Positive invariant regions for a modified Van Der Pol equation modeling heart action

The dynamical characterization of the heart rate is definitely a problem of vital importance. The selection, construction and adjustment of models that reproduce the dynamic behavior of the cardiac muscle, brings us closer to the solution of the usual classification problem in medicine, i.e. decide whether or not a patient belongs to the healthy class of patients. An usual model for this dynamic is often given by a modified Van der Pol model (vdPM), whose parameters are currently optimized with machine learning techniques or evolutionary algorithms. In any of these cases, the proper selection of the initial condition of the parameters drastically reduces the computational cost of the optimization method used. In this work, a strategy to estimate a positively invariant region that contains the periodic orbit, associated with a set of particular parameters of the vdPM considered, is presented. An algorithm is proposed to build the positively invariant region, the numerical results confirm that the vdPM solutions starting within the positively invariant region converge to the periodic orbit. This kind of result allows to reduce the cost of searching for the optimal parameters that represent a real electrocardiographic signal, which in turn, let us reduce the dimensionality of the system allowing the design of more efficient classifiers for these signals.

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