Application of Caputo-Fabrizio derivative in circuit realization

A. M. Alqahtani; Shivani Sharma; Arun Chaudhary; Aditya Sharma; A. M. Alqahtani; Shivani Sharma; Arun Chaudhary; Aditya Sharma

doi:10.3934/math.2025113

AIMS Mathematics

2025, Volume 10, Issue 2: 2415-2443. doi: 10.3934/math.2025113

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Research article

Application of Caputo-Fabrizio derivative in circuit realization

1.
Department of Mathematics, Shaqra University, Riyadh, Saudi Arabia
2.
Department of Mathematics, Amity University Rajasthan, Jaipur, India
3.
Department of Mathematics, Rajdhani College, University of Delhi, Delhi, India
4.
Department of Electronics Science, University of Delhi, Delhi, India

Received: 17 November 2024 Revised: 06 December 2024 Accepted: 09 December 2024 Published: 11 February 2025
MSC : 34H10, 34C28, 37M05, 26A33

This work incorporates a hyperchaotic system with fractional order Caputo-Fabrizio derivative. A comprehensive proof of the existence followed by the uniqueness is detailed and the simulations are performed at distinct fractional orders. Our results are strengthened by bridging a gap between the theoretical problems and real-world applications. For this, we have implemented a 4D fractional order Caputo-Fabrizio hyperchaotic system to circuit equations. The chaotic behaviors obtained through circuit realization are compared with the phase portrait diagrams obtained by simulating the fractional order Caputo-Fabrizio hyperchaotic system. In this work, we establish a foundation for future advancements in circuit integration and the development of complex control systems. Subsequently, we can enhance our comprehension of fractional order hyperchaotic systems by incorporating with practical experiments.

Keywords:

Citation: A. M. Alqahtani, Shivani Sharma, Arun Chaudhary, Aditya Sharma. Application of Caputo-Fabrizio derivative in circuit realization[J]. AIMS Mathematics, 2025, 10(2): 2415-2443. doi: 10.3934/math.2025113

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Abstract

1. Introduction

In recent years, fractional derivatives ^[1] have emerged as a valuable tool in the modeling and analysis of complex systems in various scientific and technical disciplines. Unlike traditional integer-order derivatives, fractional derivatives provide a more flexible and accurate framework for representing the memory and hereditary characteristics of diverse processes. The study of fractional derivatives encompasses various definitions tailored to specific contexts and applications. These definitions incorporate noninteger orders, introducing nuanced interpretations crucial for modeling complex phenomena with memory and hereditary properties. From the Riemann-Liouville ^[1] definitions to Caputo ^[1], Caputo-Fabrizio ^[2], and Atangana-Baleanu ^[3] definitions, each offers distinct advantages in mathematical rigor, computational feasibility, and compatibility with diverse physical systems. An understanding of these definitions is quite essential for effectively applying fractional calculus across disciplines such as physics ^[4,5,6], engineering ^[7,8,9], biology ^[10,11,12], etc., where they are crucial in enhancing analytical precision and improving modeling efficacy. The theory of fractional derivatives can also be applied to fractals. Fractals ^[13] are often characterized by self-similarity where parts of a fractal resemble the whole structure at different scales. A fractal derivative ^[14] generalizes the concept of a derivative to the irregular, non-smooth structures that are common in fractals. In classical calculus, derivative measures the rate of change of a function. However, in fractals, due to irregularities and infinite details, traditional differentiation methods are not directly applicable. A two-scale fractal derivative ^[15] refers to the generalization of a classical derivative, particularly in the context of fractal geometry, where the concept of smoothness and differentiability becomes more nuanced due to self-similarity and irregularity of fractals. A two-scale fractional derivative analyzes the rate of change over two distinct fractional orders providing insight into the multi-scale nature of the fractal. Applications of two-scale fractal derivatives can be seen in ^[16,17].

The Caputo-Fabrizio derivative is a notable extension of the Caputo derivative, enabling a more precise depiction of physical systems that exhibit the memory effect. This derivative has a non-singular kernel, making it ideal for modeling real-world problems that exhibit an exponential decay behavior. Additionally, the Caputo-Fabrizio derivative provides precise solutions to nonlinear differential equations in a shorter amount of time. Its versatility and theoretical robustness continue to contribute significantly to advance our understanding and application ^[18,19,20] of fractional calculus in practical settings.

