  Landau-Zener-St\"uckelberg-Majorana (LZSM) interference emerges when the
parameters of a $\textit{qubit}$ are periodically modulated across an avoided
level crossing. Here, we investigate the occurrence of the LZSM phenomenon in
nonlinear multilevel bosonic systems, where the interference pattern is
determined by multiple energy levels and cannot be described by a level
crossing between only two states. We fabricate two superconducting resonators
made of flux-tunable Josephson junction arrays. The first device is very weakly
nonlinear (the nonlinearity is smaller than the photon-loss rate) and, when a
weak driving field is applied, it behaves as a linear resonator, yet shows the
same LZSM interference as in a two-level system. Notably, here the interference
originates from multiple avoided level crossings of the harmonic ladder. When
subjected to a stronger drive, nonlinear effects start playing a role, and the
interference pattern departs from the one observed in two-level systems. We
demonstrate that, when two or more LZSM interference peaks merge, dissipative
quantum chaos emerges. In the second device, where the nonlinearity surpasses
the photon-loss rate, we observe additional LZSM interference peaks due to Kerr
multiphoton resonances. When described under the light of the Floquet theory,
these resonances can be interpreted as synthetic modes of an array of coupled
cavities. We derive a simple effective model highlighting the essential
features of the entirety of these phenomena. As the control of LZSM in qubit
systems led to the implementation of fast protocols for characterization and
state preparation, our findings pave the way to better control of nonlinear
resonators, with implications for diverse quantum technological platforms.
