Differential Equation
- direction field
  - Euler’s method
first order equation
- separable equation
below old class note
ordinary differential equation ODE
- order of differential equation
- solution
  - general solution
  - initial value problem
- direction field
- solving first-order differential equation
- substitution
  - Bernoulli equation
    - solution
  - Ricatti equation
    - solution
- existence and uniqueness theorem
one-dimensional (smooth) dynamical system
- autonomous equation
  - time-shift immunity
  - equilibrium
    - phase line
    - linearization (approximate linear dynamic)
- differential equation to dynamical system
- dynamical system to differential equation
discrete dynamical system
- map $p : R \to R$
- invertible $p$
- fixed point
  - stability
    - linearization
- Poincaré map
second-order linear equation
- second-order homogeneous linear equation with constant coefficient
  - solution
    - auxiliary equation
- second-order non-homogeneous linear equation
higher order differential equation
- existence and uniqueness
- system in matrix form
  - Wronskian
    - linear dependency and Wronskian
- homogeneous linear equation with constant coefficient
  - auxiliary equation
  - general solution
autonomous planar system of differential equation
- initial value problem
- integral curve
- velocity vector
- phase plane
  - phase portrait
  - equilibrium $(x_{e}, y_{e})$
- solution
non-autonomous planar system
- periodic time-shift immunity
- Poincaré map for non-autonomous periodic dynamical system
  - example: mass on spring with external forcing
- forced duffing equation
  - Poincaré map for forced duffing equation
chaos
- Poincaré map for autonomous system
  - Henon-Heiles system
linear system
- planar linear system
- homogeneity
- convert linear differential equation to linear system
- existence and uniqueness
- linearity
- linear dependency
- Wronskian
- linear system with constant coefficient
matrix exponential
- property
- diagonal matrix
- matrix exponential and linear system
  - exponential matrix as fundamental matrix
- generalized eigenvector
  - generalized eigenvector given characteristic polynomial
- compute $e^{A t}$
planar linear system
- characteristic polynomial
almost linear system
- planar system
  - equilibrium
    - stability
- linearization theorem (Hartman-Grobman theorem)
  - hyperbolic linear system
    - hyperbolic matrix
    - hyperbolic equilibrium
energy method
- mechanical system
Lyapunov’s Method
- isolated equilibrium $(x_{e}, y_{e})$
- positive/ negative definite/ semidefinite function $W (x, y)$ in $D$
- directional derivative of $V (x)$ along vector field $f (x)$ (derivative of $V$ along the flow of $f$ )
- Lyapunov’s stability theorem
  - Lyapunov function
- Lyapunov’s instability theorem
non-constant periodic solution
- limit cycle $L$
bifurcation in one-dimensional system
- implicit function theorem
- persistence of equilibrium
- fold bifurcation (saddle-node bifurcation)
  - condition
  - stability
    - proof
bifurcation in planar system
- persistence of equilibrium
- fold bifurcation (saddle-node bifurcation)

Differential Equation

an equation that contains an unknown function and one or more of its derivatives.

order
highest derivative
solution
function that satisfy the equation
initial condition
initial value problem

direction field

sketch lines as a grid with direction from derivative from the equation

Euler’s method

$y_{n + 1} = y_{n} + h y_{n}^{'}$

$h$ step size

first order equation

below old class note

ordinary differential equation ODE

general form $F (x, y, y^{'}, y^{''}, \dots, y^{(n)}) = 0$

order of differential equation

the order of the highest order derivative

solution

any function that satisfy the equation

initial value problem

initial condition $y (x_{0}) = a_{0}, y^{'} (x_{0}) = a_{1}, \dots at x_{0} \in I$

use the general solution and the initial condition

solving first-order differential equation

numerical approximation

Euler’s method $y (x_{0} + h) \approx y_{0} + h f (x_{0}, y_{0})$ example code

f[x0_, y0_] := x0 y0
step[{x0_, y0_}] := {x0 + 0.01, y0 + 0.01 f[x0, y0]}
{x100, y100} = Nest[step, {1, 1}, 100]

separable equation

use separation of variable $y^{'} = g (x) p (y)$

consider $p (y) = 0$
integrate both side $\int \frac{d y}{p ( y )} = \int g (x) d x$

explicit solution
implicit solution relationship between variable

$p (y) = y$

$y^{'} = y g (x) \Rightarrow y = A e^{G (x)}$

linear equation

$a_{1} (x) y^{'} + a_{0} (x) y = b (x)$ where $a_{1} (x), a_{0} (x), b (x)$ are continuous, $a_{1} (x) \neq = 0$

$y^{'} + P (x) y = Q (x)$ multiply both side to construct chain rule form $let μ^{'} (x) = μ (x) P (x) \Leftarrow μ (x) = A e^{\int P (x) d x} \Rightarrow y^{'} μ (x) + y μ^{'} (x) = (y μ (x))^{'} = μ (x) Q (x)$ where $μ (x)$ is the integration factor

substitution

Bernoulli equation

$\frac{d y}{d x} + P (x) y = Q (x) y^{n}$

solution

substitution $v = y^{1 - n}$ reduce the Bernoulli equation to linear equation $\frac{d u}{d x} + (1 - n) P u = (1 - n) Q$

Ricatti equation

$\frac{d y}{d x} = P (x) y^{2} + Q (x) y + R (x)$

solution

one solution $u (x)$
substitution $y = u + \frac{1}{v}$ reduce the Ricatti equation to linear equation $\frac{d v}{d x} + (2 P u + Q) v = - P$

existence and uniqueness theorem

$f (x, y)$ is continuous on $R = [x_{0} - a, x_{0} + a] \times [y_{0} - b, y_{0} + b]$ and satisfy Lipschitz condition $\Rightarrow \exists δ > 0, {y^{'} = f (x, y) y (x_{0}) = y_{0} has unique solution y (x) for x \in [x_{0} - δ, x_{0} + δ]$

