  We report the first experimental observation of multiple transitions showing
the emergence and disappearance of slowly decaying eigenmodes in a dissipative,
Floquet electronic system with synthetic components. Conventional wisdom has it
that such transitions occur at exceptional points, and avoided-level-crossing
driven phenomena in purely dissipative systems are formerly unexplored.
Remarkably, in our system, the slowly decaying eigenmodes emerge without
exceptional points. Our experimental setup makes use of an LC oscillator
inductively coupled to an RLC oscillator, where the time-periodic (Floquet)
inductive coupling and resistive-heating losses can be independently controlled
by means of external voltage signals. With a Floquet dissipation, we observe
that slowly-decaying eigenmodes emerge at vanishingly small dissipation
strength in the weak coupling limit. With a moderate, Floquet coupling,
multiple instances of their emergence and disappearance are observed. With an
asymmetric dimer model, we show that these transitions, driven by
avoided-level-crossing in purely dissipative systems, are generically present
in both static and Floquet domains.
