A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

The main aim of this paper is to demonstrate the benefit of the application of high-performance computing techniques in the field of non-linear science through two kinds of dynamical systems as test models. It is shown that high-resolution, multi-dimensional parameter scans (in the order of millions of parameter combinations) via an initial value problem solver are an efficient tool to discover new features of dynamical systems that are hard to find by other means. The employed initial value problem solver is an in-house code written in C++ and CUDA C software environments, which can exploit the high processing power of professional graphics cards (GPUs). The first test model is the Keller–Miksis equation, a non-linear oscillator describing the dynamics of a driven single spherical gas bubble placed in an infinite domain of liquid. This equation is important in the field of cavitation and sonochemistry. Here, the high-resolution parameter scans gave us the opportunity to lay down the basis of a non-feedback technique to control multi-stability in which direct selection of the desired attractor is possible. The second test model is related to a pressure relief valve that can exhibit a special kind of impact dynamics called grazing impact. A fine scan of the initial conditions revealed a second focal point of the grazing lines in the initial-condition space that was hidden in previous studies.