Lipschitz condition

$\exists K > 0, ∣ f (x, y_{1}) - f (x, y_{2}) ∣ \leq K ∣ y_{1} - y_{2} ∣ \forall (x, y_{1}), (x, y_{2}) \in R \Leftarrow \frac{\partial f}{\partial y} is continuous (therefore bounded) on closed interval R$

metric space $(X, d)$

$\forall g, h, k \in X$

$d (g, h) \geq 0$
$d (g, h) = d (h, g)$
$d (g, h) \leq d (g, u) + d (u, h)$

$d$ denote distance
can be defined using $∥ g - h ∥ = x \in [x_{0} - δ, x_{0} + δ] max ∣ g (x) - h (x) ∣$

limit

sequence ${g_{n}}$ converge to limit $y$ in $X$ $\Leftrightarrow n \to \infty lim d (g_{n}, y) = 0$

the limit is unique

Cauchy sequence

$\forall ε > 0, \exists N, \forall n, m \geq N, d (g_{n}, g_{m}) < ε$

complete metric space

every Cauchy sequence converge to some point in $X$

fixed point

point stand for function for operator

initial value problem integral form

for $y^{'} = f (x, y)$ let operator $T$ : $T [g] (x) = y_{0} + \int_{x_{0}}^{x} f (s, g (s)) d s$ then $y (x) = T [y] (x) \Rightarrow T [y] = y$ $\Rightarrow y$ is a fixed point for operator $T$

Banach fixed point theorem

$X$ is complete metric space $T : X \to X$ is a contraction $\Rightarrow T$ has a unique fixed point in $X$

contraction $T : X \to X$ is a contraction if $\exists k < 1, \forall g, h \in X, d (T [g], T [h]) \leq K d (g, h)$

Picard iteration

sequence ${g_{n}}$ where $g_{n + 1} = T [g_{n}]$ with $g_{0} (x) = y_{0}$ $n \to \infty lim g_{n} (x) = y (x)$

one-dimensional (smooth) dynamical system

continuously differentiable function $ϕ (t, x) : R \times R \to R$ $1. ϕ (0, x_{0}) = x_{0} \forall x_{0} \in R 2. ϕ (t + s, x_{0}) = ϕ (t, ϕ (s, x_{0})) = ϕ (s, ϕ (t, x_{0})) \forall t, s, x_{0} \in R$

flow of dynamic system $ϕ$

autonomous equation

$x^{'} (t) = f (x)$ assume $f$ satisfy condition of existence and uniqueness theorem

time-shift immunity

if the solution of $x^{'} = f (x), x (0) = x_{0}$ is $x_{1} (t)$ then the solution of $x^{'} = f (x), x (t_{0}) = x_{0}$ is $x_{2} (t) = x_{1} (t - t_{0})$

equilibrium

$x_{e} \in R$ , $f (x_{e}) = 0$

asymptotically stable $x$ move towards $x_{e}$ from both side $\Leftarrow f^{'} (x_{e}) < 0$
unstable $x$ move away from $x_{e}$ from both side $\Leftarrow f^{'} (x_{e}) > 0$

phase line

draw the sign graph of $f (x)$

linearization (approximate linear dynamic)

let $λ = f^{'} (x_{e}), y = x - x_{e}$ $y^{'} = λ y \Rightarrow y = y_{0} e^{λ t}$

$λ > 0 \Rightarrow$ blow up and unstable
$λ < 0 \Rightarrow$ shrink down and asymptotically stable
$λ = 0 \Rightarrow$ cannot linearize, need higher order

differential equation to dynamical system

$ϕ (t, x_{0}) = x (t)$ where $x^{'} = f (x)$

dynamical system to differential equation

$f (x) = \frac{\partial ϕ}{\partial t} (0, x)$ and convert to initial value problem with autonomous equation

discrete dynamical system

map $p : R \to R$

$x_{k + 1} = p (x_{k}) \Rightarrow x_{k + t} = p^{k} (x_{t})$

invertible $p$

$x_{k - 1} = p^{- 1} (x_{k})$

fixed point

not change when apply $p$ $p (x_{0}) = x_{0}$

stability

linearization

$q (y) = p (x_{0} + y) - x_{0} \approx λ y λ = p^{'} (x_{0})$

dynamics $y_{n} = λ^{n} y_{0}$
unstable $\Leftarrow ∣ λ ∣ > 1$
asymptotically stable $\Leftarrow ∣ λ ∣ < 1$

Poincaré map

time-periodic non-autonomous system

$x^{'} = f (t, x)$ where $f (t + T, x) = f (t, x)$

periodic time-shift immunity

if the solution of $x^{'} = f (t, x), x (0) = x_{0}$ is $x_{1} (t)$ then the solution of $x^{'} = f (t, x), x (k T) = x_{0} (k \in Z)$ is $x_{2} (t) = x_{1} (t - k T)$

Poincaré map for the system

define function $P : R \to R$ : $P (x_{0}) = x (T) \Rightarrow x (k T) = P^{k} (x_{0})$ $\forall x_{0} \in R$ , let $x (t)$ be the solution to $x^{'} = f (t, x), x (0) = x_{0}$

periodic solution

fixed point of Poincaré map $P (x_{0}) = x_{0}$

stability

asymptotically stable $∣ λ ∣ = ∣ P^{'} (x_{0}) ∣ < 1$
unstable $∣ λ ∣ = ∣ P^{'} (x_{0}) ∣ > 1$

example code

P[h_,x0_] := NDSolveValue[
x'[t] == x[t](1-x[t]) - h x[t] Sin[t] && x[0] == x0,
x[2 Pi], {t, 0, 2 Pi}]

second-order linear equation

$a (x) y^{''} + b (x) y^{'} + c (x) y = f (x)$

second-order homogeneous linear equation with constant coefficient

$a y^{''} + b y^{'} + cy = 0$

solution

a linear combination of two solution $y_{1} (x), y_{2} (x)$ where $\frac{y _{1} ( x )}{y _{2} ( x )}$ is not constant (linearly independent) $y (x) = c_{1} y_{1} (x) + c_{2} y_{2} (x)$

