We propose a new class of operators, referred to as chaos-preserving fuzzy difference operators, which maintain key dynamical properties under the Zadeh extension.

We establish existence and consistency theorems for type-I, type-II, and Hausdorff fuzzy differences when applied in dynamic contexts.

We provide sufficient conditions under which these operations preserve topological transitivity, density of periodic points, and various forms of chaos.

We illustrate the theoretical results with explicit numerical examples, using triangular fuzzy numbers and classical chaotic maps such as the logistic function.

Definition 1 (Fuzzy dynamical system via Zadeh extension). Let (X, d ) be a compact metric space, and f: X → X be a continuous map. Denote by ℱ(X) the space of fuzzy set u: X → [0,1] that are:

I. normal: supx∈Xu(x) = 1.

II. Upper semicontinuous (usc).

III. With compact support: supp(u) ≔ ̅ {̅ x̅̅∈ ̅̅̅ X ̅|̅u ̅̅(̅x ̅̅ )̅̅ > ̅̅̅0 ̅̅ } ⊂ X is compact.

The Zadeh extension of f, denoted ̂f(u)(x) = sup{u(y)| f(y) = x}, for all x ∈ X.

Then (ℱ(X), ̂f) is a fuzzy dynamical system induced by f [2], [5].

Definition 2 ( -Level compatible binary fuzzy operations). Let ∘∶ ℱ(X) × ℱ(X) → ℱ(X) be a binary operation on fuzzy sets. For each fuzzy set A ̃ ∈ ℱ(X), define its α-level set as:

[A ̃] ≔ {x ∈ X|u (x) ≥ α}, for each α ∈ (0, 1].

The operation ° is said to be α-level set compatible if, for every α ∈ (0, 1], there exists a corresponding set °α acting on crisp subsets of X such that:

[A ̃°B ̃] ≔ [A ̃] °[B ̃] , and the resulting fuzzy set C ̃ = A ̃ °B ̃ satisfies:

u (x) = sup {α ∈ (0, 1]|x ∈ [A ̃°B ̃] },

Let A ̃, B ̃ ∈ ℱ(X), and let ° be a binary operation between sets. We define:

C ̃ ≔ A ̃ °B ̃ , for all α ∈ (0, 1], and say that the operation is level-set compatible if there exists a fuzzy set C ̃ ∈ ℱ(X) such that:

μ̃(x) = supα∈(0,1]α ∙ ̃ (x),

C Ca

Furthermore, μC ̃ must define a valid fuzzy set in ℱ(X), i.e., it is normal, upper semicontinuous, and has compact support.

Definition 3 (Chaos-preserving fuzzy difference operator). Let ⊝∶ ℱ(X) × ℱ(X) → ℱ(X) be a binary fuzzy difference operator, where A ̃, B ̃ ∈ ℱ(X) are fuzzy sets.

We say that ⊝ is a chaos-preserving fuzzy difference operator (with respect to a continuous map f: X → X ) if the following conditions hold:

⊝ is α-level set compatible as in Definition 2, i.e., for each α ∈ (0, 1], the α-level set of A ̃ ⊝ B ̃ satisfies:

[A ̃ ⊝ B ̃] = [A ̃] °[B ̃] .

For a suitable difference operator ⊝α between closed subsets of X.

The Zadeh extension ̂f: ℱ(X) → ℱ(X) induces the fuzzy dynamical system (ℱ(X), ̂f ), where:

Define the difference-induced dynamics ̂f : ℱ(X) → ℱ(X) by:

f̂ (A ̃) ≔ f̂(A ̃ ⊝ B ̃).

If f is chaotic (e.g., Li-Yorke, Devaney, or distributional chaotic) on X, then ̂f is chaotic on ℱ(X) in the same sense.

A fuzzy binary operation A ̃ ⊝ B ̃ ≔ C ̃ is said to be chaos-preserving if, for a given continuous map f: X → X, it satisfies:

Definition 4 (Natural Hausdorff difference operator ⊝ ). Let A ̃, B ̃ ∈ ℱ (X) be fuzzy sets whose α-level set A ̃ and B ̃ are closed intervals for each α ∈ (0, 1]. The natural Hausdorff difference, denoted by A ̃ ⊝ B ̃, is the fuzzy set C ̃ ∈ ℱ(X) defined level-wise by:

̃ ̃ ̃ L L U U L L U U Cα = Aα ⊝ Bα = [min(aα − bα, aα − bα), max(aα − bα, aα − bα)].

Where Aα = [aα, aα] and Bα = [bα, bα].

The membership function μC ̃: X → [0,1] is then constructed via:

μ̃(x) = supα∈(0,1]α ∙ ̃ (x).

C Ca

Where (x) denotes the characteristic function of the interval C ̃ . This construction ensures that C ̃ is a

C ̃ α

normal, upper semicontinuous fuzzy set with compact support [5].