Hyperchaotic systems have undergone significant advancements since its first introduction by Rossler ^[21]. These include the development of distinct systems such as one proposed by Erkan et al. ^[22] in two dimensions for image encryption. A hyperchaotic system in 2D was constructed by Zhu et al. ^[23] which has applications in color image encryption and pseudo-random number generation. Tang et al. ^[24] gave a 2D chaotic system for secure communications. Further advancements include 3D hyperchaotic systems studied by Li et al. ^[25] for a secure transmission. In this progression, a system was developed by Zhi et al. ^[26] through the extension of a 3D autonomous chaotic framework. Sahasrabuddhe and Laiphrakpam ^[27] studied an application of a 3D hyperchaotic system in multiple-image encryption. Wang et al. ^[28] coined a dynamics in a 4D hyperchaotic system. Nestor et al. ^[29] introduced a 4D hyperchaotic system encompassing dynamic analysis, synchronization, and its application. The complex behavior of hyperchaotic systems has been widely used in fields like information processing, electronics, neuroscience, communications, and beyond. These applications include image encryption ^[30,31], random number generation ^[32,33], secure communication ^[24,34,35], audio encryption ^[36], and video encryption ^[37].

Fractional derivatives ^[1] also unfold a vast array of applications within hyperchaotic systems, offering a dynamic toolkit to explore and harness their intricate dynamics. By incorporating derivatives of fractional order, hyperchaotic systems can model complex behaviors that integer-order models might not capture completely. This incorporation adds further complexity to the dynamics of hyperchaotic systems encompassing, anomalous diffusion, nonlocal interactions, and long-range memory effects.

Yousefpour et al. ^[38] presented a fractional-order hyperchaotic economic system. Pan et al. ^[39] studied the chaos synchronization with two different fractional orders. Shi et al. ^[40] investigated the dynamics of a fractional-order hyperchaotic system and identified its potential for use in image encryption. The aforementioned systems facilitate the development of advanced control techniques, synchronization strategies ^[41,42], and encryption techniques for a broad spectrum of real-world applications including confidential communications ^[43,44] and image encoding ^[45,46], among others.

The following hyperchaotic system was proposed by Fu et al. ^[47] in 4D

$\begin{eqnarray} &&\frac{d\, x}{d\, t} = a\, (y-x)+u, \, \, \, \, \, \, \, \, \, \, \, \, \, x(0) = x_{0}, \\ && \frac{d\, y}{d\, t} = c\, y - 10\, x\, z, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, y(0) = y_{0}, \\ && \frac{d\, z}{d\, t} = -b\, z + 10\, x\, y, \, \, \, \, \, \, \, \, \, \, \, \, \, z(0) = z_{0}, \\ && \frac{d\, u}{d\, t} = d\, y + x^{2}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, u(0) = u_{0}, \end{eqnarray}$

(1.1)

where $a$ , $b$ , $c$ are constants, $d > 0$ is a variable parameter and x, y, z, u are driving variables. This hyperchaotic system posseses stability, chaotic behavior, and many more physical properties. In their work, experimental validation was performed using Matlab, Multisim, and STM32. The outcomes of their experiments consistently corroborate the theoretical basis for employing this hyperchaotic system in hands-on applications. Subsequently, utilizing the keys derived from this system's hyperchaotic generator, audio encoding is implemented on the STM32 integrated hardware system using a cross-XOR mechanism. Their findings demonstrate the feasibility of the audio encoding technique utilizing this hyperchaotic system highlighting its simplicity of implementation, nonlinear properties, and algorithmic complexity. These attributes render it applicable not only to audio encryption but also to image encryption, video encryption, and other related domains.

As the fractional derivatives enhance complexity in the chaotic behavior of a system, in this work, we extend the hyperchaotic system proposed by Fu et al. ^[47] to a hyperchaotic system of fractional order using the Caputo-Fabrizio derivative. The existence and uniqueness of the solution of this fractional order hyperchaotic system are detailed and simulations are performed for different fractional orders. This study extends beyond theoretical exploration and aims for real-world applications. For this, we have implemented this 4D fractional order Caputo-Fabrizio hyperchaotic system into circuit equations. We use a digital oscilloscope in Mutisim to realize the circuit and Matlab to perform the simulations for the fractional order Caputo-Fabrizio hyperchaotic system. The chaotic behaviors obtained through circuit realization are compared with the phase portrait diagrams obtained by simulating the fractional order hyperchaotic system.

2. Preliminaries

Definition 1. ^[2] For an integrable function $g$ on $\mathbb{R}$ , the Caputo-Fabrizio derivative of order $\varepsilon$ is given as

$\begin{equation} ^{CF} \mathbb{D}_{t}^{\varepsilon}(g(t)) = \frac{Q(\varepsilon)}{1-\varepsilon}\int_{0}^{t}exp\bigg(\frac{-\varepsilon(t-s)}{1-\varepsilon}\bigg) g'(s)ds, \, \, \, \, \, \, \, 0 < \varepsilon < 1, \end{equation}$

(2.1)

where $Q(\varepsilon)$ is a normalizing function such that $Q(0) = 1 = Q(1)$ .

