Asymptotic analysis of continuous fuzzy flows
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Asymptotic analysis of continuous fuzzy flows M. S. Cecconello1 · Jefferson Leite2 · R. C. Bassanezi3
Received: 7 October 2015 / Revised: 3 February 2016 / Accepted: 12 February 2016 / Published online: 26 February 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this work, we analyze the asymptotic behavior of fuzzy solutions using recent results on equilibrium and periodic points. We apply this analysis in some examples to investigate interesting properties of fuzzy solutions. As we show, the fuzzy solutions considered can present a more complex behavior than the deterministic ones. In addition, we show a new interpretation to the membership function of such fuzzy solutions as well as explore some properties of projections of fuzzy solutions. Computational simulations are done to illustrate these asymptotic behaviors. Keywords Fuzzy solutions · Periodic points · Equilibrium points · Dynamical systems Mathematics Subject Classification 37C75
1 Introduction Mathematical modeling of natural phenomenon by means of dynamical systems may be subject to uncertainties in the parameters of the equations that describe it. For example, in problems of population dynamics is not always possible to know exactly the number of individuals or the carrying capacity in a given environment. In addition, it is not always possible, due to technical difficulties or lack of information, to incorporate all necessary laws to describe the studied phenomenon. Thus, subjectivity is an important factor that must be considered in the mathematical modeling. The emergence of fuzzy set theory proposed by Zadeh (1965) brought new tools that enable the incorporation of subjectivity in models that describes real phenomena. Communicated by Geraldo Diniz.
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M. S. Cecconello [email protected]
1
Federal University of Mato Grosso, Cuiabá, Brazil
2
Federal University of Piauí, Teresina, Brazil
3
Federal University of ABC, São Paulo, Brazil
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M. S. Cecconello et al.
In this work we are interested in describing the asymptotic behavior of solutions of differential equation as dx = f (x) (1) dt when the initial condition is a fuzzy value. In this case, the uncertainty is only in the initial condition and not at the model defined by the function f . We can study solutions of differential equations involving subjectivity in the initial conditions by considering the Zadeh extension of the deterministic flow generated by this differential equation. The application obtained by this approach is frequently called fuzzy flow of the differential equation. If we set a particular fuzzy initial condition then we have a fuzzy solution. This kind of fuzzy solution was earlier studied in Oberguggenberger and Pittschmann (1999) and Buckley and Feuring (2000). Using this approach, Mizukoshi et al. (2009) show that there exists a important relationship between deterministic and fuzzy solutions. Fuzzy subsets whose membership function are characteristic functions of deterministic equilibrium points are stationary po
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