The dynamic and discrete systems of variable fractional order in the sense of the Lozi structure map

: The variable fractional Lozi map (VFLM) and the variable fractional ﬂow map are two separate systems that we propose in this inquiry. We study several key dynamics of these maps. We also investigate the su ﬃ cient and necessary requirements for the stability and asymptotic stability of the variable fractional dynamic systems. As a result, we provide VFLM with the necessary criteria to produce stable and asymptotically stable zero solutions. Furthermore, we propose a combination of these maps in control rules intended to stabilize the system. In this analysis, we take the 1D- and 2D-controller laws as givens.


Introduction
A significant issue in elasticity theory is how to represent the characteristics of materials that could alter as a result of various activities. For this reason, a number of researchers have put forth variable-order fractional operators (VOFO), or operators whose order varies over time or in response to particular state variables. The interest can be traced back to the early work of Samko and Ross [22,24], and further advancements in the field of VOFOs were made at the beginning of the previous decade [17,27]. The VOFOs that explicitly depend on a temperature field are modeled as random noise were suggested [11]. As an alternative to the model [5], Beltempo et al. [4] have discussed the use of VOFO to handle the aging of materials, such as concrete and other solid materials and polymers. In order to simulate real-world structures on computers, VOFO has been used to model the aging of materials and provide relaxation functions that are mathematically consistent and can be coded finite-element specific algorithms [6,7]. VOFOs are clearly a special case of ordinary and fractional differential equations, which are the generalization of these classes when the fractional order is a constant. In reality, a lot of physics, monetary, and biological processes seem to behave in fractional orders, which can change over time and/or space [16,20,25,28].
Current investigation has focused on the stabilization and synchronization of two recently suggested fractional order ( constant fractional power) chaotic maps, the generalized 3D fractional Logistic [23], Henon and Lozi maps [8,13,14], and the 2D fractional Logistic, Henon and Lozi map [2,12,21]. In this effort, we shall consider the VOFOs to generalize the Lozi system (similarly for other maps) to deliver the variable fractional Lozi map (VFLM) and the variable fractional flow map. The stability and stabilizing of the system are studied, and some variable fractional order examples are illustrated in the sequel. Finally, an analysis is presented to study the proposed system involving the equilibrium points and the set of fixed points.

Methods
We have the following concepts:

The VOFOs
The classic arbitrary integration operator is considered by the integral formula For a general function ϕ and 0 < ℘ < 1, the classical arbitrary differentiation is given by the formula The Caputo arbitrary differential operator of order 0 < ℘ < 1 is formulated by the equation The VOFOs of the above operators are presented in [26]. Let ℘(η) be a continuous function, then the VOFO for integration is written by the equation The classical fractional derivative is formulated by the structure and for the Caputo operator is given by the formula We proceed to introduce the VOFO for discrete formula type Caputo calculus: Definition 2.1. The VOFO in terms of Caputo calculus is given by [5]: . Furthermore, the term η (℘(η)) is given by the fraction The corresponding discrete integral equation can be formulated by the sum Note that when ℘(η) is a constant function, we obtain the discrete form in [1].

The VFLM
By using the VOFOs in the above part, we have the VFLM. Rene Lozi introduced the Lozi chaotic map in [18] and it is formulated by the structure where v ∈ N, φ(v) and ψ(v) are the functions and the certain parameters α and β which are in R. It is discovered that (2.3) contains a chaotic attractor with (α, β) = (1.7, 0.5). In view of Definition 2.1, we have the following VFLM Clearly, when ℘(η) is a constant function, we obtain the system in [15] (2.5) Now the integral difference in the above section implies indicates the discrete kernel function. Note that when ℘(η) is a constant function, we obtain the system in [15], as follows: . (2.7) For numerical structure, when ℵ = 0, we get (see Figures 1 and 2) (2.8) Figure 3 shows the solution of the discrete systems, when The iteration is running from 1 to 1000.
The iteration runs from 1 to 1000.

Stability
In this section, we look into the VFLM's overall stability.