Definition 2. ^[2] Let $g$ be a function which is integrable on $\mathbb{R}$ , with $t > 0$ , $0 < \varepsilon < 1$ , and the Caputo-Fabrizio fractional integral of order $\varepsilon$ is:

$\begin{equation} \mathcal{I ^{CF}}_{t}^{\varepsilon}\{g(t)\} = \Psi\, g(t) + \vartheta \int_{0}^{t} g(s) ds, \, \, \, \, \, \, t > 0, \end{equation}$

(2.2)

where $\Psi = \frac{2\, (1-\varepsilon)}{(2-\varepsilon)\, Q(\varepsilon)}$ , $\vartheta = \frac{2\, \varepsilon}{(2-\varepsilon)\, Q(\varepsilon)}$ , and $Q(0) = Q(1) = 1$ .

3. Fractional order hyperchaotic system with Caputo-Fabrizio derivative

In this part, we extend the 4D hyperchaotic system to the fractional order hyperchaotic system by using the Caputo-Fabrizio derivative. The fractional order hyperchaotic system with the Caputo-Fabrizio derivative is as follows:

$\begin{eqnarray} _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{x(t)\}& = & a\, (y-x)+u, \, \, \, \, \, \, \, \, \, \, \, \, \, x(0) = x_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{y(t)\}& = & c\, y - 10\, x\, z, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, y(0) = y_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{z(t)\}& = & -b\, z + 10\, x\, y, \, \, \, \, \, \, \, \, \, \, \, \, \, \, z(0) = z_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{u(t)\}& = & d\, y + x^{2}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, u(0) = u_{0}, \end{eqnarray}$

(3.1)

where $a$ , $b$ , $c$ are constants, $d > 0$ is a variable parameter, and x, y, z, and u are driving variables.

3.1. Existence & uniqueness

We present a thorough proof establishing the existence of a unique solution for the 4D fractional order Caputo-Fabrizio hyperchaotic system.

Theorem 1. If for some constans $k_{1}, k_{2}, k_{3}, k_{4}$ , the functions $V_{1}(t, x)$ , $V_{2}(t, y)$ , $V_{3}(t, z)$ , $V_{4}(t, u)$ satisfy the condition given below:

$\begin{eqnarray} &&\|V_{1}(t, x_{1}) - V_{1} (t, x_{2}) \|\leq k_{1} \|x_{1}(t)-x_{2}(t)\|, \\ &&\|V_{2}(t, y_{1}) - V_{2} (t, y_{2}) \|\leq k_{2} \|y_{1}(t)-y_{2}(t)\|, \\ &&\|V_{3}(t, z_{1}) - V_{3} (t, z_{2}) \|\leq k_{3} \|z_{1}(t)-z_{2}(t)\|, \\ &&\|V_{4}(t, u_{1}) - V_{4} (t, u_{2}) \|\leq k_{4} \|u_{1}(t)-u_{2}(t)\|, \end{eqnarray}$

(3.2)

where

$\begin{eqnarray} V_{1}(t, x)& = & a\, (y-x)+u, \\ V_{2}(t, y)& = & c\, y - 10\, x\, z, \\ V_{3}(t, z)& = & -b\, z + 10\, x\, y, \\ V_{4}(t, u)& = & d\, y + x^{2}. \end{eqnarray}$

(3.3)

The conditions given by Eq (3.2) are Lipschitz conditions. The above conditions are contractions if $k_{i} < 1, i = 1, 2, 3, 4.$

Proof. To begin, we start with $V_{1}$ . Let $x_{1}, x_{2}$ be two distinct functions.

$\begin{eqnarray} &&\|V_{1}(t, x_{1}) - V_{1} (t, x_{2}) \| = \left\|(a\, y-a\, x_{1}\, +w) - (a\, y-a\, x_{2}\, +w)\right\|\\ && = \left\|a(x_{1}-x_{2})\right\|\\ &&\leq\, \, a\, \|x_{1}-x_{2}\|. \end{eqnarray}$

(3.4)

Let $a = k_{1}$ , and we have $\|V_{1}(t, x_{1}) - V_{1} (t, x_{2}) \| \leq k_{1}\, \|x_{1}-x_{2}\|$ . This shows the Lipschitz condition holds for $V_{1}(t, x)$ . In addition, if $k_{1} < 1$ , then it is a contraction.