auxiliary equation

under the hood: try $y = e^{r x}$ $a r^{2} + b r + c = 0$

$Δ > 0$ two distinct real root $r_{1}, r_{2}$ $y = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}$
$Δ = 0$ double real root $r_{0}$ two solution $y_{1} = e^{r_{0} x}, y_{2} = x e^{r_{0} x}$ $y = (c_{1} + c_{2} x) e^{r_{0} x}$
$Δ < 0$ two complex root $α \pm i β$ $y = e^{αx} (c_{1} cos (β x) + c_{2} sin (β x))$
- alternatively $y = A e^{αx} cos (β x - ϕ)$ where $A = c_{1}^{2} + c_{2}^{2}, ϕ = ar g (c_{1} + i c_{2})$
- alternatively: real-valued solution $y = a e^{(α + i β) x} + \overset{a}{ˉ} e^{(α - i β) x} (a \in C)$
  - complex-valued solution $y = a_{1} e^{(α + i β) x} + a_{2} e^{(α - i β) x} (a_{1}, a_{2} \in C)$

second-order non-homogeneous linear equation

$a y^{''} + b y^{'} + cy = f (x)$

solution

let $y_{h} (x) = y (x) - y_{p} (x)$ then $y_{h} (x)$ satisfy $a y^{''} + b y^{'} + cy = 0$
find one particular solution $y_{p} (x)$ therefore $y (x) = c_{1} y_{1} (x) + c_{2} y_{2} (x) + y_{p} (x)$

linear differential operator

given $g (x)$ , $L [g] (x)$ $L [g] = a g^{''} + b g^{'} + c g$

linear operator

apply operator $L$ on function $g (x)$ $L [g] (x) = a g (x)^{''} + b g (x)^{'} + c g (x)$ $L$ linear $\Leftarrow$

$L [g_{1} + g_{2}] = L [g_{1}] + L [g_{2}]$
$L [λ g] = λ L [g]$

find a particular solution

polynomial $f (x)$

use the degree of $f (x)$ as the degree of the particular solution

exponential $f (x)$

$f (x) = P (x) e^{λ x}$ where $P (x)$ is polynomial same as the polynomial method but multiply the same exponential

when $λ$ is one root of auxiliary equation, multiply the polynomial solution by $x$
when $λ$ is double root of auxiliary equation, multiply the polynomial solution by $x^{2}$

trigonometric $f (x)$

$f (x) = P_{1} (x) e^{λ x} cos (μx) + P_{2} (x) e^{λ x} sin (μx)$ same as polynomial but degree needs to be the maximum between $P_{1} (x), P_{2} (x)$ and need two polynomial

when $λ + i μ$ is one root of auxiliary equation, multiply the polynomial solution by $x$

combination of above case

split $f (x)$ to $f_{1} (x), \dots$ that match above case

higher order differential equation

$n$ -th order $a_{n} (x) y^{(n)} + \dots + a_{0} (x) y^{(0)} = f (x)$ where $a_{i} (x), f (x)$ are continuous on $I \subseteq R$

existence and uniqueness

above equation with $y (x_{0}) = y_{0}, \dots, y^{(n - 1)} (x_{0}) = y_{n - 1}$ has unique solution $\Leftarrow \forall x \in I, a_{i} (x), f (x)$ are continuous, $a_{n} (x) \neq = 0$

system in matrix form

$M [y_{1}, \dots, y_{n}] (x_{0}) c = y_{0} y_{1} (x_{0}) ⋮ y_{1}^{(n - 1)} (x_{0}) \dots ⋱ \dots y_{n} (x_{0}) ⋮ y_{n}^{(n - 1)} (x_{0}) c_{1} ⋮ c_{n} = y_{0} ⋮ y_{n - 1}$

Wronskian

determinant $W [y_{1}, \dots, y_{n}] = det M [y_{1}, \dots, y_{n}]$

linear dependency and Wronskian

$y_{1}, \dots, y_{n}$ linearly independent in $I$ $\Leftrightarrow \exists x_{0} \in I, W [y_{1}, \dots, y_{n}] (x_{0}) \neq = 0$ $\Leftrightarrow \forall x \in I, W [y_{1}, \dots, y_{n}] (x) \neq = 0$

linearly dependent function $y_{1}, \dots, y_{n}$ linearly dependent in $I$ $\Leftarrow \exists λ_{1}, \dots λ_{m} \in R, \forall x \in I, λ_{1} y_{1} (x) + \dots λ_{m} y_{m} (x) = 0$

homogeneous linear equation with constant coefficient

$a_{n} y^{(n)} + \dots + a_{0} y = 0$

auxiliary equation

$a_{n} r^{n} + \dots + a_{0} = 0$ $n$ complex root counting multiplicity

general solution

linear combination of $n$ solution time $x$ repeatedly if a solution has multiplicity $2$ or more

autonomous planar system of differential equation

$x^{'} = f (x, y) y^{'} = g (x, y)$ with respect to $t$

initial value problem

$x (t_{0}) = x_{0} y (t_{0}) = y_{0}$

autonomous $t$ can start wherever wanted, let $t_{0} = 0$

integral curve

parameterized curve of $(x (t), y (t))$

velocity vector

$\frac{d}{d t} (x (t), y (t)) = (f (x, y), g (x, y))$

phase plane

$x y$ -plane

phase space higher dimension

phase portrait

phase plane with several solution

equilibrium $(x_{e}, y_{e})$

$f (x_{e}, y_{e}) = g (x_{e}, y_{e}) = 0$

equilibrium solution $(x (t), y (t)) = (x_{e}, y_{e})$

solution

convert to differential equation with one variable
conserved quantity / integral of motion
- reduce to non-autonomous first order equation $\frac{d y}{d x} = \frac{y ^{'}}{x ^{'}} = \frac{g ( x , y )}{f ( x , y )}$

non-autonomous planar system

$x^{'} = f (x, y, t) y^{'} = g (x, y, t)$ where $f, g$ are periodic in $t$ with period $T$

periodic time-shift immunity

similar to periodic time-shift immunity in one dimension if $(x_{1} (t), y_{1} (t))$ is solution to $x^{'} = f (x, y, t), y^{'} = g (x, y, t) (x (0), y (0)) = (x_{0}, y_{0})$ then $(x_{2} (t), y_{2} (t)) = (x_{1} (t - k T), y_{1} (t - k T))$ is solution to $x^{'} = f (x, y, t), y^{'} = g (x, y, t) (x (k T), y (k T)) = (x_{0}, y_{0})$