Linear system
We begin with the definition below, which can be expanded into the n-dimensional space.
Definition 3.1. Suppose that g(η) = g(η; η 0 , g 0 ) is an outcome of the following equation Then g is known as a stable outcome if there arises a positive real number w > 0 for all outcomes g(η) = g(η; η 0 , g 0 ) ∈ S owing the inequality and for a given number ε > 0 there occurs 0 < ≤ w with then the solution g is asymptotically stable.
The following are the consequences: Then all its outcomes are stable if and only if they are bounded. Moreover, if the characteristic polynomial corresponds to Ξ is stable then the outcomes are asymptotically stable.
Proof. Via creating a matrix-valued function with two variables, Υ, as follows: such that I d presents the identity matrix. In view of ℘ , we conclude that Now, let the outcomes of system (3.2) be bounded. As a consequence, there occurs a fixed number κ > 0 satisfying the inequality Υ < κ, where . represents the max norm. This implies that Consequently, we obtain Similarly, we have Thus, all the solutions of system (3.1) are stable.
Contrariwise, the stability of the outcomes, including the zero solution yields that for a positive number ε > 0 there is a positive constant υ satisfying the inequality Which leads to all solutions are bounded. Now, since the characteristic polynomial corresponding to Ξ is stable then the outcomes are asymptotically stable, because Similarly, for ψ, we have which implies the asymptotically stable outcomes. Proof. If Ξ < 1 and all its eigenvalues are in the interval [0, 1], then it is an invertible positive contraction [19]. Then Ξ −1 2×2 − I d is positive semi-definite, with Det(Ξ 2×2 ) > 0 (see the proof of Proposition 3.5 [9]). This suggests that the Ξ characteristic polynomial is real stable. We have the solutions that are asymptotically stable in light of Theorem 3.2.
Non-homogeneous case is given in the next outcome.
An application of Theorem 3.4, is in the following outcome Corollary 3.5. Assume that the sup norm Ξ < 1 and all its eigenvalues are in the interval [0, 1] and X < ϑ, ϑ > 0. Consequently, system (3.3) admits asymptotically stable solutions whenever

Stability of VFLM
In this part, we discuss the stability of VFLM (continuous case) using the above results. We deliver the sufficient condition on the coefficients of the system in the following result: Theorem 3.6. Consider the continuous system VFLM If the following inequalities are satisfied Proof. In matrix form, the system (3.3) becomes The characteristic polynomial takes the formula −β + αη + η 2 = 0, with the two differences roots By letting α 2 + 4β ≥ 0, we have r 1 < 0 providing α > 0. Moreover, we have Hence, the characteristic polynomial is stable. All the conditions of Theorem 3.4 are achieved, then all the solutions are asymptotically stable. Figure 4 shows the dynamic of the characteristic polynomials for different values of α > 0 and β > 0. Moreover, we have the following generalization result, with a proof similar to the one in [10]. Theorem 3.7. Consider the linear fractional-order discrete-time system, as follows: Then the zero equilibrium is asymptotically stable if and only if for all the eigenvalues λ of Π n×n . Moreover, if all the complex eigenvalues are in the open unit disk or all the real eigenvalues in the unit interval [0,1], then the system is stable.
Proof. The first part of the theorem is similar to the proof of Theorem 1.4 [10]. For the second part, since all the eigenvalues are in the open unit disk, then the characteristic polynomial is stable which leads to the stability of the system.
Theorem 3.8. Assume that Xi's characteristic polynomial is stable. If then all the solutions of system (3.6) are stable. Moreover, if κµ < e, then the zero solution of system (3.6) is asymptotically stable.
Proof. In view the variable fractional integral formula, we have . This yields that the solutions are bounded, then in view of the proof of Theorem 3.2, the solutions are stable. For the second part, we have Similarly, for the variable ψ, we have Hence, the zero solution is asymptotically stable.

Stabilizing
The exploration of chaotic systems, whether in discrete time or continuously, revolves around the development of control mechanisms to achieve stability. In this part, we will discuss some potential nonlinear control rules for stabilizing the aforementioned arbitrary order discrete-time systems. Stabilization involves applying a novel time-altering parameter, ð(η), to the particular system's states and devising an adaptive closed-form method for these parameters to quickly push the system's states to zero.
Proof. In the controlled VFLM, which is represented by the symbol, the time-altering control parameter ðφ(η) is employed. (4.1) The simplified dynamics are obtained by substituting the proposed control law ð φ (x) into (4.1) Proof. The time-varying control parameter (U φ (η), V ψ (η)) is employed in the 2D-controlled VFLM, which is formulated as follows: The simplified dynamics are obtained by substituting the proposed 2D-control law (U φ (η), V ψ (η)) into (4.3) ).

(4.4)
Thus, the set of eigenvalues is bounded, with Ξ < 1, where Therefore, the zero solution is asymptotically stable in light of Corollary 3.3.
Theorem 4.2 is a one-dimensional parametric 2D-control law of the VFLM, which is α. The next theorem describes the two-dimensional parametric 2D-control law VFLM's stabilizing parameters, α and β.

Analysis of system VFLM (2.4)
System (2.4) can be viewed as two dynamical systems The solutions of these systems occurred in two different domains.

Conclusions
We suggested two distinct systems in this study: The variable fractional Lozi map (VFLM) and the variable fractional flow map. We looked into a few of these maps' crucial dynamics. Additionally, we looked into the prerequisites that the variable fractional dynamic systems must meet to be stable and asymptotically stable. To obtain a stable and asymptotically stable zero solutions, we therefore imposed the VFLM with the essential requirements. To stabilize the system, we also suggested combining these maps with control rules. The 1D and 2D controller rules were taken as givens in this analysis. For future works, one can extend this analysis into other types of variable fractional calculus such as the ABC-operator [3].