Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

The main purpose of this paper is to introduce a concept of L -fuzzifying topological vector spaces (here L is a completely distributive lattice) and study some of their basic properties. Also, a characterization of such spaces in terms of the corresponding L -fuzzifying neighborhood structure of the zero element is given. Finally, the conclusion that the category of L -fuzzifying topological vector spaces is topological over the category of vector spaces is proved.


Introduction and preliminaries
The theory of topological vector spaces is an important branch of modern analysis.There are abroad applications in mathematics and other subjects.So it is natural to consider the reasonable generalization of the topological vector spaces in the setting of many valued sets.In 1977, Katsaras and Liu combined the structure of fuzzy topology with the structure of vector spaces in [6] for the first time and introduced the concept of fuzzy topological vector spaces.In 1997, Fang and Yan in [2] generalized it to the concept of L-topological vector spaces (where L is a Hutton algebra).From then on, many properties of L-topological vector spaces were discussed by Yan and Fang.On the other hand, Höhle [4] introduced in 1980 the concept of fuzzy measurable spaces with the idea of giving degrees in [0,1] to some topological terms rather than 0 or 1. Sharing similar ideas to those above, Ying in [9] gave a new approach for fuzzy topology from a logical point of view, that is, a concept of fuzzifying topology was given.Recently, the relations between fuzzifying topological spaces and L-topological spaces have received wide attention [7,12].Specially Liu and Zhang proved the category of L-fuzzifying topological spaces is concretely isomorphic to the category of Lowen spaces in [7].Moreover, Höhle [5] and Zhang and Xu [12,13] introduced the concept of topological Lfuzzifying neighborhood structures, respectively.The equivalences between L-fuzzifying topologies and topological L-fuzzifying neighborhood structures have also been independently established by them.In this paper, we will combine the structure of L-fuzzifying topologies with the structure of vector spaces and introduce the concept of L-fuzzifying topological vector spaces and some basic properties of these spaces are studied.Then we obtain a characterization of L-fuzzifying topological vector spaces in terms of the Lfuzzifying neighborhood structure of the zero element.We will prove that the category of L-fuzzifying topological vector spaces L-FYTVS is topological over the category of vector spaces VEC.
First we fix some notations throughout this paper.L denotes a completely distributive lattice if not otherwise stated.0 and 1 are its top element and bottom element, respectively.M(L) denotes the set of all nonbottom coprimes L. K is a nondiscrete-valued field.For a,b ∈ L, we say a is wedge below b, in symbols, a b, if for every arbitrary subset D ⊆ L,∨D ≥ b implies that a ≤ d for some d ∈ D. For every x ∈ X, ẋ denotes the principal filter generated by x, and for all Definition 1.1 (Höhle [5], Ying [9], Zhang [12]).An L-fuzzifying topology on the set X is a function τ : 2 X → L which fulfills, for all U,V ,U j ⊂ X( j ∈ J), the following: If τ is an L-fuzzifying topology on X, then (X,τ) is an L-fuzzifying topological space.The value τ(U) is interpreted as the degree of openness of U. A continuous function between L-fuzzifying topological spaces is a function f Definition 1.2 (Höhle [5], Liu and Zhang [7,12]).An L-fuzzifying neighborhood structure on a set X is a family of functions P = {p x : 2 X → L|x ∈ X} with the following conditions, for all x ∈ X, U,V ⊂ X: The pair (X,P) is called an L-fuzzifying neighborhood space, and it will be called topological if it satisfies moreover the following condition, for all x ∈ X, U ⊂ X: A continuous function between L-fuzzifying neighborhood spaces (X,P) and (Y ,Q) is a map f : Suppose τ : 2 X → L is an L-fuzzifying topology, for all x ∈ X, U ⊂ X and let p τ x (U) = V ∈ ẋ|U τ(V ); then we have the following lemma.
From [10, Definition 3.1], if τ 1 and τ 2 are two L-fuzzifying topologies on X 1 and X 2 , respectively, then a subbase ϕ of the L-fuzzifying product topology τ 1 × τ 2 may be defined as follows: ; otherwise its value is 0. At the same time, we easily find that the L-fuzzifying neighborhood structure (p x ) x∈X corresponding to the L-fuzzifying product topology may be described as follows: (1.1) Lemma 1.6 (Gierz et al. [3]).Let L be a completely distributive lattice; then the following relations hold:  2) is a corollary of Lemma 1.6(1), (3).

