I urgently need advice and help.
I have a system of differential equations like this:
$\begin{cases} \frac{dx}{dt} == y[t] \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t)) \\ \frac{dy}{dt} + h \cdot y(t) == \frac{d}{dt}(e^{-(x[t] - 2)^2}) \end{cases}$
Parameters: $\alpha = 0.3, h = 1, \omega = 2 \pi 0.5, x(0)=1/4, y(0)=0$
It corresponds to the following structural scheme:
The code that simulates such a system is shown below: ClearAll["Global`*"]
pars = {\[Alpha]1 = 0.3, h1 = 1, \[Omega]1 = 2 Pi 0.5}
extr = Exp[-(x[t] - 2)^2]
sys =
NDSolve[{x'[t] ==
hpf1[t] \[Alpha]1 Sin[\[Omega]1 t] +
D[\[Alpha]1 Sin[\[Omega]1 t], t],
y'[t] + h1 y[t] == D[extr, t], x[0] == 1/4, y[0] == 0},
x, {t, 0, 500}]
The numerical solution is presented below:
Plot[{Evaluate[x[t] /. sys]}, {t, 0, 150}, PlotRange -> Full,
PlotPoints -> 50]
It can be seen that the transition process is a transition from the initial point to the final one with a certain character.
I need to change this character i.e. make the transition from one point to another exponentially. Like this:
What are the ways to solve this problem? What to do, add a regulator or manipulate the system of differential equations?
Please help me!
The rate at which you converge is proportional to the gradient of your process, in your case $e^{-(x(t)-2)^2}$. However, for values of $x(t)$ far away from $2$ the gradient very quickly becomes tiny. For example if you use $x(0)=0$ the behaviour that you are trying to remove gets even more pronounced. In this case you could take the logarithm of the output of process before feeding it into the washout filter. However, this only works for processes for which you know that their outputs are always positive.