Inconsistencies in Numerical Simulations of Dynamical Systems Using Interval Arithmetic

Abstract

Over the past few decades, interval arithmetic has been attracting widespread interest from the scientific community. With the expansion of computing power, scientific computing is encountering a noteworthy shift from floating-point arithmetic toward increased use of interval arithmetic. Notwithstanding the significant reliability of interval arithmetic, this paper presents a theoretical inconsistency in a simulation of dynamical systems using a well-known implementation of arithmetic interval. We have observed that two natural interval extensions present an empty intersection during a finite time range, which is contrary to the fundamental theorem of interval analysis. We have proposed a procedure to at least partially overcome this problem, based on the union of the two generated pseudo-orbits. This paper also shows a successful case of interval arithmetic application in the reduction of interval width size on the simulation of discrete map. The implications of our findings on the reliability of scientific computing using interval arithmetic have been properly addressed using two numerical examples.

Computational chaos control based on small perturbations for complex spectra simulation

“…These and other problems were recently analyzed in several studies on computational chaos. 29–32 It has been shown that one of the ways of reducing these drawbacks is through the arithmetic interval technique. 31 Nevertheless, we developed some algorithms of arbitrary precision created exclusively for the modified logistic map x n + 1 = r x n ( 1 − x n ) + P .…”

“…29–32 It has been shown that one of the ways of reducing these drawbacks is through the arithmetic interval technique. 31 Nevertheless, we developed some algorithms of arbitrary precision created exclusively for the modified logistic map x n + 1 = r x n ( 1 − x n ) + P . This allowed us to eliminate any problem of chaos degradation from our computational simulations using thousands of significant digits as needed without a relevant computational cost.…”

“…In the context of chaos theory, 100% pure chaos can be degraded and yield pseudo orbits, [29][30][31] in other words, non-natural orbits that should not exist under a chaotic state. In the context of this work, the degradation of chaos can be understood as a type of butterfly effect caused by computational limitations.…”

“…But in fact, it is not 100% natural chaos, it is degraded chaos due to computational limitations. [29][30][31][32] This can cause irreproducible and unreliable results, Nazare ét al. 33 studied this topic.…”

A method to find approximate solutions of first‐order systems of nonlinear ordinary equations

“…Our approach is based in a generalization of the one‐step matrix method developed by Demidovich and other authors 1–3 some time ago, valid for systems of the form . One clear presentation of this method appears in the textbook by Farkas, 4 and it is interesting to compare it with some related procedures; see for instance previous studies 5,6 . It is quite important to remark that, while the Demidovich matrix method is applied to linear equations (with variable coefficients), our matrix method is a generalization to nonlinear systems.…”