Abstract
In this paper, the newly developed FractalFractional derivative with power law kernel is used to analyse the dynamics of chaotic system based on a circuit design. The problem is modelled in terms of classical order nonlinear, coupled ordinary differential equations which is then generalized through FractalFractional derivative with power law kernel. Furthermore, several theoretical analyses such as model equilibria, existence, uniqueness, and Ulam stability of the system have been calculated. The highly nonlinear fractalfractional order system is then analyzed through a numerical technique using the MATLAB software. The graphical solutions are portrayed in two dimensional graphs and three dimensional phase portraits and explained in detail in the discussion section while some concluding remarks have been drawn from the current study. It is worth noting that fractalfractional differential operators can fastly converge the dynamics of chaotic system to its static equilibrium by adjusting the fractal and fractional parameters.
Introduction
In recent years, chaos theory has risen its importance, and numerous investigations on chaotic systems have been observed. Different chaotic systems^{1,2} have been studied, particularly hidden^{3} and multistability attractors^{4}. Chaosbased applications is a hot topic in science and engineering. Oscillators^{5}, steganography^{6}, synchronization^{7}, control^{8}, and parameter estimation have all employed chaotic systems. Some chaotic systems have a unique feature. They can have two or more coexisting attractors^{9}. Every attractor can be achieved because of the same range of parameters, depending on the initial condition chosen. Multistable chaotic systems^{10} are such systems, and they have gotten a lot of attention in the recent decade because of their potential applications^{11}. Several parameters of a multistable dynamical system are particularly sensitive to noise, initial conditions, and system parameters^{12}. Although multistability makes some engineering fields more challenging, such as bridge vibration and wing design, chaotic systems with multistability are extremely beneficial in secure communication^{13}. The appearance of hidden attractors has been associated with multistability^{14}. Stable equilibrium systems and systems with hidden attractors are examples of multistable systems^{15}. In multistable systems, selfexcited attractors can be found using the typical computational process, but hidden attractors cannot be predicted through the typical computational approach^{16}.
In recent decades, fractional calculus has been a significant mathematical technique used to represent critical challenges in various areas, including science, technology, and engineering such as, optimal power flow problems^{17}, nonlinear outputerror systems^{18}, recommender systems with chaotic ratings behavior^{19}, parameter estimation of nonlinear control autoregressive systems^{20}, power management involving windload chaos and uncertainties^{21}, Schrodinger equations^{22,23} and shallow water waves^{24}. The powerlaw function is involved in the Liouville–Caputo fractional derivative. As a result, it's common that the physical meaning of the fractional derivative is elusive or nonexistent. On the other hand, many researchers identified the physical significance of fractional derivatives^{25,26,27,28,29,30,31,32,33}. The nonlocality property of fractional derivatives is useful in predicting certain materials' memory and heredity properties in certain circumstances. As a result, the fractional derivative^{34,35,36,37} can be defined physically as a memory search. Scholars from virtually all fields of science, technology, and engineering have been studying nonlocal operators of differentiation over the past few decades since they have the capability to incorporate more complex phenomena. Fractional order chaotic systems have also been a highly prominent and attractive topic in recent years. They've been studied by many researchers^{38}. Chaotic systems, particularly fractional orderbased chaotic systems^{39}, have influenced attention as circuit design capability has improved with the introduction of integrated circuits^{40}.
The more complex the physical problems, the more complex the mathematical differentiation operators that were developed. In order to perform a single differentiation, the fractalfractional operator combines fractional differentiation with fractal derivative. These physical processes, for example, exhibit traits of a fractal nature. There is a sort of fractal derivative that may be found in practical mathematics, and it is one in which the variable is scaled in accordance with t^{a}. We thought that by coming up with this nonstandard derivative, we would be able to replicate aspects of the physical world that was no longer governed by the conventional physical rules^{26}. Two parameters representing powerlaw, exponential decay, or Mittag–Leffler memory processes were presented by Atangana^{41}, and the order parameter order stands for fractal derivative, which can be used to define the fractal dimension. This new set of derivatives also included an order parameter that stands for fractal derivative. Some of the previous work that has been presented in this new field has been applied either to a system of fractional ordinary differential equations or, in a few exceptional cases, to a straightforward equation for heat or diffusion^{42,43,44,45,46,47}.
Motivated from the above literature, it have been found that fractional and fractalfractional differential operator is a best tool to include the memory effect, crossover behavior and fractal characteristics all at once. Thus, there is a need to modify the existing chaotic systems through fractional or fractalfractional differential operators of various singular and nonsingular kernels. Therefore, in this research we have considered a chaotic system which is based on a circuit design with the following considerations:

The considered chaotic problem is modelled in terms of nonlinear system of ODEs.