Next, consider

$\begin{eqnarray} &&\|V_{2}(t, y_{1}) - V_{2} (t, y_{2}) \| = \left\|(c\, y_{1}-10\, x\, z) - (c\, y_{2}-10\, x\, z)\right\|\\ && = \left\|c\, y_{1}-10\, x\, z-c\, y_{2}+10\, x\, z\right\|\\ &&\leq\, \, c\, \|y_{1}-y_{2}\|. \end{eqnarray}$

(3.5)

Taking $c = k_{2}$ , we have $\|V_{2}(t, y_{1}) - V_{2} (t, y_{2}) \| \leq k_{2}\, \|y_{1}-y_{2}\|$ . This proves the Lipschitz condition for $V_{2}(t, y)$ . Also, if $k_{2} < 1$ , then it is a contraction.

Now, consider

$\begin{eqnarray} &&\|V_{3}(t, z_{1}) - V_{3} (t, z_{2}) \| = \left\|(-b\, z_{1}+10\, x\, y) - (-b\, z_{2}+10\, x\, y)\right\|\\ &&\|-b\, z_{1}+10\, x\, y+b\, z_{2}-10\, x\, y\|\\ &&\|-b(z_{1}-z_{2})\|\\ &&\leq{b\|z_{1}-z_{2}}\|. \end{eqnarray}$

(3.6)

Taking $b = k_{3}$ , we have $\|V_{3}(t, z_{1}) - V_{3} (t, z_{2}) \| \leq k_{3}\, \|z_{1}-z_{2}\|$ . This proves the Lipschitz condition for $V_{3}(t, z)$ . Further, if $k_{3} < 1$ , then it is a contraction.

Also,

$\begin{equation} \|V_{4}(t, u_{1}) - V_{4} (t, u_{2}) \| \leq{k_{4}\|u_{1}-u_{2}}\|. \end{equation}$

(3.7)

This shows that the functions $V_{1}$ , $V_{2}$ , $V_{3}$ , and $V_{4}$ satisfy the Lipschitz condition. □

We now demonstrate the existence of the model configurations by applying a fixed-point hypothesis. To achieve this, we employ the method outlined in ^[48]. Consider the system given by Eq (3.1)

$\begin{eqnarray} x(t)-x(0)& = &_{0}^{CF}I_{t}^{\alpha} {V_{1}(t, x)}, \\ y(t)-y(0)& = &_{0}^{CF}I_{t}^{\alpha} {V_{2}(t, y)}, \\ z(t)-z(0)& = &_{0}^{CF}I_{t}^{\alpha} {V_{3}(t, z)}, \\ u(t)-u(0)& = &_{0}^{CF}I_{t}^{\alpha} {V_{4}(t, u)}. \end{eqnarray}$

(3.8)

Employing (2.2), the equations given by (3.8) evolve as:

$\begin{eqnarray} x(t)& = &x(0) + \Psi V_{1}(t, x) + \vartheta \int_{0}^{t} V_{1}(s, x) ds, \\ y(t)& = &y(0) + \Psi V_{2}(t, y) + \vartheta \int_{0}^{t} V_{2}(s, y) ds, \\ z(t)& = &z(0) + \Psi V_{3}(t, z) + \vartheta \int_{0}^{t} V_{3}(s, z) ds, \\ u(t)& = &u(0) + \Psi V_{4}(t, u) + \vartheta \int_{0}^{t} V_{4}(s, u) ds. \end{eqnarray}$

(3.9)

This is then followed by the presentation of the recursive method:

$\begin{eqnarray} x_{n}(t)& = &\Psi V_{1}(t, x_{n-1}) + \vartheta \int_{0}^{t} V_{1}(s, x_{n-1}) ds, \\ y_{n}(t)& = &\Psi V_{2}(t, y_{n-1}) + \vartheta \int_{0}^{t} V_{2}(s, y_{n-1}) ds, \\ z_{n}(t)& = &\Psi V_{3}(t, z_{n-1}) + \vartheta \int_{0}^{t} V_{3}(s, z_{n-1}) ds, \\ u_{n}(t)& = &\Psi V_{4}(t, u_{n-1}) + \vartheta \int_{0}^{t} V_{4}(s, u_{n-1}) ds, \end{eqnarray}$

(3.10)

with $x(0) = x_{0},$ $y(0) = y_{0},$ $z(0) = z_{0}$ , and $u(0) = u_{0}$ .