L-fuzzifying topological vector spaces
Definition 2.1.Let X be a vector space over K and τ an L-fuzzifying topology on X.The pair (X,τ) is said to be an L-fuzzifying topological vector space, if the following two mappings are continuous: ( , where X × X and K × X are equipped with the corresponding L-fuzzifying product topologies τ × τ and K × τ (here K is an L-fuzzifying topology determined by the crisp neighborhood structure on K), respectively.Remark 2.2.A usual topological vector space can be regarded as an L-fuzzifying topological vector space (with the L-fuzzifying topology determined by the crisp neighborhood structure).When L = [0,1], our definition of L-fuzzifying topological vector spaces is different from that in [11].In [11], the author gave the definition with the fuzzifying neighborhood structure directly.Our definition begins with the continuity of the operations on the vector structure.From [11, Definition 3] and Proposition 2.3 in our paper, it is easy to verify that the two definitions are equivalent with L = [0,1].In other words, our definition is a generalization of that in [11].
Proposition 2.3.The mapping f in Definition 2.1 is continuous if and only if condition (LFN1) holds, that is, for every x, y ∈ X,W ⊆ X, U+V ⊆W p x (U) p y (V ) ≥ p x+y (W) (here Proof.Necessity.Since f : (X,τ) × (X,τ) → (X,τ) is continuous, it follows from Lemma 1.3 that the mapping f : (X,P) × (X,P) → (X,P) is continuous.Thus for every x, y ∈ X,W ⊆ X, we have p (x,y) ( This means that the necessity is proved.Sufficiency.From the above proof, for each x, y ∈ X, W ⊆ X, we have So the mapping f : (X,P) × (X,P) → (X,P) is continuous; it follows from Lemma 1.4 that the mapping f : (X,τ) × (X,τ) → (X,τ) is continuous.This completes the proof.

Proposition 2.4. The mapping g in Definition 2.1 is continuous if and only if condition (LFN2) holds, that is, for every
The proof of Proposition 2.4 is similar to that of Proposition 2.3, so we leave it to readers.
Propositions 2.3 and 2.4 give the conditions of the mappings f and g in Definition 2.1 with the help of the L-fuzzifying neighborhood structure.Moreover, we may obtain the following equivalent conditions by L-fuzzifying topologies directly.

is continuous if and only if condition (LFN2) holds, that is, for every
The meaning of M is the same as in Proposition 2.4.
Proof.(1) It is sufficient to prove that condition (LFN1) is equivalent to condition (LFN1) .
In fact, if (LFN1) is satisfied, then for every x, y ∈ X, A ⊆ X with x + y ∈ A, we have (2.4) Conversely, suppose that (LFN1) is satisfied.Then for all x, y ∈ X,W ⊆ X, we have (2.5) (2) It suffices to prove that condition (LFN2) is equivalent to (LFN2) .
If condition (LFN2) is satisfied, then for every (2.6) On the contrary, if condition (LFN2) is satisfied, then for all k 0 ∈ K, x ∈ X,W ⊆ X, we have (2.7) This completes the proof.
Definition 2.6.Let (X,τ) and (Y ,η) be two L-fuzzifying topological spaces.A function f : X → Y is called an L-fuzzifying homeomorphism, if f is a bijection and f , f −1 are both continuous.
Proof.Since T −1 x0 : X → X, x → x − x 0 , it is sufficient to prove that T x0 is continuous.For each x ∈ X,W ⊆ X, from p x (•) preserving order and (X,τ) L-fuzzifying topological vector space, we have (2.9) Thus T x0 is continuous.
In addition, for fixed k 0 ∈ K, k 0 = 0, the inverse of mapping S k0 is a function S −1 k0 : X → X, x → (1/k 0 )x.It suffices to prove that S k0 : X → X, x → k 0 x is continuous.For all x ∈ X,U ⊆ X, we have (2.10) So the conclusion holds.

