This study presents nonlinear differential equations capable to generate continuous functions similar to pulse sequences. First are studied some basic properties of second-order differential equations with time-dependent coefficients generating bounded oscillating functions

As was presented in [

For a limited time interval it was shown [

Let us check the possibility of obtaining a more general positive-definite oscillating function as a result of a differential equation. We consider the function

By substituting in the previous equation

Next step consists in studying the possibilities of generating a function

which implies

Since

Substituting

By studying possible solutions for the equations presented at (i), (ii) and (iii), it results that (

A more general second-order differential equation generating a bounded positive-definite oscillating function

By equating the terms which do not depend on

Since

It is also possible to equate just the functions multiplying

It can be noticed that the function

It can be noticed that the previous differential equations can be easily implemented (using either analogue circuits if a high working frequency is necessary or digital circuits if more accurate results are required). In case of analogue circuits, mathematical operations as multiplying and/or dividing functions are available due to high performances of operational amplifiers. The errors between the required function and the function generated by analogue circuits cannot be avoided; however, the cumulative errors can be easily set to zero at the end of each working interval

The same aspect is valid also if digital circuits are used. A higher accuracy can be obtained, but numerical errors cannot be avoided and thus numerical values of

The mathematical expression for the second-order equation presented in previous paragraph should be used for generating functions similar to pulse sequences or time series. For this purpose, at first step it will be presented a differential equation able to generate a function with a constant slope for its amplitude, this means a function which can be written as

Since

For

The second term on right-hand side can be written as

The third term on right side can be written under the form

So the second derivative

Two important aspects should be emphasized.

The first term of the external command function

The second term of the external command function

This suggests that a set of differential equations

Such a set of differential equations can be also implemented using analogue or digital circuits for multiplying and/or dividing functions. The errors between the required set of functions and the functions generated by analogue or digital circuits become larger as

This study has presented nonlinear differential equations capable to generate continuous mathematical functions similar to pulse sequences. First were studied some basic properties of second-order differential equations