The difference between the successive terms is given as:

$\begin{eqnarray} &&\tau_{1n} = x_{n}(t)-x_{n-1}(t) = \Psi [V_{1}(t, x_{n-1})-V_{1}(t, x_{n-2})] + \vartheta \int_{0}^{t} [V_{1}(s, x_{n-1})-V_{1}(s, x_{n-2})] ds, \\ &&\tau_{2n} = y_{n}(t)-y_{n-1}(t) = \Psi [V_{2}(t, y_{n-1})-V_{2}(t, y_{n-2})] + \vartheta \int_{0}^{t} [V_{2}(s, y_{n-1})-V_{2}(s, y_{n-2})] ds, \\ &&\tau_{3n} = z_{n}(t)-z_{n-1}(t) = \Psi [V_{3}(t, z_{n-1})-V_{3}(t, z_{n-2})] + \vartheta \int_{0}^{t} [V_{3}(s, z_{n-1})-V_{3}(s, z_{n-2})] ds, \\ &&\tau_{4n} = u_{n}(t)-u_{n-1}(t) = \Psi [V_{4}(t, u_{n-1})-V_{4}(t, u_{n-2})] + \vartheta \int_{0}^{t} [V_{4}(s, u_{n-1})-V_{4}(s, u_{n-2})] ds.\\ && \end{eqnarray}$

(3.11)

Note that

$\begin{equation} x_{n}(t) = \sum\limits_{i = 1}^{n} \tau_{1i}(t), \, \, y_{n}(t) = \sum\limits_{i = 1}^{n} \tau_{2i}(t), \, \, z_{n}(t) = \sum\limits_{i = 1}^{n} \tau_{3i}(t), \, \, u_{n}(t) = \sum\limits_{i = 1}^{n} \tau_{4i}(t). \end{equation}$

(3.12)

Proceeding in a similar manner, we will evaluate

$\begin{array}{l} \|\tau_{1n}(t)\| = \|x_{n}(t) - x_{n-1}(t) \| \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left\|\Psi [V_{1}(t, x_{n-1})-V_{1}(t, x_{n-2})] + \vartheta \int_{0}^{t} [V_{1}(s, x_{n-1})-V_{1}(s, x_{n-2})] ds\right\|. \end{array}$

(3.13)

Applying the triangular inequality on Eq (3.13) gives:

$\begin{equation} \|x_{n}(t) - x_{n-1}(t) \|\leq \Psi\left\|[V_{1}(t, x_{n-1}) - V_{1}(t, x_{n-2})]\right\| + \vartheta \int_{0}^{t} \left\|[V_{1}(s, x_{n-1})-V_{1}(s, x_{n-2})] ds\right\|. \end{equation}$

(3.14)

Since the function $V_{1}(x, t)$ satisfies the Lipschtiz condition, we have:

$\begin{equation} \|x_{n}(t) - x_{n-1}(t) \|\leq \Psi k_{1} \left\|x_{n-1}) - x_{n-2}]\right\| + \vartheta k_{1} \left\|\int_{0}^{t} [V_{1}(s, x_{n-1})-V_{1}(s, x_{n-2})] ds \right\|. \end{equation}$

(3.15)

This gives

$\begin{equation} \left\|\tau_{1n}(t)\right\| \leq \Psi k_{1} \left\|\tau_{1(n-1)}(t)\right\| + \vartheta k_{1} \int_{0}^{t} \left\|\tau_{1(n-1)}(s)\right\| ds. \end{equation}$

(3.16)

The following results can be obtained by proceeding the same approach

$\begin{eqnarray} && \left\|\tau_{2n}(t)\right\| \leq \Psi k_{2} \left\|\tau_{2(n-1)}(t)\right\| + \vartheta k_{2} \int_{0}^{t} \left\|\tau_{2(n-1)}(s)\right\| ds, \\ && \left\|\tau_{3n}(t)\right\| \leq \Psi k_{3} \left\|\tau_{3(n-1)}(t)\right\| + \vartheta k_{3} \int_{0}^{t} \left\|\tau_{3(n-1)}(s)\right\| ds, \\ && \left\|\tau_{4n}(t)\right\| \leq \Psi k_{4} \left\|\tau_{4(n-1)}(t)\right\| + \vartheta k_{4} \int_{0}^{t} \left\|\tau_{4(n-1)}(s)\right\| ds. \end{eqnarray}$

(3.17)

Theorem 2. The fractional order hyperchaotic sysyem (3.1) has a solution, if we are able to find $t_{0}$ with the following constraint:

$\begin{equation*} \Psi k_{i} + \vartheta k_{i}\, t_{0} < 1, \, \, \, i = 1, 2, 3, 4. \end{equation*}$

Proof. As proved that the Lipschtiz condition holds for $V_{1}(t, x)$ , $V_{2}(t, y)$ , $V_{3}(t, z)$ , $V_{4}(t, u)$ . Therefore, from Eqs (3.16) and (3.17), using the recursive method, we have:

$\begin{eqnarray} &&\left\|\tau_{1n}(t)\right\| \leq \left\|x_{n}(0)\right\|\, \, \left[\Psi\, k_{1} + \, \vartheta \, k_{1}\, t\right]^{n}, \\ &&\left\|\tau_{2n}(t)\right\| \leq \left\|y_{n}(0)\right\|\, \, \left[\Psi\, k_{2} + \, \vartheta \, k_{2}\, t\right]^{n}, \\ &&\left\|\tau_{3n}(t)\right\| \leq \left\|z_{n}(0)\right\|\, \, \left[\Psi\, k_{3} + \, \vartheta \, k_{3}\, t\right]^{n}, \\ &&\left\|\tau_{4n}(t)\right\| \leq \left\|u_{n}(0)\right\|\, \, \left[\Psi\, k_{4} + \, \vartheta \, k_{4}\, t\right]^{n}. \end{eqnarray}$

(3.18)

This confirms that the previously mentioned solutions do indeed exist. Furthermore, to validate that the specified functions are solutions of the fractional order hyperchaotic system with the Caputo-Fabrizio derivative, let us assume

$\begin{eqnarray} &&W_{n} = x_{n}(t)-(x(t)-x(0)), \\ &&G_{n} = y_{n}(t)-(y(t)-y(0)), \\ &&H_{n} = z_{n}(t)-(z(t)-z(0)), \\ &&E_{n} = u_{n}(t)-(u(t)-u(0)).\\ \end{eqnarray}$

(3.19)

So then,

$\begin{eqnarray} && x(t) - x(0) = x_{n}(t) - W_{n} (t), \\ && y(t) - y(0) = y_{n}(t) - G_{n} (t), \\ && z(t) - z(0) = z_{n}(t) - H_{n} (t), \\ && u(t) - u(0) = u_{n}(t) - E_{n} (t). \\ \end{eqnarray}$

(3.20)

Therefore,

$\begin{eqnarray} &&\left\|W_{n}(t)\right\| = \left\|\Psi[V_{1}(t, x_{n})-V_{1}(t, x_{n-1})] +\vartheta \int_{0}^{t} [V_{1}(s, x_{n})-V_{1}(s, x_{n-1})] ds\right\| \\ && \leq \Psi \left\|V_{1}(t, x_{n})-V_{1}(t, x_{n-1})\right\| + \vartheta \int_{0}^{t} \left\|V_{1}(s, x_{n})-V_{1}(s, x_{n-1})\right\| ds \\ && \leq \Psi\, \, k_{1} \left\|x_{n}-x_{n-1}\right\| + \vartheta k_{1} \left\|x_{n}-x_{n-1}\right\| t. \end{eqnarray}$

(3.21)

Applying the results in a recursive manner gives

$\begin{equation} \left\|W_{n}(t)\right\| \leq (\Psi + \vartheta\, \, t)^{n+1}\, \, k_{1}^{n+1}. \end{equation}$

(3.22)

Then, at $t_{0}$ we have:

$\begin{equation} \left\|W_{n}\right\|_{t = t_{0}} \leq (\Psi + \vartheta\, \, t_{0})^{n+1}\, \, k_{1}^{n+1}. \end{equation}$

(3.23)

At $n \to \infty$ in Eq (3.23),

$\begin{equation*} \left\|W_{n}(t)\right\| \to 0 . \end{equation*}$

In the similar manner,

$\begin{equation*} \left\|G_{n}(t)\right\| \to 0, \left\|H_{n}(t)\right\| \to 0 , \left\|E_{n}(t)\right\| \to 0. \end{equation*}$

Further, to establish the uniqueness of the solution for the system (3.1), we assume that there exists another solution to this system, given by $x_{1}, y_{1}, z_{1}, u_{1}$ . Then,

$\begin{equation} x(t)-x_{1}(t) = \Psi[V_{1}(t, x)-V_{1}(t, x_{1})] + \vartheta \int_{0}^{t}[V_{1}(s, x)-V_{1}(s, x_{1})] ds. \end{equation}$

(3.24)

Taking norm on Eq (3.24) gives:

$\begin{equation} \left\|x(t)-x_{1}(t)\right\| \leq \Psi\left\|[V_{1}(t, x)-V_{1}(t, x_{1})]\right\| + \vartheta \int_{0}^{t}\left\|[V_{1}(s, x)-V_{1}(s, x_{1})]\right\| ds. \end{equation}$

(3.25)

As $V_{1}(t, x)$ satisfies the Lipschtiz condition, we have

$\begin{equation} \left\|x(t)-x_{1}(t)\right\| \leq \Psi k_{1}\left\|[x(t)- x_{1}(t)]\right\| + \vartheta \int_{0}^{t} k_{1}\, \, \, \left\|x(t)-x_{1}(t)]\right\| ds, \end{equation}$

