Estudo da dinâmica de um oscilador amortecido com retroalimentação retardada

Abstract

The dynamics of the delay differential equation ẍ + 2aẋ + bx = f(xτ), for the nonlinear function f(xτ) = tanh(αxτ), has been analyzed as a function of the parameters a, b, α and the delay τ , where xτ = x(t − τ). This model describes a damped harmonic oscillator subject to feedback with delay τ . Here, we have examined the cases of negative feedback (α 0). The method of steps have been used to show the property of solutions smoothing, for the nonlinear delay differential equation, for the increasing t. We have analyzed the local stability, made the stability charts, and showed that the spectrum of eigenvalues is discrete and at most enumerable. We have constructed the bifurcation diagrams that showed the occurrence of supercritical Hopf bifurcation, the supercritical pitchfork bifurcation and double Hopf bifurcation. For some points of resonant and non-resonant double Hopf bifurcation we have numerically calculated the time series, produced the phase space, and generated the first return map for a given Poincare section. Finally, we have performed a discretization of the equation and made a brief analysis of the dynamics of the resulting nonlinear difference equation.