Discrete dynamical systems and their stabilitiy Flashcards
In notes 9 what did we do?
We looked at sequences and recurrences equations, with investment schemes,, and we were only interested in what happens to that investment over some finite time ie finite number of investment periods.
However are all sequences finite?
No there can be infinite terms which we will explore, we will see what happens when t ggets aribitary large.
What does the first order recurrence equation say again?
With y* ( the time independent solution), when it equals 0, for all values of t,what is the value of y*, providing a doesnt equal 1?
You can see with the equation when y* = y0 a^t is nothing so we are left with yt = y*, for all levels of t.
How many types of behaviour are there we can analyse of the the a^t?
9
Describe these 2 diagrams
describe what happens when y*
when y*-y0 ( already went over this)
when y*>0?
When a>1 we find that a^t is getting larger e.g. 2^4 and 4^4
so when y*
when y*=y0 then with the equation yt = y*, so we get a horizontal line, for all values of t.
When y*>y0, then y0 - y* is negative and a^t gets larger and larger, but become negative, so as time goes each value will be further away from y* ( we say its decreasing to minus infinity)
Describe this diagram, this is when a =1 so when y* doesnt exist.
Describe when b>0
describe when b=0
describe when b <0?
use the equation y1 = yo+ tb
if b>0 then y1 = y0 +tb ( you are mulitiplying by a bigger t, this means that as time moves on the line will become linear, so each time period we are adding a value)
if b =0 then we have a horizontal line because y1 = y0.
when B <0 ( you are multiplying by a negative, so means when time moves by, you are taking away a value, each time period.)
We know that when 1>a>0 we find a^t is getting smaller and samller so given that yt=y* + (y0-y*)a^t, the sign of (y0-y*), will affect a^t?, what are the 3 affects?
if y*<0 then yo-y* is positive and so decreasing a^t will make yt tend to y*, ( decrease to y*) ( e.g. 0.5 x 0.5 = 0.25 x 0.5 = 0.125, the value is getting smaller and when you times it by y0-y*, the value of yt gets smaller)
if y* = 0 then we have our time independent solution ( horizontal line)
if y*>y0 thene y0-y* is negative and so decreasign a^t will make y^t tend to y*( increasing to y*)
Explain what is happening here when a is beterrn
0>a>-1?
so yt = y* +(y0-y*)a^t
e.g. -0.5 x -0.5 = 0.25 x -0.5 = -0.125 ( when you mutliply the the same number but is a negative, the magnitude getting smaller)
so when the modulus of a^t will be decreasing but it will be osciallitng between positive and negative and getting closer to 0.
we say its oscilating decreasingly to y*
What happens when a = -1?
a^t will either be 1 or -1
so yt = y* + (yo-y*)a^t ( the y* will either be added by a positive or negative, hence the osiciation between below y* and above y*)
it never gets any closer or further away to y*
What happens whhen -1>a? e.g. a = -2?
when you time by -2 each time, lets say the value of a^t is osciallting between a negative and a positive but increasing in magnitude or modulus, this means, its getting further away from 0, it is oscillating.
For the 9 ways in which a sequence yt can behave as t gets larger, lets do an example?
verify any answers you can.
so use the diagrams to explain.
What is the most simple type of dynamical system?
cobweb model
What is the cobweb model ?
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lets say we have a demand function q^d(p) = k - 1p
and have a supply function q^s(p) = mp-n
what draw this on the diagram, and how do we work out equibrium price and quanitity?
What about if producers launch a product and misjudge the market and charge a price higher than equibrium, what do we want to do?
we want to see how the market evolves over time.
So lets say we have this long description how will the cobweb look? , dont try this.
If we go in and out, we can the remove the irrelvant bits
So as we can see with this cobweb, consumers reply to suppliers actions, so what can we do with our equations
we can first rewrite the equations in terms of qt we know that the supply function itells us the quanitiy supplied based on previous month price
and for the demand, it is related to the supply we have this month.
2) we replace some letters with pt-1 and pt
3) we equate to solve for pt, which gives us our first order recurrence equation.
What do we do here?
We find our time independent solution, then we find our general solution.
What do we notice about the y* or p* ( the time independent solution?
Its just the equibrium price.
Now we can ask ourselves how prices operate in the long run ( e.g. to the equibrium price or not. ?
we look at the value of a^t or in this case -m/1^t and see what happens when it is greater than 1, equal 1 and less than 1?
If m/l< 1 what does this mean?
If m/1 = 1 what does this mean?
If m/l >1 what does this mean?
Cobweb model example
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