  We propose a scheme for obtaining highly-sensitive oscillators in a
coupled-resonator system with an exceptional point of degeneracy (EPD) and a
small instability. The oscillator with the exceptional degeneracy is realized
by using two coupled resonators with an almost balanced small-signal gain and
loss, that saturates due to nonlinear effects of the active component,
resulting in an oscillation frequency that is very sensitive to a perturbation
of the circuit. Two cases are investigated, with two parallel LC resonators
with balanced small-signal gain and loss that are either coupled wirelessly by
mutual inductance or coupled-wired by a capacitor. This paper demonstrates
theoretically and experimentally the conditions to obtain a second-order EPD
oscillator and analyzes the ultrasensitivity of the oscillation frequency to
components' perturbation, including the case of asymmetric perturbation that
breaks PT-symmetry. We discuss the effects of nonlinearity on the performance
of the oscillator and how the proposed scheme improves the sensing's
sensitivity of perturbations. In contrast to previous methods, our proposed
degenerate oscillator can sense positive or negative changes of a circuit
component. The degenerate oscillator circuit may find applications in various
areas such as ultrasensitive sensors, tunable oscillators and modulators.