Poincaré map for non-autonomous periodic dynamical system

$P : R^{2} \to R^{2}$ $P (x_{0}, y_{0}) = (x (T), y (T)) \Rightarrow P^{k} (x_{0}, y_{0}) = (x (k T), y (k T))$

also known as stroboscopic map

example: mass on spring with external forcing

forced duffing equation

$x^{'} = y, y^{'} = - b y + x - x^{3} + F sin (γ t)$ period $\frac{2 π}{γ}$

Poincaré map for forced duffing equation

P[b_,F_,γ_][{x0_, y0_}]:=
 NDSolveValue[
     x'[t]==y[t]&&y'[t]==-by[t]+x[t]-x[t]^3+F Sin[γ t]&&x[0]==x0&&y[0]==y0,
     {x[2Pi/γ],y[2Pi/γ]},
     {t,0,2Pi/γ}
     ]

chaos

sensitive dependence on initial condition

two initial condition close to each other deviate exponentially fast
the exact state of the system is fundamentally unpredictable though the system is deterministic
chaotic attractor of the two initial condition look identical

Poincaré map for autonomous system

Henon-Heiles system

$x_{1}^{'} y_{1}^{'} x_{2}^{'} y_{2}^{'} = y_{1} = - x_{1} - 2 x_{1} x_{2} = y_{2} = - x_{1}^{2} + x_{2}^{2} - x_{2}$

conserved quantity $H = \frac{1}{2} (y_{1}^{2} + y_{2}^{2}) + \frac{1}{2} (x_{1}^{2} + x_{2}^{2} + 2 x_{1}^{2} x_{2} - \frac{2}{3} x_{2}^{3})$
hyperplane $Σ \in R^{4}$ $Σ : x_{1} = 0$ on $Σ$ $H = \frac{1}{2} (y_{1}^{2} + y_{2}^{2}) + \frac{1}{2} (x_{2}^{2} - \frac{2}{3} x_{2}^{3})$ consider an initial condition on $Σ$ corresponding to value $h$ for $H$ , when the solution reach $Σ$ again $h$ still conserve $\Rightarrow$ it is enough to know $(x_{2}, y_{2})$ to know the rest $\Rightarrow$ define Poincaré map for $(x_{2}, y_{2})$

poincareMap[h_][{x0_,y0_}]:=Module[{stopTime},
    NDSolveValue[
        x1'[t]==y1[t]
        &&y1'[t]==-x1[t]-2x1[t] x2[t]
        &&x2'[t]==y2[t]
        &&y2'[t]==-x1[t]^2-x2[t]+2[t]^2
        &&x1[0]==0
        &&y1[0]=Sqrt[2h-y0^2-x0^2+2/3 x0^3]
        &&x2[0]=x0
        &&y2[0]=y0
        &&WhenEvent[x1[t]==0&&y1[t]>0,stopTime=t;"StopIntegration"],
        {x2[stopTime],y2[stopTime]},
        {t,0,Infinity}
        ]
    ]

linear system

$x^{'} = A (t) x + f (t)$

planar linear system

$x_{1}^{'} = a_{11} (t) x_{1} + a_{12} (t) x_{2} + f_{1} (t) x_{2}^{'} = a_{21} (t) x_{1} + a_{22} (t) x_{2} + f_{2} (t)$ where $a_{ij} (t), f_{i} (t), i, j \in {1, 2}$ continuous $x = [x_{1} x_{2}], f (t) = [f_{1} (t) f_{2} (t)], A (t) = [a_{11} (t) a_{21} (t) a_{12} (t) a_{22} (t)]$

homogeneity

$f (t) \equiv 0 \Rightarrow$ homogeneous
otherwise $\Rightarrow$ non-homogeneous

convert linear differential equation to linear system

linear differential equation $y^{(n)} + p_{n - 1} (t) y^{(n - 1)} + \dots + p_{0} (t) y = g (t)$ define $x_{1} = y x_{2} = y^{'} ⋮ x_{n} = y^{(n - 1)}$ get a linear system

existence and uniqueness

If $A (t)$ , $f (t)$ are continuous functions in an interval $I \subseteq R$ and $t_{0} \in I$ then for any initial vector $x_{0} \in R_{n}$ there exists a unique solution of $x^{'} = A (t) x + f (t)$ in $I$ that satisfies the initial condition $x (t_{0}) = x_{0}$ .

linearity

linear combination of solution are also solution

linear dependency

linearly dependent vector $\exists c_{1}, c_{2}, \dots, c_{m}, i = 1 \sum m (c_{i}^{2}) \neq = 0, \forall t \in I, c_{1} x_{1} (t) + \dots + c_{m} x_{m} (t) \equiv 0$

Wronskian

for $n$ solution $x_{1}, \dots, x_{n}$ $X (t) = [x_{1} \dots x_{n}] = x_{1, 1} ⋮ x_{1, n} \dots ⋱ \dots x_{n, 1} ⋮ x_{n, n} W [x_{1}, \dots, x_{n}] (t) = det X (t)$

linear dependency and Wronskian

solution $x_{1} (t), \dots, x_{n}$ are linearly dependent in $I$
$\Leftrightarrow \forall t \in I, W [x_{1}, \dots, x_{n}] (t) = 0$
$\Leftrightarrow \exists t_{0} \in I, W [x_{1}, \dots, x_{n}] (t_{0}) = 0$

fundamental solution

collection of $n$ linearly independent solution $x_{1}, \dots, x_{n}$

fundamental matrix

corresponding matrix $X (t) = [x_{1} (t) \dots x_{n} (t)]$

invertible
general solution for the linear system $y (t) = c_{1} x_{1} (t) + \dots + c_{n} x_{n} (t) = X (t) c$
$X^{'} (t) = A (t) X (t)$
solution for the initial value problem with $x (t_{0}) = x_{0}$ $y (t) = X (t) X (t_{0})^{- 1} x_{0}$
$\forall Y (t)$ that is another fundamental matrix $\exists C, Y (t) = X (t) C$
- $det C \neq = 0$
$\forall C, det C \neq = 0, X (t) C$ is a fundamental matrix
let $Y (t) = X (t) X (t_{0})^{- 1}$ , then $Y (t_{0}) = I$ $y (t) = Y (t) x_{0}$

linear system with constant coefficient

$x^{'} = A x$ where $A \in R^{n \times n}$ is constant

find a solution

$x (t) = e^{r t} u$ where $u$ satisfy $(A - r I) u = 0$

eigenvalue problem

$u$ is the solution to eigenvalue equation $u$ is non-zero eigenvector corresponding to $r \Leftarrow r$ is a root of characteristic polynomial $p (r) = det (A - r I)$