.13)
The other relation may be obtained by the same method.
Proof.For each x ∈ X, since 0 • x = θ, it follows from Proposition 2.4 and U ⊆ X that p θ (U) ≤ MV⊆U 0∈M p x (V ).Owing to p θ (U) > 0, there exist 0 First we notice that conditions (3) and (4) imply that Now we fix an element x ∈ X and a subset A of X. Further, we choose an arbitrary subset B of X and a subset V of X with V + V ⊆ U = A − x.Then we distinguish the following cases.
which is a contradiction to the choice of V !Hence Case 1 and Case 2.1 only occur.Referring to (3.3) this leads to the estimation Hence (u4) holds.
By Definition 1.2, there exists an L-fuzzifying topology τ p on X and (p x ) x∈X with p x (A) = p(A − x) a topological L-fuzzifying neighborhood structure on X with respect to τ p .Specially, p(•) is an L-fuzzifying neighborhood structure of the zero element in X.
Then we will prove that the mappings f and g are continuous with respect to Lfuzzifying topology τ p .
For each x, y ∈ X,W ∈ 2 X , by condition (4) and p(•) preserving order, we have  Finally we prove that the mapping g in Definition 2.1 is continuous with respect to τ p .For each k 0 ∈ K, x ∈ X, U ⊆ X, suppose that p k0x (U) = 0. Then for each a p k0x (U) = p(U − k 0 x), it follows from conditions ( 4) and ( 6) that there exists a balanced set V 0 ⊆ X with V 0 + V 0 ⊆ U − k 0 x such that a p(V 0 ).By condition (5), we have a 0 < t 0 ∈ K such that x ∈ tV 0 , so (k . By condition (5) again, we have an 4) and ( 6), there exists a balanced set V 2 ⊆ X with a p(V 2 ) such that ( We denote (3.9) By the arbitrariness of a, we have p k0x (U) ≤ MB⊆U k 0 ∈M p x (B).This means that the mapping g in Definition 2.1 is continuous with respect to τ p .
Therefore (X,τ p ) is an L-fuzzifying topological vector space, and p(•) is an L-fuzzifying neighborhood structure of the zero element.
At the end of this section, we will give a natural class of examples of L-fuzzifying topological vector spaces.
Example 3.2.Let (X,Ᏺ,T) be a Menger probabilistic normed space [8] with T = min.For each ε > 0 and λ ∈ (0,1], N(ε,0) and N(ε,λ) are defined as follows: Proof.Let X be an object in VEC, (X i ,τ i ) an object in L-FYTVS, and f i : X → U(X i ) a morphism in VEC for all i ∈ I.By [1, Definition 21.1 and Definition 21.9], it follows to prove that there exists an L-fuzzifying topology τ on X with (X,τ) ∈ L-FYTVS such that τ is the weakest topology with respect to which each mapping f i : (X,τ) → (X i ,τ i ) is continuous.Suppose that p i (•) : 2 Xi → L is an L-fuzzifying neighborhood structure of the zero element in (X i ,τ i ) and I denotes the set of all finite subsets of I. Put First we prove that the mapping p(•) : 2 X → L satisfies conditions (1), ( 2), ( 3), ( 4), ( 5), and (6) in Theorem 3.1.
It is easy to find that conditions (1) and (2) hold.
(3) By the definition of p(•), the mapping p(•) preserves order.Then for all A,B ∈ 2 X , p(A B) ≤ p(A) p(B).Conversely, for all a p(A) p(B), we have a p(A) and a p(B).By Lemma 1.6, there exist . This means that (3) holds.
(4) For each W ⊆ X and every a p(W), there exists

and a ≤ p(A) p(B). Hence p(W) ≤ ∨ A+B⊆W p(A) p(B).
(5) For all A ⊆ X, x ∈ X, if p(A) > 0, then there exists (6) For each A ⊆ X and every a p(A), there exists and W is a balanced set.Since L is locally multiplicative, we obtain a i∈J0 p i (W i ).So a p(W).Hence there exists an L-fuzzifying topology τ on X such that (X,τ) ∈ L-FYTVS.Second we prove that τ is the weakest L-fuzzifying topology such that f i is continuous for all i ∈ I.
For each V i ⊆ X i and each x ∈ X, This means that f i : (X,τ) → (X i ,τ i ) is continuous.
In the following we will prove that τ is coarser than each L-fuzzifying topology τ which makes f i continuous for all i ∈ I.In fact, suppose that p is an L-fuzzifying neighborhood structure of the zero element with respect to τ.By Lemma 1.3, we prove the next relation p(A) ≤ p(A) for all A ⊆ X.For each a p(A), there exists Hence τ ⊆ τ.This means that each U-structured source (X → U(X i )) i∈I has a unique U-initial lift (X → X i ) i∈I .This completes the proof.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: ) for a ∈ L and D ⊂ L, if a D, then there exists d ∈ D such that a d; (3) for all a,b ∈ L, a b implies (there exists c ∈ M(L)) that a c b; furthermore, a = b and a b together imply (there exists c ∈ M(L),a = c) that a c b.In fact, Lemma 1.6(1) is exactly the one on [3, Section 2.29, Exercise (iii), page 204], and Lemma 1.6(3) is the one on [3, Section 2.29, Exercise (iv), page 204].In particular, Lemma 1.6(

(3. 6 )
So the mapping f in Definition 2.1 is continuous with respect to τ p .