Numerical integration of differential equations in the presence of first integrals: observer method

Abstract

Abstract. We introduce a simple and powerful procedure-the observer method-in order to obtain a reliable method of numerical integration over an arbitrary long interval of time for systems of ordinary differential equations having first integrals. This aim is achieved by a modification of the original system such that the level manifold of the first integrals becomes a local attractor. We provide a theoretical justification of this procedure. We report many tests and examples dealing with a large spectrum of systems with different dynamical behaviour. The comparison with standard and symplectic methods of integration is also provided.

On the overfly algorithm in deep learning of neural networks

“…We follow now the idea of the proof of Main Lemma from [6], p. 377. and derive r 2 with respect to time along the solution of (2. 19) to obtain a simple differential equation:…”

“…The function Firstly, while using the standard gradient descent method, instead of dealing with the system (2.7), one can solve the observer equations (2.19) with some initial condition M(0) ∈ Γ 0 and use then Lemma 2.2 to compute X as corresponding to M(t) for some sufficiently large t > 0. It is well known that applying the Euler method (2.8) to solve (2.7), i.e following the conventional backpropagation algorithm, leads to accumulation of a global error proportional to the step size h. At the same time, the numerical integration of the observer system (2.19), as due to the existence of the attractor set Γ 0 , is much more stable numerically since the solution is attracted by the integral manifold Γ 0 (see [6] for more details and examples).…”

“…That brings more complexity to the iterative method, since the number of parameters rises considerably, but at the same time, the training data becomes built up into network in a quite natural way. The so obtained generalised gradient system is later converted to the observer one (see [6]).…”

Nonexistence of the final first integral in the Zipoy-Voorhees space-time

“…On the other hand, the results of [2] indicate chaotic behaviour of the system, but the region where that happens is very small when compared to the phase space dominated by invariant tori, and the integration was performed with the Runge-Kutta method of the 5th order only. Since it is known [4,5] that integrable systems can exhibit numerical chaos (particularly for the R-K method), the results of [2] should be taken cautiously. Our own numerical integration produced a Poincaré section visibly shifted from the one in [2] (see the end of section IV), and since we used a more accurate method, it poses the question of whether the picture would be further deformed as the precision was increased.…”