The following
study investigates regimes of self-excitation in a classical self-excited
oscillator, the Van Der Pol (VDP) oscillator, with an attached nonlinear energy
sink (NES). This assembly presents various possible bifurcation regimes of the
response, depending on the system parameters and the initial conditions (ICs).
Initial equations are reduced by averaging to a 3 dimensional system. An
assumption of a relatively small mass of the NES allows treating this averaged
system as a singularly perturbed system and allows for separation of variables
into two "slow" variables and one "super-slow" variable.
Such approach, in turn, yields a complete analytical description of the
possible response regimes. The system exhibits various response regimes; from
almost unperturbed limit cycle oscillations (LCOs) to a complete elimination of
the self-excitation, small-amplitude LCOs, chaotic-like responses (based on the
multiple points in the Poincar'e' map) and excitation of quasiperiodic strongly
modulated response (SMR). The latter can be excited by three distinct
bifurcation mechanisms: The Canard explosion, a Shil'nikov bifurcation and a
heteroclinic bifurcation. Some of the above oscillatory regimes can co-exist
for the same values of system parameters. In this case, it is possible to
establish the basins of attraction for the co-existing regimes. Direct numeric
simulations demonstrate good coincidence with the analytic predictions.
