  Existence of periodical solutions, i.e. cycles, in the Impulsive Goodwin's
Oscillator (IGO) with the continuous part of an arbitrary order m is
considered. The original IGO with a third-order continuous part is a hybrid
model that portrays a chemical or biochemical system composed of three
substances represented by their concentrations and arranged in a cascade. The
first substance in the chain is introduced via an impulsive feedback where both
the impulse frequency and weights are modulated by the measured output of the
continuous part. It is shown that, under the standard assumptions on the IGO, a
positive periodic solution with one firing of the pulse-modulated feedback in
the least period also exists in models with any m >= 1. Furthermore, the
uniqueness of this 1-cycle is proved for the IGO with m <= 10 whereas, for m >
10, the uniqueness can still be guaranteed under mild assumptions on the
frequency modulation function.
