RussellAndrewEdson

;;; Plots the bifurcation diagram, using the given starting point
;;; x-init (using 100 iterations, and plots the values for the
;;; last 30.)
(define (plot-bifurcation-diagram x-init)
  (define (steady-state-points)
    (let ((points '())
          (step-size 0.001)
          (iterations 100)
          (keep-last-points 30))
      (for ((r (in-range 0 4 step-size)))

RussellAndrewEdson

;;; Plots the cobweb diagram for the given number of steps, parameter r
;;; and starting point x-init.
(define (plot-cobweb-diagram steps r x-init)
  (define (cobweb)
    (let ((endpoints (list (vector x-init 0)))
          (x x-init)
          (y '()))
      (for ((i (in-range 0 steps)))
        (set! y (logistic-equation r x))
        (set! endpoints 

RussellAndrewEdson

;;; Returns a list of iterations of the logistic equation for a given 
;;; parameter r and starting point x-init.
(define (iterate-logistic-equation steps r x-init)
  (let ((xs (list x-init)))
    (for ((i (in-range 0 steps)))
      (set! xs (cons (logistic-equation r (car xs)) xs)))
    (map vector (stream->list (in-range 0 (+ 1 steps))) (reverse xs))))

;;; Plots the graph of x(n) for the given number of steps, parameter r
;;; and starting point x-init.

RussellAndrewEdson

;;; The logistic equation, x(n+1) = r*x(n)*(1-x(n))
(define (logistic-equation r x)
  (* r x (- 1 x)))

RussellAndrewEdson

;;;; Racket code to explore the chaotic behaviour of the logistic equation.
;;;; Date: 13/02/2016

#lang racket
(require plot)

;;; The logistic equation, x(n+1) = r*x(n)*(1-x(n))
(define (logistic-equation r x)
  (* r x (- 1 x)))

RussellAndrewEdson

;;;; Scheme code to plot the chaotic behaviour of the Lorenz attractor.
;;;; Date: 09/01/2016

#lang racket
(require plot)

;;; The parameters for the attractor (these values were used by Lorenz to
;;; demonstrate the chaotic behaviour.
(define rho 28)
(define sigma 10)

RussellAndrewEdson

;;; Code for the Simulated Annealing Cryptanalysis example.
;;;
;;; Code Author: Russell Edson
;;; Date: 28/05/2015

;; The letters for the key alphabet (ie. A-Z.)
(defconstant +key-alphabet+ "ABCDEFGHIJKLMNOPQRSTUVWXYZ")

;; Letter frequencies (from "Cryptological Mathematics", R.E. Lewand 2000)
(defconstant +english-letter-frequencies+

RussellAndrewEdson

(defn g
  "The recursive G function from Godel, Escher, Bach: takes
  in an integer n and returns G(n) = n - G(G(n-1)), with G(0) = 0."
  [n]
  (if (= n 0)
    0
    (- n (g (g (- n 1))))))

RussellAndrewEdson

(declare f m)

(defn f
  "The F function from Godel, Escher, Bach: takes in
  an integer n and returns F(n) = n - M(F(n-1)), with F(0) = 1."
  [n]
  (if (= n 0)
    1
    (- n (m (f (- n 1))))))

RussellAndrewEdson

(defn h
  "The recursive H function from Godel, Escher, Bach: takes
  in an integer n and returns H(n) = n - H(H(H(n-1))), with H(0) = 0."
  [n]
  (if (= n 0)
    0
    (- n (h (h (h (- n 1)))))))