03. Bifurcations¶

Contents¶

03.00. Introduction
03.01. Saddle-Node Bifurcation
03.02. Transcritical Bifurcation
- Example 03.02.01.
- Example 03.02.02.
03.03. Laser Threshold
- 03.03.01. Physical Background
- 03.03.02. Model
03.04. Pitchfork Bifurcation
03.05. Overdamped Bead on a Rotating Hoop
- 03.05.01. Analysis of the First-Order System
- 03.05.02. Dimensional Analysis and Scaling

03.00. Introduciton¶

birfurcation

03.01. Saddle-Node Bifurcation¶

$\dot{x} = r + x^2 \tag{1}$

03.01.01. Graphical Conventions¶

bifurcation diagram

03.01.01.01. Terminology¶

fold bifurcation (aka, tuning-point bifurcation, blue sky bifurcation)

[Abraham, Shaw (1988)][1988_Shaw_Abraham]

$\dot{x} = r - x^2 \tag{2}$

Example 03.01.01.¶

linear stability analysis of fixed point: $\dot{x} = f(x) = r - x^2$

Solution

$x^\ast = \pm \sqrt{r}$

Example 03.01.02.¶

$\dot{x} = r - x - e^{-x}$

Solution:

derivative:

$e^{-x} = r - x$

and

$\frac{d}{dx} e^{-x} = \frac{d}{dx} (r - x)$

$-e^{-x}=-1$ , $x=0$

$r_c=1$

03.01.02. Normal Forms¶

Taylor expansion

$\begin{align*} \dot{x} &= r - x - e^{-x} \\ &= r - x - \bigg[ 1 - x + \frac{x^2}{2!} + \cdots \bigg] \\ &= (r - 1) - \frac{x^2}{2} + \cdots \end{align*}$

bifurcation at $x=x^\ast$ , $r=r_c$ Taylor's expansion

$\begin{align*} \dot{x} &= f(x, r) \\ &= f(x^\ast, r_c) + (x - x^\ast) \frac{\partial f}{\partial x} \Bigg|_{(x^\ast, r_c)} \\ &+ (r - r_c) \frac{\partial f}{\partial r} \Bigg|_{(x^\ast, r_c)} \\ &+ \frac{1}{2} (x - x^\ast)^2 \frac{\partial^2 f}{\partial x^2} \Bigg|_{(x^\ast, r_c)} + \cdots \end{align*}$

$f(x^\ast, r_c) = 0$ (⇐ $x^\ast$ is fixed point), $\frac{\partial f}{\partial x}|_{(x^\ast, r_c)} = 0$

$\dot{x} = a(r - r_c) + b(x - x^\ast) + \cdots \tag{3}$

where $a = \frac{\partial f}{\partial x}|_{(x^\ast, r_c)}$ , $b = \frac{1}{2} \frac{\partial^2 f}{\partial x^2}|_{(x^\ast, r_c)}$

normal forms

Guckenheimier, Holmes (1983), Wiggins (1990)

03.02. Transcritical Bifurcation¶

transcritical bifurcation

$\dot{x} = rx - x^2 \tag{1}$

Example 03.02.01.¶

show the first-order system $\dot{x} = x(1 - x^2) - a(1 - e^{-hx})$

Solution:

$\begin{align*} 1 - e^{-hx} &= 1 - \Big[ 1 - bx + \frac{1}{2} b^2 x^2 + O(x^3) \Big] \\ &= bx - \frac{1}{2} b^2 x^2 + O(x^3) \end{align*}$

$\begin{align*} \dot{x} &= x - a (bx - \frac{1}{2} b^2 x^2) + O(x^3) \\ &= (1 - ab) x + (\frac{1}{2} ab^2) x^2 + O(x^3) \end{align*}$

$x^\ast \approx \frac{2 (ab - 1)}{ab^2}$

Example 03.02.02.¶

analyze

$\dot{x} = r\ln{x} + x - 1$

near $x = 1$

Solution:

$u := x-1$

$\begin{align*} \dot{u} &= \dot{x} \\ &= r\ln{(1 + u)} + u \\ &= r \Big[ u - \frac{1}{2} u^2 + O(u^3) \Big] + u \\ &\approx (r + 1) u - \frac{1}{2} ru^2 + O(u^3) \end{align*}$

$\dot{v} = (r + 1) v - (\frac{1}{2}ra) v^2 + O(v^3)$

$a=\frac{2}{r}$

$\dot{v} = (r + 1)v - v^2 + O(v^3)$

$R := r + 1$ and $X := v$

$\dot{X} \approx RX - X^2$

$X = v = \frac{u}{a} = \frac{1}{2} r(x-1)$

(Guckenheimer and Holmes (1983), Wiggins (1990), Manneville (1990))

03.03. Laser Threshold¶

Hanken (1983)

03.03.01. Physical Background¶

solid-state laser

03.03.02. Model¶

Milonmi and Eberly (1988)

Hanken (1983)

