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Difference Equations and
Period Doubling
un+1 = ρun(1-un)
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
un+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 = ….
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 xaxis, which is represented between the line y = x and
the parabola
Repeat this process
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
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
As ρ increases, periodic solutions of 2,4,8,16,… appear
This is called bifurcation
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
Bifurcation
The END