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Discontinuity, Nonlinearity, and Complexity 7(2) (2018) 165--172 | DOI:10.5890/DNC.2018.06.005

H. M. Rodrigues Junior; M. L. C. Peixoto; M. E.G. Nepomuceno.; S.A.M. Martins

Control and Modelling Group (GCOM), Department of Electrical Engineering, Federal University of São João del-Rei, MG, 36307-352, Brazil

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Abstract

The results obtained by numerical computation are not always precise. It happens because a computer has storage limitations and its set of numbers is finite. In this work, the interval analysis is used to give bounds around exact solution of the logistic map function. The connection between computer and interval mathematics makes possible to solve problems that can not be solved efficiently using floating point arithmetic. We use the intersection of different pseudo-orbits obtained by interval extensions to reduce the bounds of the exact solution. The method is applied using the Intlab toolbox. Without any substantial computational effort, we show a reduction of up to 26% in the width of interval by applying the method proposed in this paper.

Acknowledgments

We would like to thank CAPES, CNPq/INERGE, FAPEMIG and Federal University of São João del-Rei by support.

References

1. [1]	Galias, Z. (2013), The dangers of rounding errors for simulations and analysis of nonlinear circuits and systems and hoe to avoid them, Circuits and Systems Magazine, 13(3), 35-52.

2. [2]	Nepomuceno, E.G. andMartins, S.A.M. (2016),A lower-bound error for free-run simulation of the polynomial narmax, Systems Science & Control Engineering, 4(1), 50-58.

3. [3]	Hammel, S., Yorke, J., and Grebogi, C. (1987), Do numerical orbits of chaotic dynamical processes represent true orbits?, Journal of Complexity, 3(2), 136-145.

4. [4]	Nepomuceno, E.G. (2014), Convergence of recursive functions on computers, The Journal of Engineering, 1(1), 1-3.

5. [5]	Overton, M.L. (2001), Numerical Computing with IEEE floating point arithmetic, Philadelphia: Siam.

6. [6]	May, R.M. (1976), Simple mathematical models with very complicated dynamics, Nature, 261, 459-467.

7. [7]	Nepomuceno, E.G. and Mendes, E.M.A.M. (2017), On the analysis of pseudo-orbits of continuous chaotic nonlinear systems simulated using discretization schemes in a digital computer, Chaos, Solutions & Fractals, 95, 21-32.

8. [8]	Moore, R.E. and Bierbaum, F. (1979), Methods and applications of interval analysis, Vol.2, Philadelphia: Siam.

9. [9]	Ruetsch, G. (2005), An interval algorithm for multi-objective optimization, Structural and Multidisciplinary Optimization, 30(1), 27-37.

10. [10]	Alefeld, G. and Herzberger J. (1983), Introduction to interval computation, New York: Academic Press.

11. [11]	Moore, R.E., Kearfott, R.B., and Cloud, M.J. (2009), Introduction to interval analysis, Philadelphia: Siam.

12. [12]	Rudin, W. (1976), Principles of mathematical analysis, International Student Edition (3rd edn.), New York, McGraw- Hill.

13. [13]	Klatte, R., Kulisch, U., Wiethoff, A., and Rauch, M. (2012), C-XSC: A C++ class library for extended scientific computing, Dordrecht: Springer Science & Business Media.

14. [14]	Neher, M. and Eble, I. (2004), CoStLy: A validated library for complex functions, PAMM, 4(1), 594-595.

15. [15]	Rump, S.M. (1999), INTLAB - INTerval LABoratory, Developments in Reliable Computing, 77-104.

16. [16]	Rodrigues Junior, H.M. and Nepomuceno, E.G. (2015), Uso da computac? ?ao por intervalos para cálculo de ponto fixo de um mapa discreto (In Portuguese), Proceedings of Brazilian Conference of Dynamics, Control and Applications (DINCON 2015), Natal-RN, Brazil.

17. [17]	Feigenbaum, M.J. (1978), Quantitative universality for a class of non-linear transformations, Journal of Statistical Physics, 19(1), 25-52.