Affiliations: School of Mechanical Engineering, Hebei University of
Technology, Tianjin, China | School of Astronautics, Harbin Institute of
Technology, Harbin, China
Abstract: In this paper, the dynamics of Duffing-van der Pol oscillators under
linear-plus-nonlinear position feedback control with two time delays is studied
analytically and numerically. By the averaging method, together with truncation
of Taylor expansions for those terms with time delay, the slow-flow equations
are obtained from which the trivial and nontrivial solutions can be found. It
is shown that the trivial solution can be stabilized by appropriate gain and
time delay in linear feedback although it loses its stability via Hopf
bifurcation and results in periodic solution for uncontrolled systems. And the
stability of the trivial solution is independent of nonlinear feedback.
Different from the case of the trivial solution, the stability of nontrivial
solutions is also associated with nonlinear feedback besides linear feedback.
Non-trivial solutions may lose their stability via saddle-node or Hopf
bifurcation and the resulting response of the system may be quasi-periodic or
chaotic. The feedback gains and time delays have great effects on the amplitude
of the periodic solutions and their bifurcation control. The simulations,
obtained by numerically integrating the original system, are in good agreement
with the analytical results.
