# Changing the quality of the transient process in a nonlinear system (in Mathematica)

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 == 1/4, y == 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?

• What is "hpf1"? Nov 17, 2020 at 9:42
• I'm sorry, it's y(t)
– dtn
Nov 17, 2020 at 9:43

You might try to play with the parameters using Manipulate. Like this, for example,

extr = Exp[-(x[t] - 2)^2];
ω1 = 2 Pi 0.5;

Manipulate[
sys = NDSolve[{x'[t] ==
y[t] α1 Sin[ω1 t] +
D[α1 Sin[ω1 t], t], y'[t] + h1 y[t] == D[extr, t],
x == 1/4, y == 0}, x, {t, 0, 150}];
Plot[{Evaluate[x[t] /. sys]}, {t, 0, 150},
PlotRange -> All], {α1, 0, 1}, {h1, 0.5, 1.5}]
`

with the following effect: I do not know the limits within which you can vary the parameters, but you should know them. Then you can see what these parameters could do.

Have fun!

• This decision does not suit me. It is necessary to introduce something into the system (link or additional control signal), which provides a given transient process (in this case, exponential).
– dtn
Nov 17, 2020 at 15:45
• But that's up to you. You know your system, I do not. It is you who decides what element to add. Nov 17, 2020 at 15:55
• It consists of simple elements. ES controller is a scheme for measuring the gradient of a function known from the literature. I was hoping that someone with a more experienced and fresh outlook would suggest something. I'm stumped. The transition from one point to another must be preserved, but to provide an exponential transient (instead of what we have now) with a time constant T = 1, for simplicity.
– dtn
Nov 17, 2020 at 15:58
• process - it's our function $e^{-(x(t)-2)^2}$
– dtn
Nov 17, 2020 at 16:00
• Here we are not specialists in electronic systems. With such questions you should go to another forum. Nov 18, 2020 at 10:55