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Nepomuceno, Erivelton G.; Peixoto, Márcia L. C.; Martins, Samir A. M.; Rodrigues, Heitor M. Junior; Perc, Matjaž

Inconsistencies in numerical simulations of dynamical systems using interval arithmetic. (English) Zbl 1476.37093

Int. J. Bifurcation Chaos Appl. Sci. Eng. 28, No. 4, Article ID 1850055, 11 p. (2018).
Summary: 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.

Cited in 2 Documents

MSC:

37M05 Simulation of dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics

Keywords:

simulation; modeling; interval arithmetic; interval extensions; chaos; dynamical systems; error propagation

Software:

INTLAB; C-XSC 2.0; CoStLy
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Full Text: DOI

References:

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