \\ 0 & - 1 / 2 \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \end{matrix})$ $(TeX formula: 𝐱'=A𝐱= \begin{pmatrix} 2 & 0 \\ 0 & -1/2 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix})$ are sketched in the following phase portrait (which is simply the flow of following any of the previous vectors to their immediate successor point in the continuous vicinity around the starting point):

Ordinary Differential Equations

• First examples

• Application: Newton's equation and Kepler's law

• Linear systems: constant coefficients, real eigenvalues

• Linear systems: constant coefficients, complex eigenvalues

• Linear systems: exponentials of operators

• Linear systems: canonical forms of operators

• Contractions and generic properties of operators

• Fundamental theory

• Stability of equilibria

• Application: electrical circuits

• Poincaré-Bendixson theorem

• Application: ecology

• Periodic attractors

• Application: classical mechanics

• Nonautonomous equations and differentiability of flows

• Perturbation theory and structural stability