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Abstract
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.
Highlights
Non-linear dynamics has received a lot of attention since the discovery of the chaotic Lorenz attractor [1]
We demonstrate that the application of parameter scans with quite high resolution using initial value problem solver (IVP) solvers can be the source for new ideas and discoveries
Apart from mapping the dynamics of bubbles to obtain approximate information about their chemical activity, the bifurcation structure of the high-resolution plots led to a discovery of a new technique to control multi-stability
Summary
Non-linear dynamics has received a lot of attention since the discovery of the chaotic Lorenz attractor [1]. It might seem impractical to try to solve a two-dimensional problem with high-resolution IVP computations, since many clever techniques exist (e.g. the pseudo-arclength continuation using a boundary value problem solver (BVP) [33]) that can explore the evolution of bifurcation points even in two dimensions fast and . In this way, valuable information can be obtained about the bifurcation structures [4, 20, 34,35,36,37]. Meccanica (2020) 55:2493–2504 computations are carried out on two quite different test models, for details see Sect. 2
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