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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: I had two bugs in my simulations: 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

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 $y(t,f)$): $$\lim_{n \to \infty} \frac{1}{n}\cdot y\left(nt,\frac{f}{n}\right).$$ Update: Initial simulations suggested that this converged to $$y(t) = \frac{t}{1+\exp(z)}.$$ 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

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

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: I had two bugs in my simulations: 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

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 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

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