1. Differential equations for the elliptic Jacoby functions:

2. Two bodies problem equations:

3. Van der Paul’s equations.

The phenomenon of the limited cycles was theoretically described by Poincare and Bendicsson and since then has been widely used in physics, chemistry and biology. The first problems of such type were investigated by Rayleigh, and later by Van der Paul in a number of articles on nonlinear oscillator: the equation

solutions get damped at and are unstable at . The point is to make changes (for example, with the help of the triode connected with the circuit) so that at small y values the result should be , and at big – values – we get .The following system can be used as the elementary expression describing the physical situation in a slightly idealized form:

4. "Brusselator".

Chemical kinetics laws generates differential equations which become nonlinear for two or more molecules reactions and possess interesting properties. Some of these equations have periodic solutions, they are important in the explanation of biological phenomena. Lefeveur and Nickolis model is so-called "brusselator" is in the focus of attention:

where: are the concentration of the substances participating in the reaction.

5. The nonlinear independent stationary model with the selected linear part which is described by the following equation corresponds to one of the test problems of the non-standard analysis:

The equations solutions of the given class are called “duck solutions” by mathematicians. “Duck solutions” allow to show visually calculating possibilities and specific features of the analytical-numerical method. In the closed form it has the following exact solutions (alive duck solutions):

  
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