The integer ODEs system is generalized through the fractalfractional differential operator of power law kernel.

The system dissipation and equilibria have been calculated.

The existence and uniqueness of the solutions have been proved.

The Ulam stability have been proved.

Numerical algorithm for the nonlinear fractalfractional system have been stated.

The obtained graphical results have been portraited through 2D and 3D phase portraits.
Mathematical preliminaries
This section provides an overview of some essential ideas and prepositions that will be used in subsequent parts of the investigation^{41,48,49}.
Definition 2.1
If \(g\left( \tau \right)\) is differentiable on \(\left( {a,b} \right)\) with fractional order α and fractal order \(\beta\) and \(g\left( \tau \right)\) is differentiable on the opened interval \(\left( {a,b} \right)\), then the FractalFractional derivative of \(g\left( \tau \right)\) with fractional order \(\alpha\) and fractal order β in Riemann–Liouville sense with power law can be written as:
where \(m  1 < \alpha \le m,\,\,\,0 < m  1 < \beta \le m.\)
Definition 2.2
If \(g\left( \tau \right)\) is fractal differentiable on \(\left( {a,b} \right)\) with fractal order \(\beta\) and \(g\left( \tau \right)\) is differentiable on the opened interval \(\left( {a,b} \right)\), then the FractalFractional derivative of \(g\left( \tau \right)\) with fractional order \(\alpha\) and fractal order β in Caputo sense with power law is:
where \(m  1 < \alpha \le m,\,\,\,0 < m  1 < \beta \le m.\)
Definition 2.3
The fractalfractional integral of Power law kernel can be expressed as follows:
Definition 2.4
Numerical solution to the Caputo ODE of fractional order is defined as follows^{50}:
Consider nonlinear fractional ODE:
The numerical algorithm for Eq. (6) is as follows:
Mathematical formulation
In this study, we investigate a new type of chaotic system that was presented in^{51}:
with initial conditions:
Here \(a,b\) and \(c\) are the three nonnegative parameters.
The integer order system given in Eq. (8) can be transformed to the fractalfractional order chaotic system by applying the definition given in Eq. (3). Thus, by applying the procedure given in^{41}, Eq. (8) will take the following form:
The fractional order and fractal dimension are represented by \(\alpha\) and \(\beta\), respectively, with \(0 < \alpha ,\beta \le 1\).
Dissipation and equilibria
The model in the system (8) is dissipative because:
For the fixed points of the chaotic system (8), we consider \(\frac{{dx_{1} }}{d\tau } = \frac{{dy_{1} }}{d\tau } = \frac{{dz_{1} }}{d\tau } = 0\). Hence system (8) can be written as:
Here we have only one equilibrium point, i.e., \(\left( {x_{1} ,y_{1} ,z_{1} } \right) = \left( {0,  \frac{1}{b},0} \right)\).
Existence and uniqueness of the fractalfractional model
The presence of a unique solution to the fractalfractional model, provided in Eq. (10), is established in this portion of the study. We could reformulate the Eq. (10) in the following manner because the integral is differentiable:
where \(0 < \alpha ,\beta \le 1\) and,
We can rewrite system (13) as:
where
Now, by replacing \({}_{0}^{RL} \wp_{\tau }^{\alpha }\) with \({}_{0}^{C} \wp_{\tau }^{\alpha }\) and applying the fractional integral, we get:
Let \({\rm P}\left( \psi \right)\) be a Banach space of the realvalued continuous functions with supremum norm defined on the interval \(\psi = \left[ {0,\Upsilon } \right]\) and \(\chi = {\rm P}\left( \psi \right) \times {\rm P}\left( \psi \right) \times {\rm P}\left( \psi \right)\) with the norm \(\left\ \Gamma \right\ = \sup \left\{ {\left {\Gamma \left( \tau \right)} \right:\,\tau \in \psi } \right\}\).
Now, transform the problem (10) into a fixedpoint problem. Define an operator \(\coprod :\chi \to \chi\) as:
For the existence theory, we use the following theorem^{52,53}.
Theorem 5.1
Assume that the operator \(\coprod :\chi \to \chi\) is completely continuous and the set defined by:
\(\mho \left( \coprod \right) = \left\{ {\Phi \in \chi :\,{\rm H} = \delta \coprod \left( {\rm H} \right);\,\delta \in \left[ {0,1} \right]} \right\}\) be bounded. Then \(\coprod\) has a fixed point \(\chi\).
Theorem 5.2
Let \(\Phi :\psi \times \chi \to {\mathbb{R}}\) is a continuous function. Then the operator \(\coprod\) is compact.
Proof
Let \({\rm M}\) us a bounded set in \(\chi\). Then there is \(\vartheta_{\Phi } > 0\) with \(\left {\Phi \left( {\tau ,{\rm H}\left( \tau \right)} \right)} \right \le \vartheta_{\Phi } ,\,\,\forall {\rm H} \in {\rm M}.\) So for any \({\rm H} \in {\rm M}\) one can get:
where \(B\left( {\alpha ,\beta } \right)\) is the beta function. Thus, \({\rm M}\left( \coprod \right)\) it is uniformly bounded.
Next, for the equicontinuity of the operator \(\coprod\), for any \(\tau_{1} ,\tau_{2} \in \psi\) and \({\rm H} \in {\rm M},\) we have:
Therefore, \(\coprod\) it is equicontinuous. Since \(\coprod\) it is a bounded and continuous operator. So, by ArzelaAscoli theorem is completely continuous.
Theorem 5.3
Let for all \(\tau \in \psi\) and \({\rm H} \in {\mathbb{R}}\), there is a real number \(\vartheta_{\Phi } > 0\) with \(\left {\Phi \left( {\tau ,\coprod \left( \tau \right)} \right)} \right \le \vartheta_{\Phi }\) . Then the considered model (10) has at least one solution in the given space \(\chi .\)
Proof
Consider a set \(\mho \left( \coprod \right) = \left\{ {\Phi \in \chi :\,{\rm H} = \delta \coprod \left( {\rm H} \right);\,\delta \in \left[ {0,1} \right]} \right\}\) and show that \(\mho\) it is bounded. Suppose \({\rm H} \in \mho\), then \({\rm H} = \delta \coprod \left( {\rm H} \right)\). For \(\tau \in \psi\), one can easily obtain:
Hence \(\mho\) is bounded. So, Theorems 5.1 and 5.2\(\coprod\) has at least one fixed point. Thus, the considered system (10) has at least one solution.
For further analysis, suppose the following hypothesis:
Hypothesis 5.1
There is a constant \(\hbar_{\Phi } > 0\) such that for every \({\rm H},\overline{\rm H} \in \chi\), we have:
For uniqueness, we use Banach's Contraction Theorem^{52,53}:
Theorem 5.4
Under Hypothesis 5.1and if \(\Xi < 1\). Then the solution of the considered system (10) is unique, where
Proof
Let we define \(\mathop {\max }\limits_{\tau \to \psi } \left {\Phi \left( {\tau ,0} \right)} \right = \omega_{\Phi } < \infty\) such that \(\varpi \ge \frac{{\beta {\rm T}^{\alpha + \beta  1} B\left( {\alpha ,\beta } \right)\omega_{\Phi } }}{{\Gamma \left( \alpha \right)  \beta {\rm T}^{\alpha + \beta  1} B\left( {\alpha ,\beta } \right)\hbar_{\Phi } }}\).
We show that \(\coprod \left( {\Re_{\omega } } \right) \subset \Re_{\omega }\) where \(\Re_{\omega } = \left\{ {{\rm H} \in \chi :\left\ {\rm H} \right\ \le \omega } \right\}\). For \({\rm H} \in \Re_{\omega }\), we have:
Let the operator \(\coprod :\chi \to \chi\) define it by (18). Then in view of Hypothesis 5.1, for every \(\tau \in \psi\) and for every \({\rm H},\overline{\rm H} \in \chi\), we have:
Hence \(\coprod\) is, a contraction from (25). Thus, the integral Eq. (17) has a unique solution, and so does the system (10) has a unique solution.
Ulam stability
In this section, we develop and present some results on the stability of the model (10). We will consider a small perturbation \(\Upsilon \in P\left( \psi \right)\) that depends only on the solution and \(\Upsilon \left( 0 \right) = 0\). Further:
Lemma 6.1
The modified problem's solution is as follows:
Satisfies the following relation:
Theorem 6.1
Under Hypothesis 5.1 and Lemma 6.1, the integral Eq. (17) solution is UlamHyers stable. Consequently, the considered system is UlamHyers stable if \(\Xi < 1\), where \(\Xi\) is given by (23).
Proof
Suppose \({\rm K} \in \chi\) be a unique solution and \({\rm H} \in \chi\) be any solution of (10), then:
We can write the above inequality as:
From (31), we can write:
Therefore, the result (32) concludes that the solution of (10) is UlamHyers stable, and therefore this supposed problem is UlamHyers stable as well.
Numerical scheme for the fractalfractional model
Consider the system (10)
We can also write:
Now by replacing \({}_{0}^{RL} \wp_{\tau }^{\alpha }\) with \({}_{0}^{C} \wp_{\tau }^{\alpha }\) and applying the procedure given in Eq. (7), we obtain the numerical algorithm in the following form:
Discussion and results
This section of the article contains graphical representations of the results obtained from the recent study. This study offers the dynamics of a newly designed circuit under the consideration of fractalfractional derivative of power law kernel. The problem has been analized for both fractional order parameter \(\alpha\) and fractal dimension \(\beta\). Simulations are performed and results are computed using MATLAB software. For the initial values, we have considered \(x_{1} \left( 0 \right) = 5.4,\,\,y_{1} \left( 0 \right) =  1.8\) and \(z_{1} \left( 0 \right) = 3.3\) while \(a = 0.1,\,b = 5\,\) and \(c = 0.3\).