Main Theorem (Preliminary Form) Theorem 1 (Chaos preservation via natural Hausdorff difference). Let f: X → X be a continuous map exhibiting Li-Yorke chaos. Suppose A ̃, B ̃ ∈ ℱ (X) are fuzzy sets whose α-level sets are closed intervals. Then:

The natural Hausdorff difference A ̃ ⊝ B ̃ exists and belongs to ℱ(X).

NH The Zadeh extension satisfies:

NH NH

If f is Li-Yorke chaotic, then so is ̂f acting on ℱ(X) under the operator ⊝ .

NH

Use the decomposition theorem to define the membership function:

μ (x) = sup α ∙ (x), with M ≔ A ̃ ⊝ B ̃ .

C ̃ α∈(0,1] C ̃ α α H α

Show that if A ̃ and B ̃ are closed intervals, then M is also a closed interval.

Prove that μ is upper semicontinuous, and that C ̃ is a fuzzy number.

Theorem 2 (Chaos preservation via natural Hausdorff difference). Let f: X → X be a continuous map on a compact metric space (X, d) , and suppose that f is Li-Yorke chaotic. Let A ̃, B ̃ ∈ ℱ (X) be fuzzy sets such that for each α ∈ (0, 1], the α-level sets A ̃ ,B ̃ are closed and bounded intervals.

Then: The natural Hausdorff difference A ̃ ⊝ B ̃ exists and belongs to ℱ(X).

NH

The Zadeh extension satisfies:

NH NH

If f is Li-Yorke chaotic, then so is ̂f on ℱ(X) under the operator ⊝ .

NH

̃ ̃ L L U U L L U U

Proof: Let us define the family : Mα ≔ Aα ⊝H Bα = [min{aα − bα, aα − bα }, max{aα − bα, aα − bα}], where

Aα = [aα, aα], Bα = [bα, bα ].

Since A ̃, B ̃ ∈ ℱ (X), each M is a nonempty compact interval. Define:

cc α

μ̃(x) = supα∈(0,1]α ∙ M (x).

Each Mα being a closed interval implies that its characteristic function M is upper semicontinuous, and thus μC ̃ as a supremum of such functions is also upper semicontinuous.

Additionally, μ is normal because for some x ∈ M = A ̃ ⊝ B ̃ , we have μ (x) ≥ 1 ∙ (x) = 1 . Hence, C ̃ ∈

C ̃ 1 1 1 C ̃ M1

ℱ(X).

Observe that for each α, we have:[̂f(A ̃)] = f(A ̃ ), [̂f(B ̃)] = f(B ̃ ) so by properties of a continuous function

α α α α

on intervals: [̂f(A ̃)] ⊝ [̂f(B ̃)] = f(A ̃ ) ⊝ f(B ̃ ) = f(A ̃ ⊝ B ̃ ). Since ̂f acting level-wise, we have

α H α α H α α H α

̂f(A ̃ ⊝ B ̃) = ̂f(C ̃) ∈ ℱ(X), with [̂f(C ̃)] = f(M ). But f(M ) = f(A ̃ ⊝ B ̃ ) = f(A ̃ ) ⊝ f(B ̃ ) =

NH α α α α H α α H α

[̂f(A ̃)] ⊝ [̂f(B ̃)] .

α H α

Thus: ̂f(C ̃) = ̂f(A ̃) ⊝ ̂f(B ̃).

NH

Recall that if f is Li-Yorke chaotic, then there exists an uncountable scrambled set S ⊂ X. Define S ̃ ≔ { : x ∈ S} ⊂ ℱ(X), which is uncountable. Since ̂f( ) = , it follows that the Li-Yorke pair condition

{x} {x} {f(x)}

is preserved under ̂f, and hence ̂f is also Li-Yorke chaotic on ℱ(X).

Corollary 1 (Chaos Preservation for Type-I and Type-II Generalized Differences). Let f: X → X be a continuous map on a compact metric space, and suppose f is Li-Yorke chaotic. Let A ̃, B ̃ ∈ ℱ (X).Then the Type I and Type II generalized differences A ̃ ⊝ B ̃, A ̃ ⊝ B ̃, defined via:

G1 G2

(1) (2)

C ̃ = cl(∪ A ̃ ⊝ B ̃ ) , C ̃ = cl (conv(∪ A ̃ ⊝ B ̃ )).

α β∈[α,1] β H β α β∈[α,1] β H β

̃ (1) ̃ (2)

Define fuzzy sets C , C ∈ ℱ(X) that preserve chaos under the Zadeh extension:

f̂(A ̃ ⊝ B ̃) = f̂(A ̃) ⊝ f̂(B ̃) , f̂(A ̃ ⊝ B ̃) = f̂(A ̃) ⊝ f̂(B ̃)

G1 G1 G2 G2

Hence, if f is Li-Yorke chaotic, so are the generalized fuzzy different systems under ̂f.

Let C ̃ = A ̃ ⊝ B ̃, with level sets defined as:

̃ L L U U L L U U Cα = [min(aα − bα, aα − bα), max(aα − bα, aα − bα)], where Aα = [aα, aα], Bα = [bα, bα], and α ∈ (0, 1]. We define the membership function via the decomposition theorem:

μ̃(x) = supα∈(0,1]α ∙ ̃ (x).

