Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

Abstract

It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and roundoff error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e. the Clean Numerical Simulation (CNS), is briefly described, together with its applications to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems such as three-body problem, and thus should have many applications in various fields. We find that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions. In addition, it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are timedependent. Thus, our CNS results strongly suggest that one had better to be very careful on DNS results of statistically unsteady turbulent flows, al- * Corresponding author.though DNS results often agree well with experimental data when turbulent flows are in a statistical stationary state.

Sparse Symplectically Integrated Neural Networks

“…Initial states were randomly initialized within the interior triangular region with chaotic Hamiltonian values. We computed this dataset via Clean Numerical Simulation, a Taylor series scheme for approximating chaotic dynamical systems [19]. An additional noisy dataset was created by adding independent Gaussian noise (σ = 0.005) to all states in the initial dataset.…”

Maximum Energies of Trapped Particles Around Magnetized Planets and Small Bodies

“…We trace the trajectories using an existing fourth order Runge-Kutta solver and apply it to the relativistic Lorentz force ( Section S5.1 in Supporting Information S1 ). Chaotic trajectories are highly sensitive to the initial conditions and therefore also to roundoff errors when they are simulated numerically ( Li & Liao, 2018 ) and the exact trajectory cannot be reliably simulated on standard computers with double-precision floating point representation ( Li & Liao, 2018 ). We therefore do not attempt to capture the exact time of loss ( or particle lifetimes ) but rather examine whether the particle exhibits pitch angle variations and sensitivity to initial conditions that imply a chaotic regime ( Section S5.3 in Supporting Information S1 ) Since protons are chaotic for all conceivable asteroids ( Figure 3a ), we focus on electron motion.…”

Dynamics analysis in a tumor-immune system with chemotherapy*

“…Many nonlinear dynamical systems found in nature exhibit routes to chaos [35,36] and some studies in the literature show that the cancer models also exhibit chaos. [22,37] In this section, we present computer simulation of the system to show that the system exhibits chaotic dynamics with the selected parameter set.…”