  Transmon qubits are the predominant element in circuit-based quantum
information processing, such as existing quantum computers, due to their
controllability and ease of engineering implementation. But more than qubits,
transmons are multilevel nonlinear oscillators that can be used to investigate
fundamental physics questions. Here, they are explored as simulators of excited
state quantum phase transitions (ESQPTs), which are generalizations of quantum
phase transitions to excited states. We show that the spectral kissing
(coalescence of pairs of energy levels) experimentally observed in the
effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We
explore the dynamical consequences of the ESQPT, which include the exponential
growth of out-of-time-ordered correlators, followed by periodic revivals, and
the slow evolution of the survival probability due to localization. These
signatures of ESQPT are within reach for current superconducting circuits
platforms and are of interest to experiments with cold atoms and ion traps.