$r$ is a complex eigenvalue $\Rightarrow \overset{r}{ˉ}$ is eigenvalue
- $u$ is eigenvector corresponding to $r \Rightarrow \overset{ˉ}{u}$ is eigenvector corresponding to $\overset{r}{ˉ}$

linearly independent eigenvector and general solution

$A \in R^{n \times n}$ has linearly independent eigenvector $u_{1}, \dots, u_{n}$ corresponding to real eigenvalue $r_{1}, \dots, r_{n}$ $\Rightarrow$ general solution for $x^{'} = A x$ is $x (t) = c_{1} e^{r_{1} t} u_{1} + \dots + c_{n} e^{r_{n} t} u_{n}$

real distinct eigenvalue

$r_{1}, \dots, r_{m} (1 \leq m \leq n)$ are real distinct eigenvalue of $A \Rightarrow$ corresponding eigenvector $u_{1}, \dots, u_{n}$ are linearly independent $\Rightarrow$ a fundamental matrix is $[e^{r_{1} t} u_{1} \dots e^{r_{n} t} u_{n}]$

complex eigenvalue

eigenvalue $r = α + i β$ with eigenvector $u = a + i b$ eigenvalue $\overset{r}{ˉ}$ with eigenvector $\overset{ˉ}{u}$ $\Rightarrow$ solution $x_{1} (t) = e^{α t} (cos (βt) a - sin (βt) b) x_{2} (t) = e^{α t} (sin (βt) a + cos (βt) b)$

matrix exponential

$e^{A} = I + A + \frac{1}{2 !} A^{2} + \dots = k = 0 \sum \infty \frac{1}{k !} A^{k}$

property

$\forall A$ , series $e^{A}$ converge
$e^{O} = I$
$det e^{A} = e^{tr A} > 0$
- trace of $A$ $tr A = i = 1 \sum n a_{ii} = i = 1 \sum n λ_{i}$
$A B = B A \Rightarrow e^{A + B} = e^{A} e^{B} = e^{B} e^{A}$
$e^{A} e^{- A} = I$
$A u = r u \Rightarrow e^{A} u = e^{r} u$
$e^{U^{- 1} A U} = U^{- 1} e^{A} U$
$e^{I t} = e^{t} I$
$e^{A (t + s)} = e^{A t} e^{A s}$

diagonal matrix

diagonal matrix $R = a 0 ⋮ 0 b ⋮ \dots \dots ⋱ \Rightarrow e^{R} = e^{a} 0 ⋮ 0 e^{b} ⋮ \dots \dots ⋱$

matrix exponential and linear system

unique solution to the initial value problem $x^{'} = A x$ with $x (t_{0}) = x_{0}$ is $x (t) = e^{A (t - t_{0})} x_{0}$

derivative $\frac{d}{d t} (e^{A t}) = A e^{A t} = e^{A t} A$

exponential matrix as fundamental matrix

linear system $x^{'} = A x, x (0) = x_{0}$ has fundamental matrix $X (t)$ $\Rightarrow x (t) = X (t) X (0)^{- 1} x_{0} = e^{A t} x_{0} \Rightarrow e^{A t} = X (t) X (0)^{- 1}$

generalized eigenvector

a non-zero vector $u \in C^{n}$ $\exists r \in C,$ generalized eigenvector of $A$ associated with $r$ , $m \in N^{+}$ $(A - r I)^{m} u = 0$

$m = 1 \Rightarrow$ generalized eigenvector are also standard eigenvector
$r$ is eigenvalue of $A$ with eigenvector $max_{m} (A - r I)^{m} u \neq = 0$

generalized eigenvector given characteristic polynomial

$A \in R^{n \times n}$ has characteristic polynomial $p (r) = (r - r_{1})^{m_{1}} \dots (r - r_{k})^{m_{k}} (m_{1} + \dots + m_{k} = n)$ $\Rightarrow \forall j = 1, \dots, k, \exists$ linearly independent generalized eigenvector $u_{j, 1}, \dots, u_{j, m_{j}}$ $\forall l = 1, \dots, m_{j}, (A - r_{j} I)^{m_{j}} u_{j, l} = 0$

compute $e^{A t}$

find generalized eigenvector $u_{1}, \dots, u_{n}$
compute solution for $x (t) = e^{A t} u_{j} (j = 1, \dots, n)$
fundamental matrix $X (t) = [x_{1} (t) \dots x_{n} (t)]$
$e^{A t} = X (t) X (0)^{- 1}$

planar linear system

$x^{'} = A x$ where $A \in R^{2 \times 2}, x = [x_{1} x_{2}]$

characteristic polynomial

$p (r) Δ = det (A - r I) = r^{2} - r tr A + det A = r^{2} - T r + D = T^{2} - 4 D$ equilibrium type with different T-D

real distinct eigenvalue $Δ > 0$

$U = [u_{1} u_{2}], R = [r_{1} 0 0 r_{2}] \Rightarrow A U = [u_{1} u_{2}] [r_{1} 0 0 r_{2}] = U R \Rightarrow A = U R U^{- 1} \Rightarrow e^{A t} = U e^{Rt} U^{- 1}$

real eigenvalue $\frac{T \pm T ^{2} - 4 D}{2}$

transformation

define $z$ by $x = U z \Rightarrow z^{'} = R z \Rightarrow z_{1}^{'} = r_{1} z_{1}, z_{2}^{'} = r_{2} z_{2} \Rightarrow {z_{1} (t) = z_{1} (0) e^{r_{1} t} z_{2} (t) = z_{2} (0) e^{r_{2} t}$