(3.26)

which gives

$\begin{equation} \left\|x(t)-x_{1}(t)\right\|(1-\Psi\, \, k_{1} - \vartheta\, \, k_{1}\, t) \leq 0. \end{equation}$

(3.27)

□

Theorem 3. If the condition given below holds, then the hyperchaotic system has a unique solution

$\begin{equation} (1-\Psi\, \, k_{1} - \vartheta\, \, k_{1}\, t) > 0. \end{equation}$

(3.28)

Proof. From Eq (3.27), we have

$\begin{equation*} \label{1_b_21} \left\|x(t)-x_{1}(t)\right\|(1-\Psi\, \, k_{1} - \vartheta\, \, k_{1}\, t) \leq 0, \end{equation*}$

and if the following also holds:

$\begin{equation*} \label{1_b_22} (1-\Psi\, \, k_{1} - \vartheta\, \, k_{1}\, t) > 0, \end{equation*}$

then this is possible only if

$\begin{equation*} \left\|x(t)-x_{1}(t)\right\| = 0. \end{equation*}$

Hence, $x(t) = x_{1}(t)$ . Similarly, $y(t) = y_{1}(t),$ $z(t) = z_{1}(t)$ , and $u(t) = u_{1}(t)$ . □

3.2. Numerical solutions for defined model

To solve the fractional order 4D hyperchaotic system with the Caputo-Fabrizio derivative numerically, we follow the method as briefed in ^[49].

Consider the 4D hyperchaotic system with the Caputo-Fabrizio derivative of fractional order $\varepsilon$ , $0 < \varepsilon < 1$ ,

$\begin{eqnarray} _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{x(t)\}& = & a\, (y-x)+u, \, \, \, \, \, \, \, \, \, \, \, \, \, x(0) = x_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{y(t)\}& = & c\, y - 10\, x\, z, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, y(0) = y_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{z(t)\}& = & -b\, z + 10\, x\, y, \, \, \, \, \, \, \, \, \, \, \, \, \, z(0) = z_{0}, \\ _{0}^{CF}\mathbb{D}_{0}^{\varepsilon(t)} \{u(t)\}& = & d\, y + x^{2}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, u(0) = u_{0}. \end{eqnarray}$

(3.29)

We take into account the following general equation:

$\begin{equation} ^{CF} \mathbb{D}_{t}^{\varepsilon}P(t) = \omega(t, P(t)), \;\; t \geq 0, \, \, \, \, \, \, P(0) = P_{0}, \end{equation}$

(3.30)

where $P(t) = \{x(t), y(t), z(t), u(t)\}$ , and $P(0) = \{x(0), y(0), z(0), u(0)\}$ .

Proceeding as referred in ^[50], we rewrite Eq (3.30) as

$\begin{equation} P(t)- P(0) = \frac{1-\varepsilon}{M(\varepsilon)} \omega(t, P(t)) + \frac{\varepsilon}{Q(\varepsilon)} \int_{0}^{t} \omega\, (s, P(s))ds. \end{equation}$

(3.31)

At $t = t_{n+1}$ , Eq (3.31) is written as

$\begin{equation} P(t_{n+1})- P(0) = \frac{1-\varepsilon}{Q(\varepsilon)} \omega (t_{n}, P(t_{n})) + \frac{\varepsilon}{Q(\varepsilon)} \int_{0}^{t_{n+1}} \omega(s, P(s))ds. \end{equation}$

(3.32)

Also at $t = t_{n}$ , Eq (3.31) gives

$\begin{equation} P(t_{n})- P(0) = \frac{1-\varepsilon}{Q(\varepsilon)} {\omega}\, (t_{n-1}, P(t_{n-1})) + \frac{\varepsilon}{Q(\varepsilon)} \int_{0}^{t_{n}} {\omega}\, (s, P(s))ds. \end{equation}$

(3.33)

From Eqs (3.32) and (3.33), we get

$\begin{eqnarray} P(t_{n+1})-P(t_{n}) = \frac{1-\varepsilon}{Q(\varepsilon)}[{\omega}\, (t_{n}, P(t_{n})) - \omega\, (t_{n-1}, P(t_{n-1}))] + \frac{\varepsilon}{Q(\varepsilon)} \int_{t_{n}}^{t_{n+1}}\, \, {\omega}(s, P(s))ds. \end{eqnarray}$

(3.34)

Lagrange polynomial interpolation on $\omega(s, P(s))$ gives

$\begin{equation} \omega\, (s, P(s)) = \frac{s-t_{m-1}}{t_{m}-t_{m-1}} \, \omega\, (t_{m}, P_{t_{m}})+\frac{s-t_{m}}{t_{m-1}-t_{m}}\, \omega\, (t_{m-1}, P_{t_{m-1}}). \end{equation}$

