A new approach to find the multi-fractal dimension of multi-fuzzy fractal attractor sets based on the iterated function system
Küçük Resim Yok
Tarih
2019
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Maltepe Üniversitesi
Erişim Hakkı
CC0 1.0 Universal
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Özet
In nature, objects are not single fractal sets but are a collection of complex multiple fractals that characterise the multifractal space, a generalisation of fractal space. While fractal space includes a fractal set, a multi-fractal space includes the union of
fractals. A fuzzy fractal space is a fuzzy metric space and is an approach for the construction, analysis, and approximation of sets
and images that exhibit fractal characteristics. The finite Cartesian product of fuzzy fractal spaces is called the multi-fuzzy fractal
space. We propose in this paper, a theoretical proof to define the multi-fractal dimensions FD of a multi- fuzzy fractal attractor
of n objects for the self-similar fractals sets A = n
i=1 Ai = (A1, A2,... An) of the contraction mapping W∗∗ : n
i=1 H(F(Xi)) → n
i=1 H(F(Xi)) with contractivity factor r = max{ri, i = 1, 2,... n} where H(F(Xi) is a fuzzy fractal space for each i = 1, 2,..., n ;
over a complete metric space (n
i=1 H(F(Xi)), D∗) then for all Bi that belong toH(F(Xi)), there exists B∗ belonging to (n
i=1 H(F(Xi))
such that W∗∗(B∗ = n
i=1 Bi) = n
i=1 (
n
j=1
k(i, j)
k=1 ω∗k
i j (Bj) = n
i=1 Wi(B∗)). By supposing that M (t) =
k (r∗k
i j )
FD
n×n is the matrix
associated with the the contraction mapping ω∗k
i j with contraction factor r∗k
i j , ∀i, j = 1, 2,..., n, ∀k = 1, 2, ..., k(i, j), for all t ≥ 0,
and h (t) = det(M (t) − I) . Then, we prove that if there exists a FD such that; h(FD) = 0, then FD is the multi fractal dimension
for the multi fuzzy-fractal sets of IFS; and M(FD) has a fixed point in Rn.
Açıklama
Anahtar Kelimeler
Fractal space, multi-fuzzy fractal space, IFS, box-counting dimension, fractal dimension
Künye
Mohammed, A. J. (2019). A new approach to find the multi-fractal dimension of multi-fuzzy fractal attractor sets based on the iterated function system. Third International Conference of Mathematical Sciences, Maltepe Üniversitesi. s. 1-4.