$n(t)$ : num of photons

$\begin{align*} \dot{n} &= \text{gain} - \text{loss} \\ &= GnN - kn \end{align*}$

simulated emission

$N(t)$ : num of excited atoms
$G (>0)$ : gain coef
$k (>0)$ : rate constant;
$\tau = \frac{1}{k}$ : lifetime of photon in laser

$N(t) = N_0 - \alpha n$

$\alpha$ : rate at which atoms drop back to graund state

Substitute: $\begin{align*} \dot{n} &= Gn(N_0 - \alpha n) - kn \\ &= (GN_0 - k) n - \alpha G n^2 \end{align*}$

03.04. Pitchfork Bifurcation¶

Pitchfork bifurcation

symmetry

03.04.01. Supercritical Pitchfork Bifurcation¶

$\dot{x} = rx - x^3 \tag{1}$

Example 03.04.01.¶

$\dot{x} = - x + \beta \tanh{x}$

Example 03.04.02.¶

Plot potential $V(x)$ for the system $\dot{x} = rx = x^3$

Solution

$f(x) = \frac{-dV}{dx}$

$V(x) = -\frac{1}{2}rx^2 + \frac{1}{4} x^4$

03.04.02. Subcritical Pitchfork Bifurcation¶

$\dot{x} = rx + x^3 \tag{2}$

$\dot{x} = rx + x^3 - x^5 \tag{3}$

03.04.03. Terminology¶

Supercritial pitchfork
- aka, Forward bifurcation
- soft/safe
Subcritical pitchfork
- aka, Backward bifurcation
- hard/dangerous

03.05. Overdamped Bead on a Rotating Hoop¶

Let

$\phi$ : angle ( $-\pi < \phi ≤ \pi$ )
$\rho := r \sin{\phi}$

$\begin{align*} mr\ddot{\phi} = - b\dot{\phi} - mg\sin \phi + mr\omega^2 \sin \phi \cos \phi \tag{3.5.1} \end{align*}$

03.05.01. Analysis of the First-Order System¶

$\begin{align*} b \dot{\phi} &= - mg \sin \phi + mr\omega^2 \sin \phi \cos \phi \\ &= mg \sin \phi \Big( \frac{r\omega^2}{g} \cos \phi - 1 \Big) \end{align*} \tag{3.5.2}$

where $\sin\phi = 0$ , $\phi^* = 0, \pi$

additional fixed points, if $\frac{r\omega^2}{g} > 1$

$\phi^* = ± \cos^{-1} \bigg( \frac{g}{r\omega^2} \bigg)$

introduce param $\gamma$ ,
$\gamma = \frac{r\omega^2}{g}$

solve $\cos{\phi^*} = \frac{1}{\gamma}$

03.05.02. Dimensional Analysis and Scaling¶

when valid to neglect the inertia term $mr\ddot{\phi}$ ?

dimensionless form (C.C. Lin and L.A. Segel (1988))

dimentionless time $\tau := \frac{t}{T}$

$\frac{d\phi}{d\tau}$ , $\frac{d^2\phi}{d\tau^2}$ $\approx O(1)$

Characteristic time scale $T$

$\dot{\phi} = \frac{d\phi}{dt} = \frac{d\phi}{d\tau} \frac{d\tau}{dt} = \frac{1}{T} \frac{d\phi}{d\tau}$

$\ddot{\phi} = \frac{1}{T^2} \frac{d^2\phi}{d\tau^2}$

substitute $T\tau$ for $t$

$\frac{mr}{T^2} \frac{d^2\phi}{d\tau^2} = - \frac{b}{T} \frac{d\phi}{d\tau} - mg\sin\phi + mr\omega^2 \sin\phi\cos\phi$

$1/mg$ $\Big( \frac{r}{gT^2} \Big) \frac{d^2\phi}{d\tau^2} = - \Big( \frac{b}{mgT} \Big) \frac{d\phi}{d\tau} - \sin\phi + \Big( \frac{r\omega^2}{g} \Big) \sin\phi\cos\phi \tag{3.5.3}$

(3)

$\frac{r\omega^2}{g}:= \gamma$

needed $\frac{b}{mgT} \approx O(1) \text{and} \frac{r}{gT^2} \ll 1$

$T := \frac{b}{mg}$

$r/gT^2 \ll 1$

$\frac{r}{g} \Big( \frac{mg}{b} \Big)^2 \ll 1 \tag{3.5.4}$

$b^2 \gg m^2gr$

dumping is strong; mass is small

$\epsilon := \frac{m^2gr}{b^2} \tag{3.5.5}$

(3) becomes $\epsilon \frac{d^2\phi}{d\tau^2} = - \frac{d\phi}{d\tau} - \sin\phi + \gamma \sin\phi\cos\phi \tag{3.5.6}$

$\epsilon \rightarrow 0$

$\begin{align*} \frac{d\phi}{d\tau} &= f(\phi) \\ f(\phi) &:= -\sin\phi + \gamma \sin\phi\cos\phi \\ &= \sin\phi (\gamma\cos\phi - 1) \end{align*} \tag{3.5.7}$