The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.

The Liénard equation x + f (x)x + g(x) = 0 appears as a model in many problems of science and engineering. Since the first half of the 20th century, many papers have appeared providing existence and uniqueness conditions for limit cycles of Liénard equations. In this paper we extend some of these results for the case of the generalized ϕ-Laplacian Liénard equation, (ϕ(x)) + f (x)ψ(x) + g(x) = 0. This generalization appears when derivations of the equation different from the classical one are considered. In particular, the relativistic van der Pol equation, x / 1 − (x /c) 2 + μ(x 2 − 1)x + x = 0, has a unique periodic orbit when μ = 0.

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Programming is a key subject in many engineering programs. Students often perceive it as a difficult skill to master. There is extensive literature on helping students learn and improve to program, most of which focuses on K-12 education. However, due to the current demand for workers with programming skills, more research must be conducted on techniques for learning programming at the higher education level. In this work, an analysis and evaluation of the usefulness of an Asterisk Private Branch Exchange (PBX) as an educational tool to improve the programming skills of students in higher education is presented. The study worked with undergraduate students in telecommunications engineering, with little work experience in programming, during the completion of their final year project. Results suggest that using Asterisk has a positive impact on the students’ perception of their programming knowledge and skills, as well as an increment in the interest and comfort regarding programming.