# differential equations – Problem with the analytical study of transient processes 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 $$x(t)$$ 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.

Probably solution:

Write the entire system as a system of ode and use the tools of Mathematica to analyze nonlinear state-space or some other method.

With this question I contacted the Math Stack Exchange and the Engineering Stack Exchange. I received no substantive answer.

I would be glad to any advice and help from specialists, which path and method to choose. Numerical methods are not considered. I need to know the analytical description of the transient process to analyze it.

Remark: The system of differential equations for such a system turns out to be insoluble. Therefore, approximate methods can be used.