C Ca

To ensure that C ̃ ∈ ℱ (X), it must satisfy:

Normality: There exists x0 ∈ X such that μC ̃(x0) = 1.

Upper semicontinuity: μC ̃ is upper semicontinuous in the topological space X.

Compact support: The set supp(μC ̃) = {x ∈ X: μC ̃(x) > 0} is compact.

If the level sets C ̃ are closed and form a nested family; these properties follow directly. For instance, the characteristic functions C ̃ are upper semicontinuous, and their weighted supremum preserves this property.

To guarantee the existence of the fuzzy difference C ̃ , we verify that:

I. The family {C ̃ } is nested: i.e., if 0 < α ≤ α ≤ 1, then C ̃ ⊆ C ̃

α α∈(0,1] 1 2 α2 α1

II. The level sets C ̃ are closed for each α, as intervals in a compact metric space are closed sets.

These properties are inherited from the fact A ̃ , B ̃ are compact intervals for all α, and the Hausdorff-type difference operation ⊝H preserves compactness and closedness of the resulting intervals.

For the Type-I and Type-II generalized differences, defined respectively by:

(1) (2)

C ̃ = cl(∪ A ̃ ⊝ B ̃ ), C ̃ = cl(conv(∪ A ̃ ⊝ B ̃ )).

α β∈[α,1] β H β α β∈[α,1] β H β

We apply similar arguments. The union of nested families of compact intervals remains bounded, and the

(1) (2)

closure ensures that C ̃ and C ̃ are closed. Furthermore, since the convex hull of a compact set is also compact, the construction yields α-level sets that are compact intervals.

Hence, the reconstructed membership functions:

μ (1)(x) = sup ( ]α ∙ (1)(x) , μ (2)(x) = sup ( ]α ∙ (2)(x).

C ̃ α∈ 0,1 C ̃ C ̃ α∈ 0,1 C ̃

̃ (1) ̃ (2)

They are upper semicontinuous and normal, ensuring C , C ∈ ℱcc(X).

The above analysis confirms that all chaos-preserving difference operators proposed in Sections 3 and 4 admit well-defined fuzzy representations. Their construction through α-level operations ensures structural compatibility with the fuzzy arithmetic framework and validates their application in fuzzy dynamical systems. These results reinforce the theoretical foundations behind the use of these operators for propagating and analyzing chaotic behavior under uncertainty.

One of the central implications of our approach is that the chaotic nature of a continuous map f: X → X, when extended to the fuzzy space ℱ(X) via the Zadeh extension f̂, can be preserved under carefully constructed fuzzy arithmetic operations. This preservation is not trivial and hinges critically on the compatibility between the α-cut of the fuzzy difference and the fact that it remains a compact interval, which guarantees that the resulting fuzzy set belongs to ℱ (X), thus making f̂ well-defined and dynamically meaningful.

From a broader perspective, this framework opens the door for investigating fuzzy chaos not as a side effect of uncertainty, but as an intrinsic property that can be explicitly structured and propagated within the fuzzy domain. Fuzzy chaos has potential applications in modeling complex systems where uncertainty is inherent, such as biological growth, economic fluctuations, or real-time control systems with noisy inputs.

Classical difference operations between fuzzy sets, such as pointwise arithmetic or level -wise subtraction without closure or convexity adjustments, often fail to preserve the necessary properties (e.g., upper semi-continuity, compact support) required for the result to remain in ℱ(X). In contrast, the operators ⊝NH, ⊝G1, ⊝G2 are specifically designed to maintain the topological and metric coherence of the fuzzy system.

Non-compatible operators may lead to α-level sets that are not nested, or even to fuzzy sets that do not satisfy normality. Non-compatible operators disrupt the decomposition theorem and undermine the possibility of establishing dynamical consistency across the fuzzy space . Our results, inspired by and extending prior work by [1], provide a corrective to these inconsistencies by ensuring a unified treatment of the operator and dynamic.

An important observation emerging from our computational experiments is the sensitivity of the resulting fuzzy dynamics to both the chosen difference operator and the structure of the α-cuts of the original fuzzy sets. For instance, when comparing ⊝NH and ⊝G2, we observe that the latter produces smoother α-cut evolution due to the convex hull operation, which tends to reduce irregularities. On the other hand, ⊝G1, while preserving more of the local variability, may generate α-cuts with larger spreads, increasing sensitivity to initial perturbations.

Moreover, the shape and positioning of the α-cuts in A ̃ and B ̃ has a direct impact on the complexity of ̂f(A ̃ ⊝ B ̃). Narrow or sharply peaked fuzzy numbers lead to more localized and potentially more stable behavior, while widespread triangular or trapezoidal fuzzy numbers increase the potential for dispersion and unpredictable growth under chaotic maps.

These findings suggest that, beyond mere compatibility, operator selection acts as a tunable parameter in shaping fuzzy chaotic behavior, opening up further research directions in control and modulation of chaos in fuzzy systems.

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.