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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.

This article presents the approach from the point of view of the transformation of Legendre to obtain Hamiltonian systems from Lagrangian systems. The main objective is to show the sequence of obtaining these systems with the Legendre transformation. The problems outlined in [5] and [7] are explained with a qualitative approach, then a weak formulation for the dynamic system of the synchronous machine is shown by means of the Legendre transformation and the Lagrangian system formulation to obtain the Hamiltonian system. which describes the dynamics of an electric machine.

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The cathodic arc discharge is a deposition technique widely used to synthesize hard coatings and thin films. The structure of the plasma generated by the electrical discharge and its interaction with neutral particles was studied using numerical simulations. Typical plasma parameters were characterized considering their spatial and temporal dependence, as well as several cathode materials that are commonly used in these systems. For the evolution of the ion density, it was observed the formation of Knudsen layer, and also a dependence of pressure gradients in the global behavior. With respect to the kinetic energy, it was found a deceleration of ions, which is represented by a shock front produced in the plasma−neutrals interaction. On the other hand, the energy releasing was generated due to the heat transference between electrons and ions. The plasma potential follows a behavior, which is similar to that of the ion density, and it is caused by the dynamics of charged particles which is directly affected by the concentration of neutrals and ions. In general, the physical quantities are directly affected by electrical and thermal conductivity of the cathode material. Our results can be applied to understand the plasma phenomena produced in a cathodic arc discharge