# Research of transients in nonlinear systems with linear dynamic links

I am having difficulties of this nature. There is a nonlinear system of the following type: I need to analyze analytically the transient process in such a system. The analytical study of the transient process is understood as the solution of the differential equation for such system and its analysis, i.e.:

1. Estimation of the time of the transient process by analyzing the properties of the solution components.

2. Assessment of the presence / absence of oscillatory components.

3. Estimation of the amplitude of the oscillatory components.

The situation is complicated by the following circumstances:

1. The presence of linear dynamic links in the system, which implies the use of LTI techniques. At the same time, the presence of a quadratic nonlinear link makes this impossible.

2. The presence of a nonlinear quadratic link forces us to consider the option of drawing up the corresponding differential equation in the system. At the same time, the presence of linear dynamic links in the system determines the presence of convolution operators in the differential equation, which turns the differential equation into an integro-differential equation with a complex structure, the methods for solving which in general form do not exist.

Problem:

It is necessary to choose the path of analytical calculation of transient processes in a nonlinear system with linear dynamic links.

I ask for help and advice from specialists, which path and method to choose. Numerical methods are not considered.

• Why are you ruling out numerical methods? Some numerical simulations (made with reliable software, or course!) will show you what is happening, and what features of the response your analytical study needs to investigate. – alephzero Sep 10 at 18:23
• I don’t know how to properly structure the numerical studies. As for the rest, the analytical solution would allow, upon substitution of the selected parameter values, to obtain an answer to the question of the quality of the transient process, without performing numerical simulation, if the computer is not at hand. – dtn Sep 10 at 18:32
• In real-world use of control engineering, computers are always at hand. Even your cell phone's processor is MUCH more powerful than a 1980s supercomputer like the Cray 1. – alephzero Sep 12 at 16:51
• All this fades when you work with a large number of such objects, and there is no computer at hand. We need a good theory! – dtn Sep 12 at 16:52