Stability and bifurcation properties of the algorithms for keeping of differential equations solutions on the required level

TL;DR

This paper investigates the stability and bifurcation characteristics of algorithms designed to control differential equations solutions, analyzing idealized and real modifications, and identifying conditions for similar numerical behavior and complex dynamics.

Contribution

It introduces a base equation for analyzing idealized algorithms and examines the effects of real system differences on stability and bifurcation properties.

Findings

One algorithm shows high reliability.

Both algorithms can exhibit bifurcations and strange attractors.

Results are similar when control time step approaches zero.

Abstract

Algorithms of control of differential equations solutions are under investigation in the article. Idealized and real modifications of the algorithms are distinguished. An equation, which can be the base equation for investigation of the idealized algorithms properties, is constructed. The difference appearing for real systems and real algorithms is for separate investigation. This difference tends to zero under tending to zero of the time step of control. If the systems of equations satisfy or almost satisfy some properties for which the algorithms are intended, then the results are similar numerically as well. One of the algorithms demonstrates high reliability. Another one is of more complex properties. Bifurcations, periodic solutions and strange attractors are possible in both algorithms in addition to stable steady states.