# A principal possibility for computer investigation of evolution of dynamical systems independent on time accuracy

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## Abstract

Extensive *N*-body simulations are among the key means for the study of numerous astrophysical and cosmological phenomena, so various schemes are developed for possibly higher accuracy computations. We demonstrate the principal possibility for revealing the evolution of a perturbed Hamiltonian system with an accuracy independent on time. The method is based on the Laplace transform and the derivation and analytical solution of an evolution equation in the phase space for the resolvent and using computer algebra.

## 1 Introduction

The appearance of powerful computers has made the *N*-body simulations among efficient tools [1] for the study of various astrophysical and cosmological problems. The *N*-body gravitating systems cover a broad class of astrophysical problems, from the evolution of the Solar system as of a nearly integrable problem, up to the dynamics of star clusters, galaxies, and galaxy clusters as non-integrable problems. The cosmological simulations include from the evolution of primordial density perturbations up to the formation of dark matter galaxy halos, etc.

*N*-body simulation activities have been developed in two conventional directions:

- 1.
Numerical integration of

*N*equations of motion in order to follow the evolution of the system. The main difficulty of these investigations based on standard iterative methods of numerical integration of differential equations (Runge–Kutta, etc.) is the inevitable storage of errors with the increasing of the number of steps. This leads to rapid loss of the reliability of the results derived, the faster, the larger is the number of equations (i.e. the number of particles) and/or the longer is the duration of the calculations. - 2.
Investigation of the character of motion in the systems, which includes the finding out of integrals of motion, analysis of appearing stochastic regions, etc. The main feature of this direction is the application of methods of dynamical systems, e.g. of Kolmogorov–Arnold–Moser (KAM) theory, Lyapunov exponents, KS-entropy, etc [2].

*N*-body gravitating systems, for example, do reveal variety of complex, chaotic properties which play a crucial role in their evolution and structure; see e.g. [3, 4, 5, 6, 7, 8].

*u*,

*v*are periodic functions [2].

It is well known that the KAM theorem with its roots goes back to the main problem of celestial mechanics, the stability and evolution of the Solar system. Although the KAM theorem does not directly reveal the fate of the Solar system, its ideas were used for efficient numerical studies of the Solar system dynamics; see [7, 8] and other studies by Laskar et al. The KAM theorem itself does not determine the value of the perturbation \(\beta \), for which its statements are true, and regarding the planetary dynamics there are also principal difficulties in checking of the necessary conditions of the theorem with respect to each consideration. Moreover, due to the Arnold diffusion – a universal instability peculiar to non-linear systems of dimension \(N > 2\) – irrespective of whether the Solar system satisfies the conditions of the KAM theorem or not, it still cannot remain stable. Finding itself after an accidental perturbation in the stochastic region of phase space, the system can remain in it for an infinitely long time and therefore, the observed picture has to be destroyed anyway – the planets must either fall onto the Sun or fly away. So, those fundamental results at least do predict that, strictly speaking, the Solar system cannot last forever anyway, although the time scale of the latter instability (Arnold diffusion), according to estimates, far exceeds the Hubble time.

The example described above shows, on the one hand, the universality of systems with a perturbed Hamiltonian; on the other hand, the difficulties in direct application of the KAM theorem for revealing of the evolution of real physical and astrophysical systems.

Connecting certain hopes with computers and speaking of their potentialities, one must mention the importance of computer algebra methods, which, in our opinion, offer new prospects for the investigation of complex dynamics.

Below, we will show that computer algebraic methods along with those of dynamical systems enable the principal possibility of investigation of the evolution of a system without the effect of the storage of errors, i.e. of an accuracy independent on time. The search of error-free numerical integration schemes can be considered among the key problems of stellar dynamics, Problem 5 in [10]; see also [11].

## 2 Evolution in the phase space

*n*-dimensional integrable system with a phase space \(R^{2n} =\{p,q\}\), e.g. in the case of free oscillators with perturbation

*n*first integrals of motion in involution, enables one to transit from the variables (

*p*,

*q*) to that of action-angle variables \((I,\vartheta )\),

*n*-tori,

*f*(

*x*) the following holds:

*N*, we can reach a given accuracy.

*g*(

*x*) in the form

*t*we find the time evolution of \(I,\vartheta \), thus determining the evolution of the Hamiltonian system for those time instances.

The principal difference between this and the standard iteration methods of integration of Hamiltonian systems is clear; here the time *t* is a parameter of *f*(*x*, *t*) and does not influence the accuracy of calculation of the latter.

## 3 Computations via the new scheme

## 4 Conclusions

We demonstrated a principal way of finding out of the evolution of a dynamical system independent on time accuracy; namely, the phase space point \(I_1(0), I_2(0), \vartheta _1(0), \vartheta _2(0)\) describing the initial state of the Hamiltonian system is transferred to an arbitrary *t*, without any storage of errors. The key feature of this method is that we transfer the function \(f(I, \vartheta , 0)\) and, therefore, solve an equation which is sufficiently easier (linear), as compared to the Hamiltonian equations. Moreover, the former equation is solved analytically and not by an iteration procedure.

Another remarkable advantage of this procedure is the following: the initial state of a non-linear system during evolution can find itself in the stochastic region of phase space. The description of this is impossible by integration (even numerical) of the Hamiltonian equations, while our method allows us to investigate even this phenomenon. This aspect is particularly important for the study of *N*-body gravitating systems in view of their well-known chaotic properties. Then this method can open new principal possibilities for cosmological and extensive astrophysical *N*-body simulations.

### References

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