Using \(\alpha = \beta = 1\) and keeping the other variables constant, Figs. 1 and 2 illustrate the classical behavior of a chaotic attractor. The influence that the fractional parameter \(\alpha\) has on the chaotic attractor behavior is seen in Figs. 3, 4, 5, 6, 7, 8. As seen in Figs. 3 and 4, the dynamics converges to its static equilibrium when fractional parameter \(\alpha\) is reduced while in the classical/integer order, it always evolve around its equilibrium. Further, it can be noticed that the time period and amplitude of the oscillations reduces by reducing \(\alpha\). The impact \(\alpha\) on the dynamic of chaotic attractors conforms to the same configuration as that which is shown in Figs. 5, 6, 7, 8. From the figures depicting the effects of effect \(\alpha\) on chaotic attractors, perhaps the current fractalfractional theory provides us with several solutions, rather than one as in classical/integer order. It provides us an alternative, and if we make the necessary adjustments, we may be able to get the best outcome possible by combining the results of our experiments with the theoretical data. Figures 9, 10, 11, 12, 13, 14 illustrates the effect that the fractal dimension \(\beta\) has on the dynamics of chaotic attractors. We made the observation that decreasing the value of the fractal dimension parameter \(\beta\) causes the chaotic attractor dynamics to persist for a longer period of time while the amplitude approximately remains the same. There are various 2D and 3D phase graphs produced in order to better understand the dynamical behavior of the considered chaotic system. For the purpose of making the differences between them more evident, we compare each figure to the classical order. We can observe the dynamics of the problemlimited cycles and periodic orbits that have been described based on the graphs that have been displayed so far.
Concluding remarks
This study has been carried out for the chaotic system based on circuit design presented by a threedimensional chaotic system. The problem is modelled in the form of nonlinear integer order ODEs along with the initial conditions. In addition, theoretical analyses of the problem such as equilibria and dissipation have been calculated while the existence and uniqueness of the solutions have been proved. In order to generalize the classical model, we used a fractalfractional differential operator of power law. To get the graphical solution, a numerical approach is also stated, after which it is implemented using the MATLAB software for simulations. Several graphs are used to illustrate the strange attractors that the chaotic system produces.
According to our observations, the chaotic system exhibits certain unusual behaviors as it fluctuates. Because of this, the outcomes of the current model can be significantly changed, and since fractionalorder parameters and fractal dimensions are involved in the current study to make it more general. In addition, it is obvious from the figures shown above that a reduction in fractional parameter \(\alpha\) will result in decrease in amplitude as well as time period of oscillations while a decrease in fractal parameter \(\beta\) would result in increase in the periods of the trajectories. Modifying these parameters is most likely to produce results that are comparable to those obtained numerically with the experimental results. It is noteworthy to notice that, when \(\alpha = \beta = 1\), we may recover the integer order results from the modified fractalfractional one.
Data availability
The data will be available from the corresponding author upon request.
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Acknowledgements
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Large Group Research Project under Grant RGP2/112/43.
Funding
National Natural Science Foundation of China (No. 71601072), the Fundamental Research Funds for the Universities of Henan Province (No. NSFRF210314) and Innovative Research Team of Henan Polytechnic University (No. T20227).
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N.K., Z.A., S.M., J.B. and H.A,, conceptualized the model, N.K., S.M., J.S. and M.D.A. draw the figures, N.K., J.B., and S.W.Y. wrote the whole manuscript, Z.A., J.B., H.A. and M.D.A. supervised the project. All the authors reviewed the final draft and approved for submission.
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Khan, N., Ahmad, Z., Shah, J. et al. Dynamics of chaotic system based on circuit design with Ulam stability through fractalfractional derivative with power law kernel. Sci Rep 13, 5043 (2023). https://doi.org/10.1038/s41598023320991
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DOI: https://doi.org/10.1038/s41598023320991
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