  Superconducting circuits are a versatile platform to implement a multitude of
Hamiltonians which perform quantum computation, simulation and sensing tasks. A
key ingredient for realizing a desired Hamiltonian is the irradiation of the
circuit by a strong drive. These strong drives provide an in-situ control of
couplings, which cannot be obtained by near-equilibrium Hamiltonians. However,
as shown in this paper, out-of-equilibrium systems are easily plagued by
complex dynamics leading to instabilities. Predicting and preventing these
instabilities is crucial, both from a fundamental and application perspective.
We propose an inductively shunted transmon as the elementary circuit optimized
for strong parametric drives. Developing a novel numerical approach that avoids
the built-in limitations of perturbative analysis, we demonstrate that adding
the inductive shunt significantly extends the range of pump powers over which
the circuit behaves in a stable manner.
