Cone of non-linear dynamical system and group preserving schemes

Abstract

The first step in investigating the dynamics of a continuous time system described by a set of ordinary differential equations is to integrate to obtain trajectories. In this paper, we convert the non-linear dynamical system x ̇ = f ( x ,t), x ∈ R n , into an augmented dynamical system of Lie type X ̇ = A ( X ,t) X , X ∈ M n+1, A ∈ so(n, 1) locally. In doing so, the inherent symmetry group and the (null) cone structure of the non-linear dynamical system are brought out; then the Cayley transformation and the Padé approximants are utilized to develop group preserving schemes in the augmented space. The schemes are capable of updating the augmented state point to locate automatically on the cone at the end of each time increment. By projection we thus obtain the numerical schemes on state space x , which have the form similar to the Euler scheme but with stepsize adaptive. Furthermore, the schemes are shown to have the same asymptotic behavior as the original continuous system and do not induce spurious solutions or ghost fixed points. Some examples are used to test the performance of the schemes. Because the numerical implementations are easy and parsimonious and also have high computational efficiency and accuracy, these schemes are recommended for use in the physical calculations.