Difference Equations and Period Doubling u n+1 = ρu n (1-u n )

Why we use Difference Equations Overview of Difference Equations Step 1: Graphical Representation Step 2: Exchange of Stability Step 3: Increasing Parameter (ρ) Step 4: Period Doubling/Bifurcation

Why we use Difference Equations Differential Equations are good for modeling a continually changing population or value. Ex: Mouse Population, Falling Object Difference Equations are used when a population or value is incrementally changing. Ex: Salmon Population, Interest Compounded Monthly

Overview of a Difference Equation u n+1 = ρu n (1-u n ) This is called a recursive formula in which each term of the sequence is defined as a function of the preceding terms. Example of recursive formula: a n = a n a 1 = 39 a 2 = a = = 46 a 3 = a = = 53 a 4 = ….

u n+1 = ρu n (1-u n ) Step 1: Graphical Representation Given our positive parameter ρ and our initial value u n, we can graph the parabola y = ρx(1-x) and the line y = x The sequence starts at the initial value u n on the x-axis The vertical line segment drawn upward to the parabola at u n corresponds to the calculation of u n+1 = ρu n (1-u n ) The value of u n+1 is transferred from the y-axis to the x- axis, which is represented between the line y = x and the parabola Repeat this process

u n+1 = ρu n (1-u n ) Step 2: Exchange of Stability u n+1 = ρu n (1-u n ) has two equilibrium solutions: u n = 0, stable for 0 ≤ ρ < 1 u n = (ρ-1)/ρ, stable for 1 < ρ < 3 When ρ = 1, the two equilibrium solutions coincide at u = 0, this solution is asymptotically stable We call this an exchange of stability from one equilibrium solution to the other

u n+1 = ρu n (1-u n ) Step 3: Increasing Parameter (ρ) For ρ > 3, neither of the equilibrium solutions are stable, and solutions of u n+1 = ρu n (1-u n ) exhibit increasing complexity as ρ increases Since neither of the solutions are stable, we have what’s referred to as a period A period is a number of values in which u n oscillates back and forth along the parabola

As ρ increases, periodic solutions of 2,4,8,16,… appear This is called bifurcation

u n+1 = ρu n (1-u n ) Step 4: Period Doubling/Bifurcation ρ, we describe the situation as chaotic When we find solutions that possess some regularity, but have no discernible detailed pattern for most values of ρ, we describe the situation as chaotic One of the features of chaotic solutions is extreme sensitivity to the initial conditions

Bifurcation

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