Global stability of local fractional Hénon-Lozi map using fixed point theory

1.   Introduction

Because the presence of these windows in certain chaotic areas implies that small changes in the parameters will end the chaos, the resilient chaotic system (map) is defined by the absence of episodic windows and simultaneous attractors in some parameter space neighborhood. This outcome reflects the fragile nature of this form of chaos. The chaotic dynamical system has been studied extensively by many academics because to its wide applications in science and engineering. The first discrete-time chaotic system ^[1] was suggested by H'enon. Lozi developed a unique discrete chaotic system utilizing the quadratic formula in the H'enon map with quasi linear term ^[2]. To improve the map in ^[3], it is advised to use a linear combination (convex type) of H'enon and Lozi maps. The H'enon-Lozi system has recently been developed using fractional calculus and the Capotu derivative operator in ^[4] and its extensions.

The notion of local fractional calculus (fractal calculus), first introduced by Kolwankar and Gangal for the regular fractional operators of Riemann-Liouville calculus^[5,6]. It was used to deal with non-differentiable formulas that appeared in engineering sciences, thermal and heat transforms another science^[6,7,8]. Various other ideas and specifics of fractal calculus were introduced, such as the geometric fractal. In the logical extensions of the definitions to the issue of local derivative on fractals, Yang et al. ^[6] constructed what is known as the cantor fractal.

Recognition of patterns using the fractal concept has been increased in the last decay. In ^[9], the authors joined procedure of altered edge detection techniques with box-counting for fractal feature extraction. Edge detection of fractal images can be studied as a significant area of investigation in fractal image processing ^[10]. License plate recognition is an emerging concept for real-time applications, is improved by using a new fractal series expansion ^[11]. Fractal entropy is developed and used in ^[12]. Finally, it employed to analyze images ^[13]. In this note, we use the fractal difference operator to define a new fractal Hénon-Lozi system (fractal map (FHLS)). Employing the FHLS to study the image recognition. Other studies are presented in^{[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]}.

To spread between the asymptotically stable variety and the chaotic array, we analyze bifurcation diagrams and present new irregular limitations on the fractal order. Utilizing bifurcation maps produced using forward and backward extension procedures, domains of fractal order spaces are confirmed, and the existence of simultaneous is demonstrated.

The paper is organized as follows: Section 2 deals with methods that we utilize in the sequel, Section 3 involves the main conditions for the global stability, Section 4 produces the test of our results and Section 5 indicates the conclusion.

2.   Methodology

In this section, we illustrate our methodology based on the fractal concept.

2.1.   Fractal concept

Let $C_\alpha(a, b), \alpha \in (0, 1]$ be a fractal set and let $f \in C_\alpha.$ For $\varepsilon > 0$ and $|\chi-\chi_0| < \delta$, the limit

is finite and available. It indicates that

The 2D-fractal differential operator is presented for $f(x, y)$ defining on the space of fractal $C_\alpha$ in $x$-direction

and in $y$-direction

For example,

has the following fractal derivatives

respectively. The results of Eq (2.1) are two vectors, one represents the fractal derivative of the image pixel in $x$-direction and the second one indicates the fractal derivative of the image pixel in $y$-direction. Note that the fractal difference term is in p.15 of ^[6].

Moreover, the definition of fractal integral is in p.38 of ^[6], as follows:

where $\delta(x_j)$ is a small number that indicates the difference between the modeled map and the real data.

More knowledge about fractal trigonometric functions is the fractal sine function

and the fractal cosine function

2.2.   Fractal Hénon-Lozi system (FHLS)

The Hénon system formulates a point $(x_n, y_n)$ in the plane and translates it to a new position (point) ^[1]

while the Lozi system indicates the following formula ^[2]

Considering the function

the following Hénon-Lozi system is recognized by ^[3]

Assuming the difference operator $\Delta x_{n} = x_{n+1}-x_n$, we obtain the difference system

Utilizing the fractal difference operator $\Delta^\alpha,$ we have FHLS

We present the next outcome.

Proposition 1. Consider FHLS in $(2.3)$. Then

● the solution is determined by the fractal integral system

● the critical points are

and

● the eigenvalues and eigenvectors of the Jacobian matrix of $(2.3)$ are

● the values of $\lambda_{1, 2}$ can be approximated in terms of $x_n$ as follows:

where $a = \ell-1-\wp \Bbbk$ and $b = \wp(1-\Bbbk).$

Proof. Applying the definition of fractal integral on the system FHLS in (2.3), we directly obtain the desired result, when $\delta_j^\alpha = 1$. For the second part, substitute $y = \ell x$ in the first equation, we have

which has two solutions when $\wp(1-\Bbbk)\neq 0.$ The third assertion is a direct application of the Jacobian matrix, while the last is a substitution of the values of the critical points in the values of the eigenvalues.

Example 1. We illustrate our example, as follow:

● Consider the following data $\wp = 1, \Bbbk = 0.5, \ell = 0.3$ then we have $a = -1.2$ and $b = 0.5.$ A computation yields $\lambda_{1, 2} = \Gamma(1+\alpha)(\pm 1.04).$

● Consider the following data $\wp = 1, \Bbbk = 0.3, \ell = 0.3$ then we have $a = -1.0$ and $b = 0.7.$ A computation yields $\lambda_{1, 2} = \Gamma(1+\alpha)(\pm 1.00).$

● Consider the following data $\wp = 2, \Bbbk = 0.3, \ell = 0.3$ then we have $a = -1.0$ and $b = 0.7.$ A computation yields $\lambda_{1, 2} = \Gamma(1+\alpha)(\pm 1.09).$

● Consider the following data $\wp = 3, \Bbbk = 0.3, \ell = 0.3$ then we have $a = -1.0$ and $b = 0.7.$ A computation yields $\lambda_{1, 2} = \Gamma(1+\alpha)(\pm 1.095).$

From the above example, for $\ell = 0.3, \Bbbk \in [0, 1]$ and for a positive constant $\wp < \infty,$ we conclude that $\lambda_{1, 2}\approx \pm\Gamma(1+\alpha).$ That is the eigenvalues depend on the fractal number $\alpha \in [0, 1].$ Also, around $\alpha = 0,$ we have

where $\gamma = 0.577$ presents the Euler constant and $\psi$ indicates the digamma function. Figure 1 shows the behavior of $\lambda_{1, 2}$ with the arch curve length

Some examples are illustrated in Figure 2.

2.3.   Chaotic system that involves FHLS

Because the presence of cyclic windows (periodic) in various chaotic indicators implies that even minor modifications in the parameters will destroy the chaos, robust chaos is defined by the absence of cycling values and coexisting attractors in a particular region of the parameter space. This impact hints at the vulnerability of this kind of disorder. Despite this, there are numerous functional applications where trustworthy chaotic mode operation is essential, such as connections and spreading the spectrum of switch-mode power supplies to avert electromagnetic interference. There are different methods to investigate robust chaos such as using polynomials, one parametric system, multi-parametric system ^[14,15].

There are two considerations of the chaotic system with robust chaos in $[0, 1]$ corresponding to FHLS (C-FHLS): 1D-parameter C-FHLS, which presents by letting $\alpha = \Bbbk \in [0, 1]$, and 2D-parameter C-FHLS, which indicates two different values of $\alpha \neq \Bbbk \in [0, 1].$

We shall study the two cases.

2.3.1.   1D-parameter C-FHLS

In this case, the C-FHLS is given by the formula

where $\alpha$ indicates the bifurcation parameter achieving the functional $\hslash_{\alpha}(x) = \alpha |x|+(1-\alpha) x^2.$ It is clear that when $\alpha = 0, \wp = 1.4$ and $\ell = 0.3,$ we have the difference Henon map and for $\alpha = 1,$ we get the difference Lozi map. It's worth noting that we get the chaotic map satisfying various forms of attractors for $\alpha \in (0, 1)$. Figure 2 shows the Lyapunov exponents and bifurcation diagram. The structure (see Figure 3) can be used to express the Lyapunov exponents.

where $\lambda_{1, 2}$ are the eigenvalues of the Jacobi matrix of (2.4),

There are two different eigenvalues

We note that the chaotic map $\Upsilon_\alpha$ is a piecewise smooth map, and that $\alpha \in [0, 1]$ will be partitioned into two domains in accordance with the shape of the chaotic map $\Upsilon_\alpha$.

where $x = 0$ is a line equation that divides the phase plane into two domains $R_1$ and $R_2.$ Note that, for $\alpha = 1,$ we have the solution ($x = \dfrac{10}{21}, y = \dfrac{ 1}{7}$).

When $\alpha \in [0, 1]$, System (2.4) is obviously chaotic. Furthermore, in System (2.4), the control parameter $\alpha$ reveals the transition of dynamical performances from the Henon to the Lozi attractors. Lastly, System (2.4) indicates the robust chaotic attractors for $\alpha \in [0.4616, 1)$ and absent when $\alpha = 1$ and $\alpha = 0.$

2.3.2.   2D-parameter C-FHLS

In this part, we consider 2D-parameter C-FHLS $(\alpha, \Bbbk),$ where $\alpha \neq \Bbbk$ in $[0, 1].$ The system becomes

where $\alpha$ and $\Bbbk$ present the bifurcation parameters satisfying the functional

Eventually, for all $\alpha \in [0, 1],$ The vector field of the chaotic map $\Upsilon_{\alpha, \Bbbk}$ owns two domains symbolized by:

where $x = 0$ is separated the plane into tow domains defining by $D_1$ and $D_2.$ Putting $\alpha = 0.4616$ then $\Gamma(1.4616) = 0.88$ and hence we obtain $\hslash_\Bbbk \approx 1$ if and only if

By the above analysis, together with $\wp = 1.4$ and $\ell = 0.3,$ we obtain the traditional Heno-Lozi system. This is true for all $\alpha \in [0.4616, 1].$ The Jacobian matrix is given by the formula

with two different eigenvalues (see Figures 4 and 5)

3.   Global stability

In this section, we investigate the stability of the suggested system. To study this fact, we check the solution under the existing of the fixed points.

3.1.   The set of the fixed points

For 1D-parameter C-FHLS, we have two fixed points when $\alpha = 0.4616$, as follows:

where $P_1 \in R_1$ and $P_2 \in R_2.$ Then Eq (2.5) yields

where in $\lambda(P_1) = -57 \in R_1.$ Similarly, we calculate the eigenvalues in $R_2$ to obtain

thus, we have $\lambda(P_2) = 5.68 \in R_2.$

Obviously, the true solutions of the system are the fixed points of the unified chaotic map (2.4)

For the 2D-parameter C-FHLS, we have the following set of fixed points

where $Q_1 \in D_2$ and $Q_2 \in D_1.$ Consequently, we get

which indicates the root $\Bbbk = \dfrac{85}{112}.$

Moreover, for the second fixed point, we have

It is clear that $0 < \Bbbk < 1,$ implies

which leads to $\lambda(Q_2)\in D_1.$ Clearly, the true solutions of the system are the fixed points of the unified chaotic map (2.6)

The next section deals with the stability conditions and global stability using the set of fixed points.

3.2.   Stability

The goal of this section is to look into the stability of the suggested systems.

Definition 1. Let $\chi$ be a solution of the fractal equation

such that $\|\chi\| < t_0.$ Then $\chi$ is called stable solution if there occurs a positive constant $\varrho > 0$ for all solution $\chi_1$ satisfying $\|\chi_1-\chi_0\| < \varrho$ and given $\rho > 0$ there occurs $0 < \delta \leq \varrho$ such that

Moreover, if

then $\chi$ is asymptotically stable.

The following are the outcomes:

Theorem 2. All solutions of the continuous linear system corresponding to $(2.4)$ (similarly for $(2.6)$)

are stable if and only if they are bounded. In addition, if the characteristic polynomial of $\Lambda$ is stable then the solutions are asymptotically stable.

Proof. Let's start by defining a two-variable matrix-valued function, which we'll call $\Theta$ as follows:

where $\mathfrak{I}$ is the identity matrix. By the formula of $\Im^\alpha,$ we obtain

If all solutions of System (3.3) are bounded, then there is a constant $\sigma > 0$ such that the fundamental matrix of (3.3) satisfies $\|\Theta\| < \sigma,$ where $\|.\|$ indicates the max norm. That is

Since the solution of (3.3) achieves

Then a computation implies

Therefore, all the outcomes are stable.

Conversely, the stability of the solutions brings that there occurs $\eta > 0$ corresponding to a given $\kappa > 0$ such that

That is all solutions are bounded.

The second part of the theorem, can be realized by the conclusion, which is an extended of Eq (1.9) in ^[6],

which leads to asymptotically stable solutions.

In the similar manner of Theorem 2, we have the following result

Theorem 3. All solutions of the continuous nonlinear system corresponding to $(2.4)$ (similarly for $(2.6)$)

are stable if and only if they are bounded and

In addition, if the characteristic polynomial of $\Lambda$ is stable then the solutions are asymptotically stable.

3.3.   Stabilization control

In this section, we'll discuss how to use control laws to stabilize the fractal map seen above.

Theorem 4. Under the 1D-control law, the 1D-parameter fractal map

can be controlled by

Proof. The time-varying control parameter $U_x(t)$ is utilized in the controlled fractional order Henon-Lozi map, which is defined as

This yields the system

In matrix form, we get

The eigenvalues corresponding to (3.9) are $\lambda_1 = \lambda_2 = -\Gamma(1+\alpha).$ Thus, the zero solution of (3.9) is bounded. According to Theorem 2, the zero solution is asymptotically stable, and so the system is stabilized.

We get the following outcome from the same conclusion.

Theorem 5. The 2D- parameter fractal map

can be governed by the 1D-control legislation

4.   Generalized testing

In this section, we generalize the zero-one test by using the fractal trigonometric functions to analyze our results. The zero-one test is formulated by proposing by Gottwald and Melbourne ^[33,34,35]. In view of Proposition 1, we select the solution

Define two translation terms

Next, we formulate the fractal square displacement for $x_n,$ where the control law is defined $\mu_\tau(n)$ by:

Hence, the asymptotic growth is formulated by

Obviously, the asymptotic growth ratio approaches to zero in the ordinary case, while heads for one in the chaotic case. For $\alpha = 1$ (the ordinary case), we have the generalized Puiseux series by

For all $\alpha \in (0, 1)$, the gamma function can be approximated by

where $\gamma$ is the Euler number and $\zeta$ is the zeta function. Thus we obtain the (see Figure 6)

5.   Conclusions

A new fractal-order chaotic map has been devised, displaying a variety of marvels such as chaos and synchronized attractions. The Yong-like difference operator was used to analyze the FHLS map. Bifurcation graphs have been provided for both $\alpha$ and $\Bbbk$ parameters. The FHLS map rests chaotic across a wide range of parameter space, as a result of the consequences. The existence of solutions is discussed using the system's fixed points; the necessary requirements for stability are shown; and lastly, the system was tested for convergence. As a result, it has a lot of applications in the engineering field, such as transistors, resistors, conductors, capacitors, and diodes in an electrical circuit. Furthermore, a spectacular 1D-control legislation for stabilizing the conditions of the recommended map was established. The fixed point theory indicates that there is global stability. The generalized zero-one test is used to assess the stability of solutions and provides testing. As future works, we aim to consider two different four parameters dimensional dynamical systems with Caputo fractional derivative or its modifications. In this case, the control system will be studied as 2D-system.

Conflict of interest

The authors declare no conflict of interest.