case 1a. $0 < r_{1} < r_{2}$ (or $D > 0, T > 0$ ) all arrow point away from origin
- unstable node
case 1b. $r_{2} < r_{1} < 0$ (or $D > 0, T < 0$ ) all arrow point towards origin
- stable node
case 1c. $r_{1} < 0 < r_{2}$ (or $D < 0$ ) all arrow point towards origin on $z_{1}$ -axis, point away from origin on $z_{2}$ -axis
- saddle
case 1d. $r_{1} = 0 < r_{2}$ (or $D = 0, T > 0$ ) all arrow point away from $z_{1}$ -axis parallel to $z_{2}$ -axis
case 1e. $r_{2} < 0 = r_{1}$ (or $D = 0, T < 0$ ) all arrow point towards $z_{1}$ -axis parallel to $z_{2}$ -axis

complex conjugate eigenvalue $Δ < 0$

conjugate eigenvalue $α \pm i β$ with corresponding eigenvector $a \pm i b$ $U = [a b], R = [α - β β α] \Rightarrow A U = U R \Rightarrow A = U R U^{- 1} \Rightarrow e^{A t} = U e^{Rt} U^{- 1}$ where $e^{Rt} = e^{α t} [cos (βt) - sin (βt) sin (βt) cos (βt)]$

complex eigenvalue $\frac{T \pm i 4 D - T ^{2}}{2}$

polar coordinate transformation

$x = U z {r = z_{1}^{2} + z_{2}^{2} θ = arctan (\frac{z _{2}}{z _{1}}) \Rightarrow {θ^{'} = - β r^{'} = α r$

case 2a. $α > 0$ (or $T > 0$ ) arrow rotate around origin moving away
- unstable spiral
case 2b. $α < 0$ (or $T < 0$ ) arrow rotate around origin moving towards
- stable spiral
case 2c. $α = 0$ (or $T = 0$ ) solution are closed curve with period $T = \frac{2 π}{β}$
- center

rotation direction

rotation direction on $z_{1}, z_{2}$ -plane
- $β > 0$ $\Rightarrow θ^{'} < 0$ , clockwise
rotation direction on $x_{1}, x_{2}$ -plane
- $det U > 0$ same direction as on $z_{1}, z_{2}$ -plane

real repeated eigenvalue $Δ = 0$

generated eigenvalue of $A$ $U = [u_{1} u_{2}]$

real eigenvalue $\frac{r _{0} = T}{2}$
case c1. $A = r_{0} I$ (or both $u_{1}$ and $u_{2}$ are eigenvalue of $A$ ) $x = [c_{1} e^{r_{0} t} c_{2} e^{r_{0} t}]$ all arrow point away from origin
- unstable node
case c2. $A \neq = r_{0} I$ (or only $u_{1}$ is eigenvalue of $A$ ) $x = [(c_{1} + c_{2} t) e^{r_{0} t} c_{2} e^{r_{0} t}]$ all arrow point away from origin
- turn to the right with $c_{2} > 0$
- turn to the left with $c_{2} < 0$
- unstable

almost linear system

almost linear system $x^{'} = f (x)$ at $x_{e}$
- $x_{e}$ is an equilibrium of $x^{'} = f (x)$
- $y^{'} = g (y)$ is an almost linear system at the origin where $y = x - x_{e}, g (y) := f (x_{e} + y)$
almost linear system $x^{'} = f (x)$ at the origin
- $0$ is an equilibrium of $x^{'} = f (x)$
- $f$ is continuous around $0$
- $\frac{∥ f ( x ) - A x ∥}{∥ x ∥} \to 0 as ∥ x ∥ \to 0$ $\Leftarrow \frac{\partial f _{i}}{\partial x _{j}}$ are continuous near $(0, 0)$
- $A = D f (0)$
- $det A \neq = 0$
jacobian of $f (x) = [f_{1} (x_{1}, x_{2}) f_{2} (x_{1}, x_{2})]$ $D f (x) = \frac{\partial f _{1}}{\partial x _{1}} (x_{1}, x_{2}) \frac{\partial f _{2}}{\partial x _{1}} (x_{1}, x_{2}) \frac{\partial f _{1}}{\partial x _{2}} (x_{1}, x_{2}) \frac{\partial f _{2}}{\partial x _{2}} (x_{1}, x_{2})$

planar system

$x^{'} = f (x)$

equilibrium

point $x_{e}, f (x_{e}) = 0$

stability

stable $\forall ε > 0, \exists δ > 0, \forall x (0) \in B_{δ} (x_{e}), \forall t \geq 0, x (t) \in B_{ε} (x_{e})$ in English, given a bigger disk, one can always find a smaller disk, so that if you start from the smaller disk, you don’t go out of the bigger disk
asymptotically stable stable and $\exists η > 0, \forall x (t), x (0) \in B_{η} (x_{e}), ∥ x (t) - x_{e} ∥ \to 0 as t \to 0$
unstable not stable

open disk

$B_{δ} (x) = {y \in R^{2} : ∥ y - x ∥_{2} < δ}$

linearization theorem (Hartman-Grobman theorem)

transform the dynamics of system $x^{'} = f (x)$ to the dynamics of system $y^{'} = A y$ where $A = D f (x_{e})$

system $x^{'} = f (x)$ is almost linear at $x_{e}$
$x_{e}$ is hyperbolic
$\Rightarrow \exists$ coordinate transformation $y = ϕ (x)$ near $x_{e}$ , $ϕ (x_{e}) = 0$

hyperbolic linear system

hyperbolic matrix

all eigenvalue have non-zero real part

hyperbolic equilibrium

equilibrium $x_{e}$ of $x^{'} = f (x)$ $D f (x_{e})$ is hyperbolic matrix

energy method

mechanical system

$x^{'} = y y^{'} = f (x)$

potential $U (x)$

antiderivative of $- f (x)$

$U^{'} (x) = - f (x)$

energy function

conserved quantity $E (x, y) = \frac{1}{2} y^{2} + U (x)$

level set $E (x, y) = h$

fix $E (x, y)$ to $h$ $\Rightarrow y = \pm 2 (h - U (x))$

graph of $y$ is only defined where $U (x) \leq h$
the two parts above and below $x$ -axis mirror each other
intersection with $x$ -axis are where $U (x) = h$
- $U^{'} (x_{i}) \neq = 0 \Rightarrow$ curve is vertical at intersection
$\frac{d ∣ y ∣}{d ( h - U ( x ))} > 0$

potential plane

$x, U (x)$ -plane example using phase plane

equilibrium $(x_{e}, 0)$

$f (x_{e}) = 0, y = 0$

$U^{'} (x_{e}) = 0$

linearization

$f (x), f^{'} (x)$ are continuous near $x_{e}$ $U^{''} (x_{e}) \neq = 0$ $\Rightarrow$ system is almost linear at equilibrium $(x_{e}, 0)$

