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This paper deals with a comparison from the precision and stability point of view of different discrete algorithms for simulating differential equation systems, applied in the case of a simple differential system: the harmonic oscillator. It points out the relation between the classical and incursive algorithms and shows the effect of incursion on...
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Citations
... A good introduction to incursion and hyperincursion is given in the following series of papers on the total incursive control of linear, non-linear and chaotic systems [3], on computing anticipatory systems with incursion and hyperincursion [4], on the computational derivation of quantum and relativist systems with forward-backward space-time shifts [5], on a review of incursive, hyperincursive and anticipatory systems, with the foundation of anticipation in electromagnetism [6], then, on the precision and stability analysis of Euler, Runge-Kutta and incursive algorithms for the harmonic oscillator [7], and finally, on the new concept of deterministic anticipation in natural and artificial systems [8]. ...
... An important difference between the incursive and the recursive discrete systems is the fact that in the incursive system, the order in which the computations are made is important: this is a sequential The two incursive harmonic oscillators are numerically stable, contrary to the classical recursive algorithms like the Euler and Runge-Kutta algorithms [7]. ...
This paper deals with a review of the properties of the hyperincursive discrete harmonic oscillator separable into two incursive discrete harmonic oscillators. We begin with a presentation step by step of the second order discrete harmonic oscillator. Then the 4 incursive discrete equations of the hyperincursive discrete harmonic oscillator are presented. The constants of motion of the two incursive discrete harmonic oscillators are analyzed. After that, we give a numerical simulation of the incursive discrete harmonic oscillator. The numerical values correspond exactly to the analytical solutions. Then we present the hyperincursive discrete harmonic oscillator. And we give also a numerical simulation of the hyperincursive discrete harmonic oscillator. The numerical values correspond also to the analytical solutions. Finally, we demonstrate that a rotation on the position and velocity variables of the incursive discrete harmonic oscillators gives rise to a pure quadratic expression of the constant of motion which is an ellipse. This result is fundamental because it gives an explanation of the effect of the discretization of the time in discrete physics. The information obtained from the incursive and hyperincursive discrete equations is richer than the information obtained by continuous physics. In conclusion, we have shown the temporal discretization of the harmonic oscillator produces a rotation similarly to the formalism of the special relativity dealing with rotations.
... A good introduction to incursion and hyperincursion is given in the following series of papers on the total incursive control of linear, non-linear and chaotic systems [13], on computing anticipatory systems with incursion and hyperincursion [14], on the computational derivation of quantum and relativist systems with forward-backward space-time shifts [15], on a review of incursive, hyperincursive and anticipatory systems, with the foundation of anticipation in electromagnetism [16], then, on the precision and stability analysis of Euler, Runge-Kutta and incursive algorithms for the harmonic oscillator [17], and finally, on the new concept of deterministic anticipation in natural and artificial systems [18]. ...
This paper begins with the formalization of the second order hyperincursive discrete Klein-Gordon equation. The temporal second order hyperincursive discrete Klein-Gordon equation is similar to the time-symmetric hyperincursive discrete harmonic oscillator and so bifurcates into a group of 4 incursive discrete real equations of first order. In this group, two equations are the discrete time reverse of the two other equations, giving an oscillator and an anti-oscillator. Firstly, the discrete Klein-Gordon equation, with one space dimension (1D), bifurcates to 4 first order incursive discrete equations that we called the Dubois-Ord-Mann real 4-spinors equations because Ord and Mann obtained the same equations from a stochastic method. Secondly, we generalize to three spatial dimensions (3D) these discrete Dubois-Ord-Mann equations. These 4 discrete equations are then transformed to real partial differential equations which can be written under the generic form of the Dirac quantum 4-spinors equation. Thirdly, we consider a change in the order of space variables and a change of indexes of the functions of the Dubois-Ord-Mann equations. With these changes we obtain the original real 4-Spinors Majorana partial differential equations. Also we obtain the 4 incursive discrete Majorana real equations.
... First, this paper presents a survey of the papers [1,2,[10][11][12][13]15] on the hyperincursive and incursive algorithms of the classical harmonic oscillator that are stable and show the conservation of energy. The hyperincursive discrete oscillator is separable into two incursive discrete oscillators. ...
... The incursive algorithms are numerically stable and the numerical simulation of the pendulum will show the conservation of the energy. Let us consider the example of the harmonic oscillator [15], with m the oscillating mass and k the spring constant, represented by the ordinary differential equations: ...
... So, for the harmonic oscillator, the conditions for obtaining an orbital stability are given by relations (8a) and (8c), rewritten as [15] (A C) 2 4 and AC BD 2 = 1 (9a,b) in using the equality from the relation (8c), in the relation (8a). Let us first consider the well-known Euler and Runge-Kutta integration methods, e.g. ...
This chapter will present algorithms for simulation of discrete space-time partial differential equations in classical physics and relativistic quantum mechanics. In simulation-based cyber-physical system studies, the main properties of the algorithms must meet the following conditions. The algorithms must be numerically stable and must be as compact as possible to be embedded in cyber-physical systems. Moreover the algorithms must be executed in real-time as quickly as possible without too much access to the memory. The presented algorithms in this paper meet these constraints. As a first example, we present the second-order hyperincursive discrete harmonic oscillator that shows the conservation of energy. This recursive discrete harmonic oscillator is separable to two incursive discrete oscillators with the conservation of the constant of motion. The incursive discrete oscillators are related to forward and backward time derivatives and show anticipative properties. The incursive discrete oscillators are not recursive but time inverse of each other and are executed in series without the need of a work memory. Then, we present the second-order hyperincursive discrete Klein–Gordon equation given by space-time second-order partial differential equations for the simulation of the quantum Majorana real 4-spinors equations and of the relativistic quantum Dirac complex 4-spinors equations. One very important characteristic of these algorithms is the fact that they are space-time symmetric, so the algorithms are fully invertible (reversible) in time and space. The development of simulation-based cyber-physical systems indeed evolves to quantum computing. So the presented computing tools are well adapted to these future requirements.
This paper presents the second order hyperincursive discrete Klein-Gordon equation in three spatial dimensions. This discrete Klein-Gordon equation bifurcates to 4 incursive discrete equations. We present the 4 incursive discrete Dubois-Ord-Mann real equations and the corresponding first order partial differential equations. In three spatial dimensions, the two oscillators given by these equations are now entangled with one spatial dimension, so the 4 equations form a whole. Then, we deduce the 4 incursive discrete Dubois-Majorana real equations and the corresponding first order partial differential equations. Next, these Dubois-Majorana real equations are presented in the generic form of the Dirac 4-spinors equation. In making a change in the indexes of the 4 functions in the Dubois-Majorana equations, the 4 first order partial differential equations become identical to the Majorana real 4-spinors equations. Then, we demonstrate that the Majorana equations bifurcate to the 8 real Dirac first order partial differential equations that are transformed to the original Dirac 4-spinors equations. Then, we give the 4 incursive discrete Dirac 4-spinors equations. Finally, we show that there are 16 discrete functions associated to the space and time symmetric discrete Klein-Gordon equation. This is in agreement with the Proca thesis on the 16 components of the Dirac wave function in 4 groups of 4 equations. In this paper, we restricted our derivation of the Majorana and Dirac equations to the first group of 4 equations depending on 4 functions.
