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This article presents an analysis of the chaotic dynamics presented by the Lorenz system and how this behavior can be eliminated through the implementation of sliding mode control. It is necessary to know about the theory of stability of Lyapunov to develop the appropriate control that allows to bring the system to the desired point of operation.

The spatial distribution of an electrical potential in a cell membrane subjected to an electric field was numerically obtained using an equivalent electrical circuit where the spatial variables that depend on the geometry are combined and an electrical circuit that relates the dynamics in the time of said excitation in four branches that represent the middle. It was observed that the potential decreases linearly in the geometry of the membrane due to the characteristics of the medium (sodium, potassium). On the other hand, the finite element method was developed for a two-dimensional domain that represents the geometry of a membrane, in such a way that it is possible to qualitatively analyze the behavior of the potential at any point of the membrane for an electrical pulse (electrode).

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In this document the dynamic model of the synchronous motor is presented, which has a typical structure of Lienard-type systems, the theory of dynamic systems is used, especially bifurcations, in this case, Hopf’s, which will be applied to the described model, to show the variations in the balance points of the system by taking the voltage of the bus to which it is connected as a variable parameter.

In the present paper a detailed analysis of the bifurcation of the Van der Pol oscillator is carried out, which starts by giving an account of the theory of chaos and dynamic systems, complements mentioning the main theoretical concepts of the dynamic systems and the different forms of analysis that exist, then continues to expose the discovery of the Van der Pol oscillator, to finally expose the results obtained from the bifurcation analysis of the oscillator.

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