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edited Jul 23 2010 at 5:26 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest Update: Initial simulations suggested that this converges converged to $$y(t) = \frac{t}{1+\exp(z)}.$$ Update: I had two bugs in my simulations: But that was wrong. In conclusion, the differential equation is indeed a good approximation to the above limit. Sorry about the confusion.
How can I rigorously show this? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 23 2010 at 5:16 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $$y(t) = \frac{t}{1+\exp(z)}.$$ Update: the above I had two bugs in my simulationswere incorrect and : In conclusion, the differential equation is not always indeed a good approximation to the above limit. Sorry about the confusion.
How can I rigorously find the solution to show thislimit? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 22 2010 at 14:05 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $$y(t) = \frac{t}{1+\exp(z)}.$$ Update: the above simulations were incorrect and the differential equation is not always a good approximation to the above limit.
How can I rigorously find the solution to this limit? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 22 2010 at 13:58 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $$y(t) = \frac{t}{1+\exp(z)}.$$ (Update: only for small $t$). Update: the above were incorrect and the differential equation is not always a good approximation to the above limit.
How can I rigorously find the solution to this limit? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 22 2010 at 13:10 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $$y(t) = \frac{t}{1+\exp(z)}.$$ (Update: only for small $t$). How can I rigorously derive find the solution to this limit? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 21 2010 at 21:55 |
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edited Jul 21 2010 at 19:18 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution is to find involves finding (making the dependence of $y$ on $f$ explicit by writing $y(t)$ as $y(t,f)$):
$$\lim_{n \to \infty} \frac{f}{n}\cdot frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $$y(t) = \frac{ft}{1+\exp(z)}.$$ frac{t}{1+\exp(z)}.$$ How can I rigorously derive this? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 21 2010 at 5:14 |
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edited Jul 21 2010 at 5:13 |
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,y(0)=0$\quad y(0)=0,$$
where z $z$ is a real number f $f$ and u $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution is to find (making the dependence of y $y$ on f $f$ explicit by writing $y(t)$ as $y(t,f)$):
$\lim_{n $\lim_{n \to \infty} \frac{f}{n} y(nt,\frac{f}{n})$frac{f}{n}\cdot y\left(nt,\frac{f}{n}\right).$$
Simulations suggest that this converges to $y(t) $y(t) = \frac{ft}{1+exp(z)}$. frac{ft}{1+\exp(z)}.$$ How can I rigorously derive this? Also appreciated are references to texts that discuss similar problems.
Thanks
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edited Jul 21 2010 at 5:07 |
I have the following discrete time dynamical system
$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} $, y(0)=0$
where z is a real number f and u are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution is to find (making the dependence of y on f explicit by writing $y(t)$ as $y(t,f)$):
$\lim_{n \to \infty} \frac{f}{n} y(nt,\frac{f}{n})$
Simulations suggest that this converges to $y(t) = \frac{ft}{1+exp(z)}$. How can I rigorously derive this? Also appreciated are references to texts that discuss similar problems.
Thanks
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asked Jul 21 2010 at 4:59 |
Limit of a discrete time dynamical system
I have the following discrete time dynamical system
$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} $
where z is a real number f and u are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution is to find (making the dependence of y on f explicit by writing $y(t)$ as $y(t,f)$):
$\lim_{n \to \infty} \frac{f}{n} y(nt,\frac{f}{n})$
Simulations suggest that this converges to $y(t) = \frac{ft}{1+exp(z)}$. How can I rigorously derive this? Also appreciated are references to texts that discuss similar problems.
Thanks
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