Jacobian corresponding to $F = [y f (x)]$ $D F (x, y) = [0 - U^{''} (x) 10]$
$U^{''} (x_{e}) > 0$ , minimum at $x_{e}$ linear system is a center linearization does not hold use Taylor series quadratic term $U (x) \approx U (x_{e}) + \frac{1}{2} U^{''} (x_{e}) (x - x_{e})^{2}$ level curve are approximate ellipse
$U^{''} (x_{e}) < 0$ , maximum at $x_{e}$ linear system is a saddle by linearization theorem, level curve also saddle

Lyapunov’s Method

isolated equilibrium $(x_{e}, y_{e})$

$\exists$ open disk $D = B_{δ} (x_{e}, y_{e})$ of radius $δ > 0$ centered at $(x_{e}, y_{e})$ $D$ does not contain other equilibrium

positive/ negative definite/ semidefinite function $W (x, y)$ in $D$

$D$ is open disk centered at $(0, 0)$ $W (x, y)$ is continuous in $D$ $W (0, 0) = 0$ $D_{*} = D </span> {(0, 0)}$

positive definite function $W (x, y)$ in $D$ $\forall (x, y) \in D_{*}, W (x, y) > 0$
positive semidefinite function $W (x, y)$ in $D$ $\forall (x, y) \in D, W (x, y) \geq 0$
negative definite function $W (x, y)$ in $D$ $\forall (x, y) \in D_{*}, W (x, y) < 0$
negative semidefinite function $W (x, y)$ in $D$ $\forall (x, y) \in D, W (x, y) \leq 0$
positive definite (/ semidefinite) function $W (x, y)$ in $D$ $\Leftrightarrow$ negative definite (/ semidefinite) function $- W (x, y)$ in $D$

directional derivative of $V (x)$ along vector field $f (x)$ (derivative of $V$ along the flow of $f$ )

planar system $x^{'} = f (x)$ real-valued function $V (x)$ $\dot{V} (x) = \frac{\partial V}{\partial x _{1}} (x) f_{1} (x) + \frac{\partial V}{\partial x _{2}} (x) f_{2} (x) = \frac{d}{d t} [V (x (t))] = f (x) \cdot \nabla V (x)$

Lyapunov’s stability theorem

planar system $x^{'} = f (x)$ $(0, 0)$ is an isolated equilibrium

$\exists$ positive definite function $V (x)$ in an open disk centered at $(0, 0)$ $\dot{V} (x)$ is negative definite in $D$ $\Rightarrow (0, 0)$ is asymptotically stable
- explanation $V$ increase going away from origin $V$ decrease along vector field $f$ going away from origin $\Rightarrow V$ will always go towards origin along vector field $f$
$\exists$ positive definite function $V (x)$ in an open disk centered at $(0, 0)$ $\dot{V} (x)$ is negative semidefinite in $D$ $\Rightarrow (0, 0)$ is stable

Lyapunov function

a function $V (x)$ that satisfy the condition of either part of Lyapunov’s stability theorem

Lyapunov’s instability theorem

planar system $x^{'} = f (x)$ $(0, 0)$ is an isolated equilibrium

$\exists$ continuous function $V (x)$ in an open disk centered at $(0, 0)$ $V (0, 0) = 0$ $\dot{V} (x)$ is positive definite in $D$ $\forall$ open disk $B$ centered at $(0, 0)$ , $\exists a \in B, V (a) > 0$ $\Rightarrow (0, 0)$ is unstable
- explanation $V$ increase along vector field $f$ going away from origin $V$ is higher somewhere than the origin $\Rightarrow V$ will go away from origin at some point
alternatively, instability for restrict system mean instability for whole system

non-constant periodic solution

limit cycle $L$

closed curve $L = {x (t) : 0 \leq t \leq T}$

planar system $x^{'} = f (x)$
$x (t)$ is a non-constant $T$ -periodic solution
- non-constant mean the periodic solution is not a single point
$\exists$ at least one other solution $x_{1} (t)$ , $t \to - \infty lim d (x_{1} (t), L) = 0$ or $t \to \infty lim d (x_{1} (t), L) = 0$
- distance $d (x, L) = min {∥ x - y ∥ : y \in L}$
- explanation this solution go to the closed curve as time roll back or go forward

example of limit cycle

$x^{'} = x - y - x f (r) y^{'} = x + y - y f (r)$

coordinate transformation $x = r cos θ y = r sin θ r r^{'} = x x^{'} + y y^{'} r^{2} θ^{'} = x y^{'} - x^{'} y$
$r \geq 0$
$θ^{'}$ is constant for the specific $f (r)$
dynamic in radial direction have equilibrium when $r^{'} = 0$
- stability can be determined using phase line $r^{'} (t)$
- each equilibrium correspond to a circular limit cycle

limit cycle enclose equilibrium

$L$ enclose at least one equilibrium

$L$ enclose $1$ equilibrium $\Rightarrow$ the equilibrium cannot be saddle

Bendixson’s negative criterion

planar system $x^{'} = f (x)$
simply connected domain $D \subseteq R^{2}$
$f_{1}, f_{2}$ have continuous partial derivative in $D$
$div f = \nabla \cdot f = \frac{\partial f _{1}}{\partial x _{1}} + \frac{\partial f _{2}}{\partial x _{2}}$ does not change sign in $D$
$\Rightarrow$ system have no non-constant periodic orbit in $D$

