Difference Equations and Period Doubling

1. Difference Equations and Period Doubling un+1 = ρun(1-un)

2. 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

3. 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

4. Overview of a DifferenceEquationun+1 = ρun(1-un) • 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: an = an-1 + 7 a1 = 39 a2 = a1 + 7 = 39 + 7 = 46 a3 = a2 + 7 = 46 + 7 = 53 a4 = ….

5. un+1 = ρun(1-un) Step 1: Graphical Representation • Given our positive parameter ρ and our initial value un, we can graph the parabola y = ρx(1-x) and the line y = x • The sequence starts at the initial value un on the x-axis • The vertical line segment drawn upward to the parabola at un corresponds to the calculation of un+1 = ρun(1-un) • The value of un+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

7. un+1 = ρun(1-un) Step 2: Exchange of Stability • un+1 = ρun(1-un) has two equilibrium solutions: un = 0, stable for 0 ≤ ρ < 1 un = (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

9. un+1 = ρun(1-un) Step 3: Increasing Parameter (ρ) • For ρ > 3, neither of the equilibrium solutions are stable, and solutions of un+1 = ρun(1-un) 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 un oscillates back and forth along the parabola

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

11. un+1 = ρun(1-un) Step 4: Period Doubling/Bifurcation • 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

12. Bifurcation

13. The END