(3.35)

Substituting $\omega\, (s, P(s))$ in Eq (3.34) gives

$\begin{array}{l} P_{n+1}-P_{n} = \frac{1-\varepsilon}{Q(\varepsilon)}[{\omega}\, (t_{n}, P(t_{n})) - {\omega}\, (t_{n-1}, P(t_{n-1})) ] + \frac{\varepsilon}{Q(\varepsilon)}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \int_{t_{n}}^{t_{n+1}}\left(\frac{{\omega}\, (t_{n}, P_{n})}{h} (s-t_{n-1}) - \frac{{\omega}\, (t_{n-1}, P_{n-1})}{h} (s-t_{n})\right) ds. \end{array}$

(3.36)

On taking $h = t_{n}-t_{n-1}$ and solving, we get

$\begin{equation} P_{n+1} = P_{0}+\left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{3 h}{2\, \, Q(\varepsilon)}\right)\, {\omega}\, (t_{n}, P(t_{n}))- \left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{\varepsilon\, \, h}{2\, \, Q(\varepsilon)}\right)\, \omega(t_{n-1}, P(t_{n-1})). \end{equation}$

(3.37)

In a similar way, we have

$\begin{equation} x_{n+1} = x_{0}+\left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{3 h}{2\, \, Q(\varepsilon)}\right)V_{1}(t_{n}, x(t_{n}))- \left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{\varepsilon\, \, h}{2\, \, Q(\varepsilon)}\right)\, V_{1}(t_{n-1}, x(t_{n-1})), \end{equation}$

(3.38)

where $V_{1}(t, x) = a\, (y-x) + u$ .

$\begin{equation} y_{n+1} = y_{0}+\left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{3 h}{2\, \, Q(\varepsilon)}\right)V_{2}(t_{n}, y(t_{n}))- \left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{\varepsilon\, \, h}{2\, \, Q(\varepsilon)}\right)\, V_{2}(t_{n-1}, y(t_{n-1})), \end{equation}$

(3.39)

where ${V_{2}}(t, y) = c\, y - 10\, x\, z$ .

$\begin{equation} z_{n+1} = z_{0}+\left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{3 h}{2\, \, Q(\varepsilon)}\right)V_{3}(t_{n}, z(t_{n}))- \left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{\varepsilon\, \, h}{2\, \, Q(\varepsilon)}\right)\, V_{3}(t_{n-1}, z(t_{n-1})), \end{equation}$

(3.40)

where $V_{3}(t, z) = -b\, z + 10\, x\, y$ .

$\begin{equation} u_{n+1} = u_{0}+\left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{3 h}{2\, \, Q(\varepsilon)}\right)V_{4}(t_{n}, u(t_{n}))- \left(\frac{1-\varepsilon}{Q(\varepsilon)} + \frac{\varepsilon\, \, h}{2\, \, Q(\varepsilon)}\right)\, V_{4}(t_{n-1}, u(t_{n-1})), \end{equation}$

(3.41)

where, $V_{4}(t, u) = d\, y + x^{2}$ .

The numerical solutions of the fractional order 4D hyperchaotic system with the Caputo-Fabrizio derivative defined by Eq (3.1) are obtained from Eqs (3.38)–(3.41).

3.3. Simulations

The simulations are performed for different fractional orders. The parameter values ^[47] are taken as $a = 35, b = 3, c = 12$ . The initial conditions are taken as $x(0) = y(0) = z(0) = u(0) = 1$ . Further, we consider three different values for the variable parameter $d$ ( $10$ , $17$ , and $22$ ).

● In –, the numerical simulations are performed for the fractional order $\varepsilon = 0.95$ , $0.97$ , and $0.99$ , respectively, with $d = 10$ .

Figure 1. Phase portrait of the 4D fractional order Caputo-Fabrizio hyperchaotic system defined by Eq (3.1) simulated at

$\varepsilon = 0.95$ with

$a = 35$ ,

$b = 3$ ,

$c = 12$ ,

$d = 10$ .

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AIMS Mathematics

Application of Caputo-Fabrizio derivative in circuit realization

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Abstract

1. Introduction

2. Preliminaries

3. Fractional order hyperchaotic system with Caputo-Fabrizio derivative

3.1. Existence & uniqueness

3.2. Numerical solutions for defined model

3.3. Simulations

4. Circuit implementation of the hyperchaotic system

5. Discussion on the results

6. Conclusions

Author contributions

Use of Generative-AI tools declaration

Acknowledgments

Conflict of interest

References

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