difference equations introduction

7/29/2019 Difference Equations Introduction

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An Introduction toFirst-Order Linear

Difference Equations

With Constant Coefficients

Courtney Brown, Ph.D.

Emory University


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Definition of a Difference Equation

y(t+1) = ay(t) + b

y(t+1) = Some constant of proportionality

times y(t) plus some constant.

Some interesting cases are

y(t+1) = ay(t) exponential growth

y(t+1) = b a horizontal line

y(t+1) = y(t) + b a straight sloping line


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Relation to Interest

yt+1 = yt + ryt = yt(1+r)

r = rate of interest


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To Do List for Difference Equations

Plot the equation over time

Get an analytical solution for the difference

equation if it is available

Describe the models behavior

Determine the equilibrium values


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Equilibrium Values

At equilibrium, y(t+1) = y(t) = y*

y* = ay* + b

y* - ay* = b

y*(1-a) = b

y* = b/(1-a) = the equilibrium value


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Stability Criteria

y(t) will be stable if |a| < 1

y(t) will be unstable if |a| > 1

y(t) will oscillate if a < 0

y(t) will change monotonically if a > 0

y(t) will converge to a stable equilibrium value

if |a| < 1


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Analytic Solution: Part I

y1 = ay0 + b y2 = ay1 + b

y2 = a(ay0 + b) + b

y2 = a2

y0 + ab + by2 = a

2y0 + (a+1)b

y3 = ay2 + b

y3

= a(a2y0

+ ab + b) + b

y3 = a3y0 + a

2b + ab + b

y3 = a3y0 + b(a

2 + a + 1)


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Part II: Proof by Induction Proves That

yn = any0 + b(1 + a + a

2+ + an-1), then

To find the sum 1 + a + a2+ + an-1 write

S = 1 + a + a2+ + an-1

-aS = -(a + a2+ + an-1 + an)

S aS = 1 an

S(1 a) = 1 a

n

S = (1 an)/(1 a)


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Part III: Thus, the solution for yn is

yn = an

y0 + b(1 an

)/(1 a)or, more conveniently,

yn = b/(1-a) + [y0 b/(1-a)]an, for a1

We like to write it this way because b/(1-a) isthe equilibrium value, y*.

If a=1, then go back to

yn = an

y0 + b(1 + a + a2

+ + an-1

) to obtainyn = y0 + b(n), the solution when a=1


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Note: These solutions for the first-order linear

difference equation with constant coefficientsare used analytically to describe the time

paths with words.

Computing the time paths with a computer isdone using the original equation,

y(t+1) = ay(t) + b

using programming loops.


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a > 1, y0 > b/(1-a)

Increasing, monotonic, unbounded

y(t)

time

y0


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a > 1, y0 < b/(1-a)

Decreasing, monotonic, unbounded

y(t)

time

y0


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0 < a < 1, y0 < b/(1-a)

Increasing, monotonic, bounded, convergent

y(t)

time

y0

y*


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0 < a < 1, y0 > b/(1-a)

Decreasing, monotonic, bounded, convergent

y(t)

time

y0

y*


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-1 < a < 0

bounded, oscillatory, convergent

y(t)

time

y0

y*


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a < -1

unbounded, oscillatory, divergent

y(t)

time

y0

y*


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a = 1, b = 0

constant

y(t)

time

y* and y0


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a = 1, b > 0

constant increasing

y(t)

time

y0


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a = 1, b < 0

constant decreasing

y(t)

time

y0


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a = -1

finite, bounded oscillatory

y(t)

time

y0

y*


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Rules of Interpretation

|a| > 1 unbounded

[repelled from line b/(1-a)]

|a| < 1 bounded

[attracted or convergent to [b/(1-a)] a < 0 oscillatory

a > 0 monotonic

a = -1 bounded oscillatoryAll of this can be deduced from the solution

yn = b/(1-a) + [y0 b/(1-a)]an, for a1


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Special Cases

a = 1, b = 0 constant

a = 1, b > 0 constant increasing

a = 1, b < 0 constant decreasing


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Words Are Important

The words used to describe the behaviors of

the first-order linear difference equation with

constant coefficients are used in your text.

These behaviors can all be deduced from thealgebra of the analytical solution to the

equation.

difference equations introduction

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