Study of teh dynamics of the damped oscillator with delayed feedback

Author(s)

Daniel Câmara de Souza

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2011-02-18

Examining board members

Coraci Pereira Malta; Ibere Luiz Caldas; Roberto Andre Kraenkel

• Advisor: Coraci Pereira Malta

Abstract

The dynamics of the delay differential equation x 2 pontos + 2ax ponto + bx = f(x ), for the nonlinear function f(x) = tanh(x), has been analyzed as a function of the parameters a, b, and the delay , where x = x(t ). This model describes a damped harmonic oscillator subject to feedback with delay . Here, we have examined the cases of negative feedback (< 0) and positive feedback ( > 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 Poincaré section. Finally, we have performed a discretization of the equation and made a brief analysis of the dynamics of the resulting nonlinear difference equation. (AU)

• FAPESP's process: 09/12512-0 - Study of the dynamics of the damped oscillator with delayed positive feedback
• Grantee: Daniel Câmara de Souza
• Support Opportunities: Scholarships in Brazil - Master