Motivated by recent experiments, where a voltage biased Josephson junction is placed in series with a resonator, the classical dynamics of the circuit is studied in various domains of parameter space. This problem can be mapped onto the dissipative motion of a single degree of freedom in a nonlinear time-dependent potential, where in contrast to conventional settings the nonlinearity appears in the driving while the static potential is purely harmonic. For long times the system approaches steady states which are analyzed in the underdamped regime over the full range of driving parameters including the fundamental resonance as well as higher and sub-harmonics. Observables such as the dc-Josephson current and the radiated microwave power give direct information about the underlying dynamics covering phenomena as bifurcations, irregular motion, up-and down conversion. Due to their tunability, present and future set-ups provide versatile platforms to explore the changeover from linear response to strongly nonlinear behavior in driven dissipative systems under well defined conditions. I. INTRODUCTIONThe nonlinear properties of Josephson junctions (JJs) have made such devices a key circuit element for classical and quantum electronics. Accordingly, there has been a long tradition of studying non-linear phenomena in driven superconducting circuits, starting as early as the 1960s with the discovery of Shapiro steps 1 . While Shapiro-steps have remained a tool in exploring new directions in Josephson physics, for instance in atomic point contacts 2-6 , other nonlinear phenomena, like synchronization, have been investigated in arrays of JJs: as a test-bed for generic theory models to capture synchronization phenomena 7,8 , but as importantly with the prospect of applications as sources of more intense coherent radiation 9 , cf. also new developments using intrinsic arrays 10-12 .More recently, the nonlinearity of the JJ was exploited as the crucial factor in enabling the high sensitivity of Josephson bifurcation amplifiers, achieving substantial improvements towards reaching quantum-limited measurement processes [13][14][15][16] . Most of the features of Josephson bifurcation amplifiers, in fact, only rely on (and can consequently be described by) any type of nonlinearity 17-22 , e.g., Duffing-type models, so that only recently the full nonlinear potential of the JJ has become of interest in this field 23 .A recent addition to the field of driven nonlinear JJs 24-26 are experiments on a dc-biased JJ connected to a resonator 27-29 . In this sort of setup, charge transfer through the JJ leads to excitations in the resonator, and therefore allows to convert energy carried by charge quanta into quantum microwave photons. In these devices measurement of both the Josephson current and the emitted microwave radiation is possible, a distinct advantage in comparison to other recently proposed transport setups 30,31 , which show similar nonlinear features like bifurcations, period multiplication and up-and downconversion 32-34 . S...

Abstract. Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis.