New Type Modelling of the Circumscribed Self-Excited Spherical Attractor

Abstract

The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions.

1. Introduction

Classification of specific systems which are able to display chaotic behavior is one of the most interesting subjects in nonlinear problems [1,2,3,4,5]. Due to the importance of chaotic performance in dynamical systems, various studies have been conducted on the special properties of chaotic and nonlinear systems. In fact, different dynamical systems have been presented. For example, some interesting multistable models can be read in [6,7,8,9]. Moreover, some works on megastable dynamical systems have been done which can be seen in [10,11,12]. Additionally, on extreme multistable dynamical systems, notable investigation have been conducted in [13,14,15]. To read more on the other kind of such systems, see [16,17,18,19,20,21,22,23]. In this study, we aim to investigate the behaviour of numerical solutions of a new circumscribed self-excited spherical attractor which have been introduced in [24] with a specific type of fractional operator. Here, we present the structure of the mentioned system, which is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \frac{\mathrm{d}P}{\mathrm{d}t}=BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A,\hfill \\ \hfill & \frac{\mathrm{d}T}{\mathrm{d}t}=S^2\left(t\right)-CT\left(t\right)S\left(t\right),\hfill \\ \hfill & \frac{\mathrm{d}S}{\mathrm{d}t}=T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right),\hfill \end{array}\end{array}$$

(1)

In (1), P indicates the radial state, T symbolizes the azimuthal state, S displays the polar state and $A,B$ and C are parameters of the system.

Noninteger operators are the most practical means to describe the real phenoms due to having memory effects. For example, modeling and analysis of fractional order Ebola virus model with Mittag–Leffler kernel was reported in [25]. Furthermore, fractional modelling and simulations of the SEIR and Blood Coagulation systems can be read in [26]. Modeling and analysis of fractional order Zika model can be read in [27]. Moreover, new fractal fractional derivative on chemistry kinetics hires problem was done in [28]. In [29], the numerical solution to malaria fractional model with temporary immunity and relapse was conducted. A numerical and analytical study of SE (Is)(Ih) AR epidemic fractional-order COVID-19 model can be seen in [30]. By using noninteger operators, we have the ability to confer real phenoms that are more beneficial than classical-order ones [31,32,33,34,35,36]. Additionally, it is confirmed that the noninteger order models are able to describe chaotic behaviors precisely; consequently, such noninteger models have appeared in different categories [37,38,39]. The fractal–fractional idea considers the memory effect, the heterogeneity, and elascoviscosity of the medium, as well as the fractal geometry of the dynamic system. So, motivated by the above statements, it is logical to concentrate more on designing new models with fractional operators. In this work, we are going to model the system (1) with a fractal–fractional operator in the Atangana-Baleanu-Caputo (ABC) sense. In order to make fractional model of the system (1), some definitions are required. For more details see [40,41,42,43,44].

Definition 1

([45]). The He’s fractional derivative defined is defined as:

$${\mathcal{D}}_t^{\alpha }\left(u\left(t\right)\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}\frac{d^n}{d{t}^n}{\int }_{t_0}^t{(\tau -t)}^{n-\alpha -1}[u_0\left(\tau \right)-u\left(\tau \right)]\mathrm{d}\tau ,0<\alpha \le 1.$$

(2)

Definition 2

([46]). Suppose that the function $u\left(t\right)$ is continuous on $(a,b)$ and fractal differentiable on $(a,b)$ of order β. So, the fractal-fractional derivative of u of order α is defined as:

$$\exists {}^{FFM}{\mathcal{D}}_t^{\alpha ,\beta }\left(u\left(t\right)\right)=\frac{AB\left(\alpha \right)}{1-\alpha }{\displaystyle \frac{d}{d{t}^{\beta }}}{\int }_a^{t}u\left(z\right)E_{\alpha }\left[{\displaystyle \frac{-\alpha }{1-\alpha }}(t-z)\alpha \right]\mathrm{d}z,0<\alpha ,\beta \le 1,$$

(3)

where the Mittag–Leffler functions are defined as:

$${\mathbf{E}}_{\rho }\left(t\right)=\sum _{q=0}^{\infty }\frac{t^q}{\mathsf{\Gamma }(q\rho +1)},\rho \in {\mathbb{R}}^{+},t\in \mathbb{R},$$

(4)

as well as,

$${\mathbf{E}}_{\rho ,\sigma }\left(t\right)=\sum _{q=0}^{\infty }\frac{t^q}{\mathsf{\Gamma }(q\rho +\sigma )},\rho ,\sigma \in {\mathbb{R}}^{+},t\in \mathbb{R}$$

(5)

and $AB\left(\alpha \right)=1-\alpha +\frac{\alpha }{\mathsf{\Gamma }\left(\alpha \right)}$ is the normalization function satisfying $AB\left(0\right)=AB\left(1\right)=1$.

Definition 3

([46]). Consider continuous function $u\left(t\right)$ on $(a,b)$. So the fractal–fractional integral of u of order α is

$$\exists {}^{FFM}{\mathcal{I}}_t^{\alpha ,\beta }\left(u\left(t\right)\right)=\frac{\beta \alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{z}^{\beta -1}u\left(z\right){(t-z)}^{\alpha -1}dz+{\displaystyle \frac{\beta (1-\alpha )t^{\beta -1}}{AB\left(\alpha \right)}}u\left(t\right),0<\alpha ,\beta \le 1,$$

(6)

Many researchers have used FFM operators in their studies. For example, analysis of fractal–fractional differential equations can be seen in [47]. Another application of FFM operator to reaction-diffusion model was reported in [48]. Numerical solutions of the fractal–fractional Benjamin–Bona–Mahony equations can be read in [49]. Now, we are in the position of modelling the system (1) using the abovementioned derivative. So, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \exists {}_0^{FFM}{\mathcal{D}}_t^{\alpha ,\beta }S\left(t\right)=BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A,\hfill \\ \hfill & \exists {}_0^{FFM}{\mathcal{D}}_t^{\alpha ,\beta }E\left(t\right)=S^2\left(t\right)-CT\left(t\right)S\left(t\right),\hfill \\ \hfill & \exists {}_0^{FFM}{\mathcal{D}}_t^{\alpha ,\beta }I\left(t\right)=T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right),\hfill \end{array}\end{array}$$

(7)

with $P\left(0\right)=P_0,T\left(0\right)=T_0$ and $S\left(0\right)=S_0$. The rest of this numerical study contains 5 sections. Section 2 is dedicated to provide the existence and uniqueness of the results for the designed model. After that, stability analysis can be seen Section 3. Numerical results are provided in Section 4 and numerical discussion can be found in Section 5. Finally, we present the conclusion of this numerical research in Section 6.

2. Existence and Uniqueness Results

Now, we show the solution for system (2) using fixed-point theorem. Consider system (2) as

$$\left\{\begin{array}{c}\exists {}_0^{ABR}{D}_t^{\alpha }P\left(t\right)=\beta t^{\beta -1}\mathcal{Q}(t,P,T,S)\hfill \\ \exists {}_0^{ABR}{D}_t^{\alpha }T\left(t\right)=\beta t^{\beta -1}\mathcal{W}(t,P,T,S)\hfill \\ \exists {}_0^{ABR}{D}_t^{\alpha }S\left(t\right)=\beta t^{\beta -1}\mathcal{E}(t,P,T,S),\hfill \end{array}\right.$$

(8)

with

$$\left\{\begin{array}{c}\mathcal{Q}(t,P,T,S)=BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A\hfill \\ \mathcal{W}(t,P,T,S)=S^2\left(t\right)-CT\left(t\right)S\left(t\right),\hfill \\ \mathcal{M}(t,P,T,S)=T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right),\hfill \end{array}\right.$$

(9)

Now, (8) can be written as:

$$\left\{\begin{array}{c}\exists {}_0^{ABR}{D}_t^{\alpha }G\left(t\right)=\beta t^{\beta -1}J(t,G\left(t\right)),\hfill \\ G\left(0\right)=G_0\hfill \end{array}\right.$$

(10)

Substituting $\exists {}_0^{ABR}D{t}^{\alpha }$ by $\exists {}_0^{ABC}{D}_t^{\alpha \beta }$ and employing fractional integral, we obtain

$$G\left(t\right)=G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}\mathsf{\Lambda }(t,G\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}\mathsf{\Lambda }(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda ,$$

where

$$G\left(t\right)=\left\{\begin{array}{c}P\left(t\right)\\ T\left(t\right)\\ S\left(t\right)\end{array}\right.,G\left(0\right)=\left\{\begin{array}{c}P\left(0\right)\\ T\left(0\right)\\ S\left(0\right)\end{array}\right.,J(t,\mathsf{\Xi }\left(t\right))=\left\{\begin{array}{c}\mathcal{Q}(t,P,T,S)\\ \mathcal{W}(t,P,T,S)\\ \mathcal{M}(t,P,T,S)\end{array}\right.$$

Due to existence hypothesis, a Banach space $\mathcal{X}=\mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}$ is assigned, where $\mathcal{Y}=\mathbb{C}[0,\mathbb{T}]$ using

$$\parallel \mathsf{\Xi }\parallel =\underset{t\in [0,\mathbb{T}]}{max}|P\left(t\right)+T\left(t\right)+S\left(t\right)|,$$

Assign $\mathcal{L}:\mathcal{B}\to \mathcal{B}$ as:

$$\mathcal{L}\left(G\right)\left(t\right)=G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,G\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-\lambda )}^{\beta -1}\mathsf{\Lambda }(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda$$

(11)

Imposing Lipschitz condition $J(t,\mathsf{\Xi }(t\left)\right)$ as:

• For $G\in \mathcal{X}$, ∃ constants ${\mathcal{G}}_J>0$ and $M_{\mathsf{\Lambda }}$ such that

  $$\left|J(t,G\left(t\right))\right|\le {\mathcal{G}}_J\left|G\left(t\right)\right|+M_J,$$

  (12)
• For each $G,\overline{G}\in \mathcal{X}$, ∃ a constant ${\mathcal{L}}_J>0$ such that

  $$|J(t,G\left(t\right))-J(t,\overline{G}\left(t\right))|\le {\mathcal{L}}_J|G\left(t\right)-\overline{G}\left(t\right)|,$$

  (13)

Theorem 1.

Consider relation (12) works. Suppose $J:[0,\mathbb{T}]\times \mathcal{X}\to \mathbb{R}$ be a continuous. So, the presented system has solution.

Proof. 

We confirm $\mathcal{L}$ expressed via (11) is continuous. Because $\mathsf{\Lambda }$ is continuous, $\mathcal{L}$ is continuous.

Consider $\mathbb{H}=\{G\in \mathcal{X}:\parallel G\parallel \le \mathcal{R},\mathcal{R}>0\}$. For each $G\in \mathcal{X}$, we own

$$\begin{array}{cc}\hfill \parallel \mathcal{L}\left(G\right)\parallel =& \underset{t\in [0,\mathbb{T}]}{max}|G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J\alpha \beta AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right){\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda |\le G\left(0\right)+\hfill \\ \hfill & \frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}(\mathcal{G}\parallel G\parallel +M_J)\hfill \\ \hfill & +\underset{t\in [0,\mathbb{T}]}{max}\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}\left|J(\lambda ,G\left(\lambda \right))\right|\mathrm{d}\lambda \le G\left(0\right)+\frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}(\mathcal{G}\parallel \mathsf{\Theta }\parallel +M_J)\hfill \\ \hfill & +\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}(\mathcal{G}\parallel \mathsf{\Theta }\parallel +M_J){\mathbb{T}}^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )\le \mathcal{R},\hfill \end{array}$$

Then $\mathcal{L}$ is limited, which $\mathcal{H}(\alpha ,\beta )$ indicates function of beta. For equicontinuity $\mathcal{L}$, we consider $t_1\le t_2\le \mathbb{T}$. Thus, take

$$\begin{array}{cc}\hfill |\mathcal{L}\left(G\right)\left(t_2\right)-\mathcal{L}\left(G\right)\left(t_1\right)|& =|\frac{\beta t_2^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t_2,\mathsf{\Xi }\left(t_2\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_2}{\lambda }^{\beta -1}{(t_2-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda \hfill \\ \hfill & -\frac{\beta t_1^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t_1,G\left(t_2\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_1}{\lambda }^{\beta -1}{(t_1-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le \frac{\beta t_2^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}({\mathcal{G}}_{\mathsf{\Lambda }}\left|G\left(k\right)\right|+M_J)+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}\left(\mathcal{G}\right|G\left(t\right)|+M_J)t_2^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )\hfill \\ \hfill & -\frac{\beta t_1^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}({\mathcal{G}}_J\left|G\left(t\right)\right|+M_J)-\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}\left(\mathcal{G}\right|G\left(t\right)|+M_J)t_1^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta ),\hfill \end{array}$$

when $t_1\to t_2$, then $|\mathcal{L}\left(G\right)\left(t_2\right)-\mathcal{L}\left(G\right)\left(t_1\right)|\to 0$. As a result, we have

$$\parallel \mathcal{L}\left(G\right)\left(t_2\right)-\mathcal{L}\left(G\right)\left(t_1\right)\parallel \to 0,as{t}_1\to t_2$$

Hence $\mathcal{L}$ is equicontinuous. Then, via the Arzela–Ascoli theorem, it is continuous. So, via Schauder’s fixed-point result, the existence of solutions for the presented system is approved. □

Theorem 2.

Consider (13) works and by $\rho <1$, which

$$\rho =\left(\frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\mathbb{T}}^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )\right){\mathcal{G}}_{\mathsf{\Lambda }},$$

So, the existence of solutions for the given model is supported.

Proof. 

For $G,\overline{G}\in \mathcal{X}$, we own

$$\begin{array}{cc}\hfill \parallel \mathcal{L}\left(G\right)-\mathcal{L}\left(\overline{G}\right)\parallel & =\underset{t\in [0,\mathbb{T}]}{max}|\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}\left(J(t,G\left(t\right))-J(t,\overline{G}\left(t\right))\right)\hfill \\ \hfill & +\frac{\gamma \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}\mathrm{d}\lambda \left[J(\lambda ,G\left(\lambda \right))-J(\lambda ,\overline{G}\left(\lambda \right))\right]|\hfill \\ \hfill & \le \left[\left(\frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\mathbb{T}}^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )\right)\right]\parallel G-\overline{G}\le \rho \parallel G-\overline{G}\parallel ,\hfill \end{array}$$

Therefore, $\mathcal{L}$ is a contraction. Then, via the Banach contraction principle, the performed system has solution. □

3. Ulam–Hyres Stability

Now, we prove the Ulam–Hyres stability for the offered system.

Definition 4.

The offered system is Ulam–Hyres stable if $\exists {\aleph }_{\alpha ,\beta }\ge 0$ for $ϵ>0$ and for $G\in \mathbb{C}\left(\right[0,\mathbb{T},\mathbb{R})$ works

$$|\exists {}_0^{FFM}{D}_t^{\alpha ,\beta }G\left(t\right)-J(t,G\left(t\right))|\le ϵ,t\in [0,\mathbb{T}],$$

and there is solution $\mathsf{\Omega }\in \mathbb{C}\left(\right[0,\mathbb{T}],\mathbb{R})$ which

$$|G\left(t\right)-\mathsf{\Omega }\left(t\right)|\le {\aleph }_{\alpha ,\beta }ϵ,t\in [0,\mathbb{T}],$$

We consider a tiny disturbance $b\in \mathbb{C}[0,\mathbb{T}]$ which $\Phi \left(0\right)=0$. Consider

• $\left|b\right(t\left)\right|\le ϵ,forϵ>0$,
• $\exists {}_0^{FFM}{D}_t^{\alpha ,\beta }G\left(t\right)=J(t,G\left(t\right))+b\left(t\right)$

Lemma 1.

The solution for the disturbed system

$$\exists {}_0^{FFM}{D}_t^{\alpha ,\beta }G\left(t\right)=J(t,G\left(t\right))+b\left(t\right),G\left(0\right)=G_0,$$

accomplish following relation

$$|G\left(t\right)-\left(G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,G\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda \right)|\le x_{\alpha ,\beta }ϵ$$

where $x_{\alpha ,\beta }=\frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\mathbb{T}}^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )$.

Proof. 

The proof is clear; thus, we disregard it. □

Lemma 2.

Using (13) solutions of the considered system is Ulam–Hyres stable regardingaccomplish following relation $\rho <1$.

Proof. 

Suppose we have unique solution as $\mathsf{\Omega }\in \mathcal{X}$, and consider $\mathsf{\Xi }\in \mathcal{X}$ as the solution of the presented model, therefore

$$\begin{array}{cc}\hfill \left|G\right(t)-\mathsf{\Omega }(t\left)\right|& =|G\left(t\right)-\left[\mathsf{\Omega }\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,\mathsf{\Omega }\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,\mathsf{\Omega }\left(\lambda \right))\mathrm{d}\lambda \right]|\hfill \\ \hfill & \le |G\left(t\right)-\left[G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,G\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda \right]|\hfill \\ \hfill & +|G\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,G\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,G\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & -|\mathsf{\Omega }\left(0\right)+\frac{\beta t^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}J(t,\mathsf{\Omega }\left(t\right))+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{\lambda }^{\beta -1}{(t-\lambda )}^{\beta -1}J(\lambda ,\mathsf{\Omega }\left(\lambda \right))\mathrm{d}\lambda |\hfill \\ \hfill & \le x_{\alpha ,\beta }ϵ+\left(\frac{\beta {\mathbb{T}}^{\beta -1}(1-\alpha )}{AB\left(\alpha \right)}+\frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\mathbb{T}}^{\alpha +\beta -1}\mathcal{H}(\alpha ,\beta )\right){\mathcal{L}}_J|G\left(t\right)-\mathsf{\Omega }\left(t\right)|\hfill \\ \hfill & \le x_{\alpha ,\beta }ϵ+\rho |G\left(k\right)-\mathsf{\Omega }\left(k\right)|,\hfill \end{array}$$

So, we have

$$\parallel G-\mathsf{\Omega }\parallel \le x_{\alpha ,\beta }ϵ+\rho \parallel G-\mathsf{\Omega }\parallel .$$

Or

$$\parallel G-\mathsf{\Omega }\parallel \le {\aleph }_{\alpha ,\beta }ϵ,$$

which ${\aleph }_{\alpha ,\beta }=\frac{x_{\alpha ,\beta }}{1-\rho }$. So, the current system’s solution is Ulam–Hyres stable. □

4. Numerical Results and Simulations

We consider the fractal–fractional SEIR system as:

$$\begin{array}{cc}\hfill \exists {}_a^{FFM}{D}_t^{\alpha ,\beta }P\left(t\right)& =BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \exists {}_a^{FFM}{D}_t^{\alpha ,\beta }T\left(t\right)& =S^2\left(t\right)-CT\left(t\right)S\left(t\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \exists {}_a^{FFM}{D}_t^{\alpha ,\beta }S\left(t\right)& =T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right),\hfill \end{array}$$

(16)

We rewrite the model (11)–(13) as

$$\begin{array}{cc}\hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^{t}P\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =\beta t^{\beta -1}(BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A),\hfill \\ \hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^{t}T\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =\beta t^{\beta -1}\left(S^2\left(t\right)-CT\left(t\right)S\left(t\right)\right),\hfill \\ \hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^{t}S\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =\beta t^{\beta -1}(T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right)),\hfill \\ \hfill \end{array}$$

Taking

$$\begin{array}{cc}\hfill D1(t,P,T,S)& =BP\left(t\right)S\left(t\right)-AP\left(t\right)-10BS\left(t\right)+10A,\hfill \\ \hfill E1(t,P,T,S)& =S^2\left(t\right)-CT\left(t\right)S\left(t\right),\hfill \\ \hfill F1(t,P,T,S)& =T^2\left(t\right)-P\left(t\right)T\left(t\right)-10T\left(t\right),\hfill \end{array}$$

We have

$$\begin{array}{cc}\hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}k}{\int }_0^{t}P\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =D1(t,P,T,S),\hfill \\ \hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^{t}T\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =E1(t,P,T,S),\hfill \\ \hfill \frac{AB\left(\alpha \right)}{1-\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^{t}S\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right)\mathrm{d}\tau & =F1(t,P,T,S),\hfill \\ \hfill \end{array}$$

Now we use the AB integral on the above system, so we have

$$\begin{array}{cc}\hfill P\left(t\right)-P\left(0\right)& =\frac{1-\alpha }{AB\left(\alpha \right)}D1(t,P,T,S)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-\tau )}^{\alpha -1}D1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \\ \hfill T\left(t\right)-T\left(0\right)& =\frac{1-\alpha }{AB\left(\alpha \right)}E1(t,P,T,S)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-\tau )}^{\alpha -1}E1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \\ \hfill S\left(t\right)-S\left(0\right)& =\frac{1-\alpha }{AB\left(\alpha \right)}F1(t,SP,T,S)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-\tau )}^{\alpha -1}F1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \end{array}$$

By discretizing at $t_{n+1}$ we obtain

$$\begin{array}{cc}\hfill P^{n+1}& =P^0+\frac{1-\alpha }{AB\left(\alpha \right)}D1(t_{n+1},P^n,T^n,S^n)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}}{(t_{n+1}-\tau )}^{\alpha -1}D1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \\ \hfill T^{n+1}& =T^0+\frac{1-\alpha }{AB\left(\alpha \right)}E1(t_{n+1},P^n,T^n,S^n)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}}{(t_{n+1}-\tau )}^{\alpha -1}E1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \\ \hfill S^{n+1}& =S^0+\frac{1-\alpha }{AB\left(\alpha \right)}F1(t_{n+1},P^n,T^n,S^n)+\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}}{(t_{n+1}-\tau )}^{\alpha -1}F1(\tau ,P,T,S)\mathrm{d}\tau ,\hfill \end{array}$$

Which results in

$$\begin{array}{cc}\hfill P^{n+1}& =P^0+\frac{1-\alpha }{AB\left(\alpha \right)}D1(t_{n+1},P^n,T^n,S^n)\hfill \\ \hfill & +\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }D1(t_s,P^n,T^n,S^n)}{\mathsf{\Gamma }(\alpha +2)}({(n+1-s)}^{\alpha }(n-s+2+\alpha )-{(n-s)}^{\alpha }(n-s+2+\alpha ))\right]\hfill \\ \hfill & -\frac{\alpha }{AB\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }D1(t_{s-1},P^{n-1},T^{n-1},S^{n-1})}{\mathsf{\Gamma }(\alpha +2)}\left({(n+1-s)}^{\alpha +1}-{(n-s)}^{\alpha }(n-s+1+\alpha )\right)\right],\hfill \\ \hfill T^{n+1}& =T^0+\frac{1-\alpha }{AB\left(\alpha \right)}E1(t_{n+1},P^n,T^n,S^n)\hfill \\ \hfill & +\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }E1(t_s,P^n,T^n,S^n)}{\mathsf{\Gamma }(\alpha +2)}({(n+1-s)}^{\alpha }(n-s+2+\alpha )-{(n-s)}^{\alpha }(n-s+2+\alpha ))\right]\hfill \\ \hfill & -\frac{\alpha }{AB\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }E1(t_{s-1},P^{n-1},T^{n-1},S^{n-1})}{\mathsf{\Gamma }(\alpha +2)}\left({(n+1-s)}^{\alpha +1}-{(n-s)}^{\alpha }(n-s+1+\alpha )\right)\right],\hfill \\ \hfill S^{n+1}& =S^0+\frac{1-\alpha }{AB\left(\alpha \right)}F1(t_{n+1},P^n,T^n,S^n)\hfill \\ \hfill & +\frac{\alpha }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }F1(t_s,P^n,T^n,S^n)}{\mathsf{\Gamma }(\alpha +2)}({(n+1-s)}^{\alpha }(n-s+2+\alpha )-{(n-s)}^{\alpha }(n-s+2+\alpha ))\right]\hfill \\ \hfill & -\frac{\alpha }{AB\left(\alpha \right)}\sum _{s=0}^n\left[\frac{h^{\alpha }F1(t_{s-1},P^{n-1},T^{n-1},S^{n-1})}{\mathsf{\Gamma }(\alpha +2)}\left({(n+1-s)}^{\alpha +1}-{(n-s)}^{\alpha }(n-s+1+\alpha )\right)\right],\hfill \end{array}$$

5. Numerical Experiments

Now, we are going to show the application of the derived numerical scheme in Section 4. We consider two cases containing different fractal, fractional, and initial conditions (ICs) to achieve this aim. The parameters of the studied model are chosen as $A=2,B=25$ and $C=10$. For the first case we consider $P\left(0\right)=1,T\left(0\right)=1$ and $S\left(0\right)=1$(t) and $\beta =0.7,0.8,0.9$ and $\beta =1$ are selected as the fractal orders. In this case, we consider $\alpha =0.99$ as the fractional order. Figures related to this case can be found in Figure 1, Figure 2 and Figure 3. Figure 1 shows the approximate solutions of the state variables. 2D diagrams showing the numerical results can be seen in Figure 3. Furthermore, the chaotic behaviour of the approximate results can be observed in Figure 3.

For the second case, we hold $A=1.5,B=18,C=10,P\left(0\right)=1,T\left(0\right)=1$ and $S\left(0\right)=1$ and $\beta =0.65,0.75,0.85$ and $\beta =0.95$ are picked as the fractal orders. We take $\alpha =1$ as the fractional order to see how the behaviours of numerical solutions change. Similar to the first case, Figures reporting the numerical results are shown in Figure 4, Figure 5 and Figure 6. Figure 4 is responsible to reveal the approximate results of the state variables under the chosen initial conditions. 2D diagrams displaying the results can be viewed in Figure 5. Furthermore, the chaotic performance of the approximate results can be seen in Figure 6. The provided Figures reveals the changes which have occurred by altering the fractal–fractional orders and initial conditions.

6. Conclusions

During this study, we designed the fractal–fractional order model of the the circumscribed self-excited spherical attractor by employing an ABC fractal–fractional operator. Furthermore, Schauder’s and Banach’s fixed-point theorems were applied to display the existence of solutions for the suggested dynamical system. We discovered the solutions of Ulam–Hyres stability for the system using nonlinear functional analysis. Moreover, to obtain the approximate solutions of the considered problem and to see how the solutions behave, we employed the effective algorithm under various amounts of fractal and fractional orders. We plotted the graphs of solutions for each case to show how the results change under different conditions.