Poincaré-Bendixson Theorem

planar system $x^{'} = f (x)$
solution $x (t)$
isolated region $R$
- closed bounded
- $\forall t \geq 0, x (t) \in R$ solution stay in $R$ forever
$f_{1}, f_{2}$ have continuous partial derivative in $R$
no equilibrium in $R$
$\Rightarrow x (t)$ is either periodic or approach a limit cycle in $R$

bifurcation in one-dimensional system

$x^{'} = f (a, x)$ $a$ is changing parameter

implicit function theorem

smooth function $g (x, y)$
$g (x_{0}, y_{0}) = 0$
$g_{y} (x_{0}, y_{0}) \neq = 0$
$\Rightarrow \exists θ > 0$ , unique $h (x)$ defined on $(x_{0} - δ, x_{0} + δ)$ , $h (x_{0}) = 0, \forall x \in (x_{0} - δ, x_{0} + δ), g (x, h (x)) = 0$

persistence of equilibrium

$f (a, x)$ is a smooth function
$f (a_{0}, x_{0}) = 0$ , $x_{0}$ is an equilibrium when $a = a_{0}$
$f_{x} (a_{0}, x_{0}) \neq = 0$
$\Rightarrow \exists δ > 0$ , unique smooth function $g (a)$ defined on $(a_{0} - δ, a_{0} + δ)$ $g (a_{0}) = x_{0}, \forall a \in (a_{0} - δ, a_{0} + δ), f (a_{0}, g (a_{0})) = 0$
- $g (a)$ is equilibrium

fold bifurcation (saddle-node bifurcation)

$\exists$ smooth curve $a = h (x)$ near $x = x_{0}$ $f (h (x), x) = 0 h (x_{0}) = a_{0} h^{'} (x_{0}) = 0 h^{''} (x_{0}) = - \frac{f _{xx} ( a _{0} , x _{0} )}{f _{a} ( a _{0} , x _{0} )} \neq = 0$ in English: curve of equilibria (bifurcation diagram) near $(a_{0}, x_{0})$ look like parabola example curve of equilibria

condition

$f (a, x)$ is a smooth function
$f (a_{0}, x_{0}) = 0$ , $x_{0}$ is an equilibrium when $a = a_{0}$
$f_{x} (a_{0}, x_{0}) = 0$ equilibrium do not persist
$f_{xx} (a_{0}, x_{0}) \neq = 0$
$f_{a} (a_{0}, x_{0}) \neq = 0$ transversality condition
$\Rightarrow$ system undergo fold bifurcation at $a_{0}$

stability

$s (x) = f_{x} (h (x), x) s^{'} (x) = f_{x a} (h (x), x) h^{'} (x) + f_{xx} (h (x), x)$ check $s^{'} (x_{0})$ $s^{'} (x_{0}) = f_{xx} (a_{0}, x_{0})$

stability depend on sign of $f_{x}$ , regardless of $h (x)$
$s^{'} (x_{0}) > 0$ equilibrium $x < x_{0}$ is stable equilibrium $x > x_{0}$ is unstable
$s^{'} (x_{0}) < 0$ equilibrium $x < x_{0}$ is unstable equilibrium $x > x_{0}$ is stable

proof

$f (a_{0}, x_{0}) = 0$ , $f_{a} (a_{0}, x_{0}) \neq = 0$ , implicit function theorem $\Rightarrow \exists$ smooth function $h (x)$ near $x_{0}$ , $f (h (x), x) = 0$
define $g (x) = f (h (x), x)$ $\Rightarrow g^{'} (x) = f_{a} (h (x), x) h^{'} (x) + f_{x} (h (x), x) = 0$
Evaluate $f_{a} (h (x), x) h^{'} (x) + f_{x} (h (x), x) = 0$ at $x_{0}$ $0 = g^{'} (x_{0}) = f_{a} (h (x_{0}), x_{0}) h^{'} (x_{0}) + f_{x} (h (x_{0}), x_{0}) = f_{a} (a_{0}, x_{0}) h^{'} (x_{0}) + f_{x} (a_{0}, x_{0}) = f_{a} (a_{0}, x_{0}) h^{'} (x_{0})$ $f_{a} (a_{0}, x_{0}) \neq = 0 \Rightarrow h^{'} (x_{0}) = 0$ .
Then we take the derivative of the relation $g^{'} (x) = f_{a} (h (x), x) h^{'} (x) + f_{x} (h (x), x) = 0$ with respect to $x$ . We find $g^{''} (x) = f_{aa} (h (x), x) [h^{'} (x)]^{2} + f_{a x} (h (x), x) h^{'} (x) + f_{a} (h (x), x) h^{''} (x) + f_{x a} (h (x), x) h^{'} (x) + f_{xx} (h (x), x) = 0$ Evaluating at $x_{0}$ and using that $h^{'} (x_{0}) = 0$ we find $0 = g^{''} (x_{0}) = f_{a} (a_{0}, x_{0}) h^{''} (x_{0}) + f_{xx} (a_{0}, x_{0}) .$ Since $f_{a} (a_{0}, x_{0}) \neq = 0$ and $f_{xx} (a_{0}, x_{0}) \neq = 0$ we can solve for $h^{''} (x_{0})$ to find $h^{''} (x_{0}) = - \frac{f _{xx} ( a _{0} , x _{0} )}{f _{a} ( a _{0} , x _{0} )} \neq = 0$

bifurcation in planar system

$x^{'} = f (a, x, y) y^{'} = g (a, x, y)$ $a \in R^{2}$ is the parameter

persistence of equilibrium

equilibrium persist $\Leftarrow det [f_{x} (a_{0}, x_{0}, y_{0}) g_{x} (a_{0}, x_{0}, y_{0}) f_{y} (a_{0}, x_{0}, y_{0}) g_{y} (a_{0}, x_{0}, y_{0})] \neq = 0$ where $(x_{0}, y_{0})$ is equilibrium when $a = a_{0}$

fold bifurcation (saddle-node bifurcation)

supercritical bifurcation stable limit cycle + unstable spiral
subcritical bifurcation unstable limit cycle + stable spiral

Steven Hé (Sīchàng)