Bifurcation diagram with critical curves / Mark McClure

I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization. I'm happy to be part of the first Observable Ambassador's cohort.

Workspace

Public

Math

Edited

Aug 19, 2024

9 stars

Math

A Julia set on the Riemann sphere The Z-Curve Barnsley's fern A stochastic digraph IFS algorithm Self-affine tiles The Twindragon The Eisenstein fractions A self-affine tile with holes Self-affine tiles via polygon merge Golden rectangle fractals

Bifurcation diagram with critical curves

The tame twindragon Illustrations for the proof of Green's theorem Non-orientability of a Mobius strip Examples of parametric surfaces Penrose tiling The extended unit circle Penrose three coloring Newtons's method on the Riemann sphere Conic sections Divisor graphs The dance of Earth and Venus Iterating multiples of the sine function Borderline fractals Self-similar intersections Box-counting dimension examples Mandelbrot by dimension Inverse iteration for quadratic Julia sets Integer Apollonian Packings Illustrations of two-dimensonal heat flow The logistic bifurcation locus The eleven unfoldings of the cube A unimodal function with fractal level curves Green's theorem and polygonal area The geometry and numerics of first order ODEs The xxx^xxx-spindle Animated beats Rauzy Fractals Hilbert's coordinate functions Pluck Not Pi Drum strike The Koch snowflake Fractalized square A Taylor series about π/4\pi/4π/4 PlotX3D Hyperboloid A PlotX3D animation Modular arithmetic in 5th grade art Simple S-I-R Model The Poisson Kernel Poly-gaskets Classification of 2D linear systems via trace and determinant Julia sets and the Mandelbrot set Water waves Fourier Series Disks for a solid of revolution Orbit detection for the Mandelbrot set Tracing a path on a sphere Plot for mathematicians Functions of two variables Partial derivatives Dijkstra's algorithm on an RGG Gradient ascent Unfolding polyhedra Tangent plane to a level surface A strange discontinuity Examples of level surfaces McMullen carpets Hills and valleys The definition of ⇒Double and iterated integrals MST in an RGG Trees are bipartite Fractal typesetting d3.hierarchy and d3.tree K23 is Planar Polar Coordinates Parametric region generator Parametric Plot 2D Contour plots Greedy graph coloring Graph6 A few hundred interesting graphs The Kings Problem First order, autonomous systems of ODEs Runge-Kutta for systems of ODEs

scales = {

let w = width < 900 ? width : 900;

let h = 0.625 * w;

let rmin = 2.6;

let rmax = 4;

let pad = 30;

let x_scale = d3

.scaleLinear()

.domain([rmin, rmax])

.range([pad, w - pad]);

let y_scale = d3

.scaleLinear()

.domain([0, 1])

.range([h - pad, pad]);

let pts_to_path = d3

.line()

.x(d => x_scale(d[0]))

.y(d => y_scale(d[1]));

return {

w: w,

h: h,

rmin: rmin,

rmax: rmax,

pad: pad,

x_scale: x_scale,

y_scale: y_scale,

pts_to_path: pts_to_path

};

}

f = (r, x) => r * x * (1 - x)

function F(n, r) {

let x = 0.5;

for (let i = 0; i < n; i++) {

x = f(r, x);

}

return x;

}

function FF(n, r, eps) {

let x = 0.5 + eps;

for (let i = 0; i < n; i++) {

x = f(r, x);

}

return x;

}

function cobweb_pic(r, opts_in = {}) {

let f = x => r * x * (1 - x);

let { w = 300, h = 300 } = opts_in;

let opts = {

xdomain: [0, 1],

ydomain: [0, 1],

marginTop: (10 * w) / 300,

marginRight: (10 * w) / 300,

marginBottom: 0, //(10 * w) / 300,

marginLeft: 0, //(10 * w) / 300,

inset: 6,

insetTop: 6,

insetRight: 6,

insetBottom: 0,

insetLeft: 0,

width: w,

height: h,

strokeWidth: 2.5,

xticks: [0],

yticks: false

};

let x_scale = d3.scaleLinear(opts.xdomain, [

opts.marginLeft + opts.insetLeft,

opts.width - opts.marginRight - opts.insetRight

]);

let y_scale = d3

.scaleLinear(opts.ydomain, [

opts.height - opts.marginBottom - opts.insetBottom,

opts.marginTop + opts.insetTop

])

.nice();

let pts_to_path = d3

.line()

.x(function(d) {

return x_scale(d[0]);

})

.y(function(d) {

return y_scale(d[1]);

});

let plot = funplot([f, x => x], opts);

let x0 = 0.5;

let cobweb = make_cobweb(f, x0, 0, 1);

let svg = d3.select(plot);

svg

.append("path")

.attr("d", pts_to_path(cobweb))

.style("stroke", "black")

.style("stroke-width", 0.4)

.style("fill", "none")

.style("opacity", 0.4);

svg

.append("line")

.attr("x1", 0)

.attr("x2", w)

.attr("y1", y_scale(0))

.attr("y2", y_scale(0))

.attr("stroke", "black");

svg

.append("line")

.attr("y1", 0)

.attr("y2", h)

.attr("x1", x_scale(0))

.attr("x2", x_scale(0))

.attr("stroke", "black");

svg

.append("circle")

.attr("cx", x_scale(x0))

.attr("cy", y_scale(x0))

.attr("fill", "#3d3")

.attr("stroke", "black")

.attr("r", 3);

return plot;

}

function make_cobweb(f, x0, xmin, xmax) {

let x = x0;

let y = x;

let bail = 1500;

let cobweb = [[x, y]];

let i = 0;

while (i < bail && xmin <= y && y <= xmax) {

y = f(x);

cobweb.push([x, y]);

x = y;

cobweb.push([x, y]);

i++;

}

return cobweb;

}

max_period = super_attracting_parameters.slice(-1)[0].period

super_attracting_parameters = await FileAttachment(

"superAttractingLogisticParameters@2.csv"

).csv({ typed: true })

import { funplot } from "@mbostock/funplot"

import { Range } from '@observablehq/inputs'

tippy_style = html`

`tippy.js/themes/light.css`

)}">

tippy = require("tippy.js@6")

Purpose-built for displays of data

Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.

Learn more