System of second order non-linear difference equations
I was working on a problem in economics and came across a system of non-linear difference equations of the form:
$$x_{n}=x_{n-1}+(y_{n-2}-y_{n-1})left(1-frac{2}{b}y_{n-1}right)$$
$$y_n = a-sqrt{a^2+bx_{n}}$$
I tried substituting $y_n$ into $x_n$ and vice versa to reduce it to just one difference equation, but it ends up 2nd order, non-linear with no obvious closed form solution. I'm aware that chances are pretty big that there is none. If there is a way to solve the system I would greatly appreciate any tips.
However, I'm not interested in solving the system per say. It would be sufficient for me to understand what happens to
$$lim_{nto infty}{x_n} text{ and } lim_{nto infty}{y_n} text{ for different } a text{ and } b.$$
Would it be sufficient to examine fixed points if I wanted to observe the behavior of limits at infinity? (analogous to system of ODEs and a phase diagram?) If so, how should I go about it?
I can't rewrite it to vector form and use the usual textbook approach (i.e. drop the subscripts and solve the normal equation). I had trouble finding any good material on how to find and classify them in case like this.
Thank you very much.
sequences-and-series numerical-methods recurrence-relations
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
add a comment |
I was working on a problem in economics and came across a system of non-linear difference equations of the form:
$$x_{n}=x_{n-1}+(y_{n-2}-y_{n-1})left(1-frac{2}{b}y_{n-1}right)$$
$$y_n = a-sqrt{a^2+bx_{n}}$$
I tried substituting $y_n$ into $x_n$ and vice versa to reduce it to just one difference equation, but it ends up 2nd order, non-linear with no obvious closed form solution. I'm aware that chances are pretty big that there is none. If there is a way to solve the system I would greatly appreciate any tips.
However, I'm not interested in solving the system per say. It would be sufficient for me to understand what happens to
$$lim_{nto infty}{x_n} text{ and } lim_{nto infty}{y_n} text{ for different } a text{ and } b.$$
Would it be sufficient to examine fixed points if I wanted to observe the behavior of limits at infinity? (analogous to system of ODEs and a phase diagram?) If so, how should I go about it?
I can't rewrite it to vector form and use the usual textbook approach (i.e. drop the subscripts and solve the normal equation). I had trouble finding any good material on how to find and classify them in case like this.
Thank you very much.
sequences-and-series numerical-methods recurrence-relations
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
add a comment |
I was working on a problem in economics and came across a system of non-linear difference equations of the form:
$$x_{n}=x_{n-1}+(y_{n-2}-y_{n-1})left(1-frac{2}{b}y_{n-1}right)$$
$$y_n = a-sqrt{a^2+bx_{n}}$$
I tried substituting $y_n$ into $x_n$ and vice versa to reduce it to just one difference equation, but it ends up 2nd order, non-linear with no obvious closed form solution. I'm aware that chances are pretty big that there is none. If there is a way to solve the system I would greatly appreciate any tips.
However, I'm not interested in solving the system per say. It would be sufficient for me to understand what happens to
$$lim_{nto infty}{x_n} text{ and } lim_{nto infty}{y_n} text{ for different } a text{ and } b.$$
Would it be sufficient to examine fixed points if I wanted to observe the behavior of limits at infinity? (analogous to system of ODEs and a phase diagram?) If so, how should I go about it?
I can't rewrite it to vector form and use the usual textbook approach (i.e. drop the subscripts and solve the normal equation). I had trouble finding any good material on how to find and classify them in case like this.
Thank you very much.
sequences-and-series numerical-methods recurrence-relations
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
I was working on a problem in economics and came across a system of non-linear difference equations of the form:
$$x_{n}=x_{n-1}+(y_{n-2}-y_{n-1})left(1-frac{2}{b}y_{n-1}right)$$
$$y_n = a-sqrt{a^2+bx_{n}}$$
I tried substituting $y_n$ into $x_n$ and vice versa to reduce it to just one difference equation, but it ends up 2nd order, non-linear with no obvious closed form solution. I'm aware that chances are pretty big that there is none. If there is a way to solve the system I would greatly appreciate any tips.
However, I'm not interested in solving the system per say. It would be sufficient for me to understand what happens to
$$lim_{nto infty}{x_n} text{ and } lim_{nto infty}{y_n} text{ for different } a text{ and } b.$$
Would it be sufficient to examine fixed points if I wanted to observe the behavior of limits at infinity? (analogous to system of ODEs and a phase diagram?) If so, how should I go about it?
I can't rewrite it to vector form and use the usual textbook approach (i.e. drop the subscripts and solve the normal equation). I had trouble finding any good material on how to find and classify them in case like this.
Thank you very much.
sequences-and-series numerical-methods recurrence-relations
sequences-and-series numerical-methods recurrence-relations
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
edited Dec 27 '18 at 18:06
Leucippus
19.6k102871
19.6k102871
asked Dec 27 '18 at 17:59
zyt
63
asked Dec 27 '18 at 17:59
zyt
63
asked Dec 27 '18 at 17:59
zyt
63
63
add a comment |
add a comment |
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