1. Introduction

Fractional calculus provides an accurate way of modeling real-world phenomena [1, 2]. It has recently been integrated into mathematical modeling, enabling researchers to open new avenues for accurately representing nonlinear dynamics across various scientific fields [3–11]. There are several ways to describe the fractional derivative since the pioneering works of Riemann in 1847 (RL derivatives) [12] and Caputo in 1967 [13]. The RL and Caputo derivatives are described by nonlocal operators with singular kernels. More recently, a modification of the Caputo operator using a nonsingular kernel has been introduced by Caputo and Fabrizio (CF) [14]. Consequently, the CF fractional derivative provides a useful way of describing nonlocal and complex phenomena across various scientific fields since it has significant applications in many interdisciplinary fields such as circuit design [15, 16], signal processing [17], DC-DC converter circuits [18], fractional Schrödinger equations [19], models of cancer [20] and brain metabolite variations in the circadian rhythm [21].

In 1900, Hilbert initiated the research on hidden oscillations [22]. Recently, the study of hidden oscillations in dynamical systems has become a topic of interest [23] due to its various applications in different fields such as climate science, aircraft control systems [24], ecosystems, financial market [25] and engineering systems [26]. Neglecting this interesting dynamical phenomenon may lead to serious industrial accidents and air disasters like the one that happened to Boeing Flight No. YF-22 in 1992 [27]. Most of the previously mentioned dynamical systems can be described via no-local operators to achieve better accuracy and adequacy in describing natural phenomena. In addition, this kind of systems exhibits the multistability property, which is associated with the hidden oscillations. More recently, Danca [28] reported the study of hidden oscillations in fractional systems governed by the Caputo derivatives. In dynamical systems, an attractor that surrounds an equilibrium state but does not intersect an open neighborhood of this equilibrium is classified as a hidden attractor. Otherwise, it is classified as a self-excited attractor. Chaos is another important dynamical phenomenon that can observed in 3D autonomous dynamical systems or higher. Chaos has potential applications in many interdisciplinary fields such as electronic circuits [29, 30], plasma perturbations [31], nonlinear oscillators such as the ADVP oscillator [32], the Hartley oscillator [33] and the 3D memristive jerk oscillator [34]. In some 3D dynamical systems, chaos is associated with bifurcations such as Hopf bifurcation [32, 35] and torus doubling bifurcation [36].

Recently, the study of chaos in fractional-order systems has become a vital research topic [37–40]. During the last decade, many studies have appeared that deal with chaos for some systems using the CF fractional derivatives. This line of research is new and has potential applications in secure communications, circuit design, and modeling of some chemical, physical, biological and engineering systems [16, 41–43]. In [43], Matouk and Botros explained the existence of hidden chaotic attractors in a novel circuit system described by four fractional-order differential equations that are governed by the CF derivatives. As far as we know, the foundation of hidden attractors in 3D fractional systems involving the CF derivatives has not been published yet.

In this work, occurrences of hidden chaotic attractors in two 3D chaotic systems are shown. The systems’ chaotic states are stabilized to their equilibrium states using a linear feedback control (LFC) scheme. Chaos synchronization is obtained in the two chaotic systems using suitable linear control functions. An adequate numerical scheme is used to simulate the two chaotic systems with CF operators. Finally, varieties of complex dynamics are illustrated such as one-scroll (O-S) chaotic attractors, hidden periodic attractors, self-excited and hidden chaotic attractors. In addition, some bifurcation diagrams are carried out to confirm the rich varieties of nonlinear phenomena. Thus, this work reports the first examples of hidden attractors in 3D systems that involve CF derivatives.

2. Fractional Calculus

Utilizing Lemma 1, one concludes that inequality (3) is the necessary and sufficient condition that makes the real parts of the eigenvalues of σAJ have negative signs. In other words, lim_t⟶∞ X(t) = 0 if and only if condition (3) holds.

Obviously, this numerical scheme is adequate and robust enough to handle other fractional systems.

3. The Two Chaotic Systems

Figure 1 illustrates the implementation of the previous circuital equations. The circuit in Figure 1 uses common electronic components, such as the LF353N operational amplifier with a voltage supply of ±15 V, to perform multiple functions simultaneously, including integration, differentiation, subtraction, and addition. We use 13 resistors, three capacitors, one multiplier, and six op-amps in the circuit schematic. The resistor and capacitor values in the previous model can be approximated as follows:

R₄ = R₅ = R₈ = R₉ = R₁₀ = R₁₁ = R₁₂ = R₁₃ = 100 kΩ, R₇ = 0.5 kΩ, R₁ = R₃ = 1 kΩ, R₂ = 10 kΩ, R₆ = 62.5 kΩ and C₁ = C₂ = C₃ = 10 nF. After simulation with the values described above, the phase portraits in the oscilloscope are shown in Figures 2, 3 and 4.

The fractional-order form of system (16) in the Caputo sense exhibits chaotic dynamics when p₁ = 10, p₂ = 40, p₃ = 1, p₄ = 2.5, p₅ = 4. The physical meaning of the last parameter values was explained in [45].

4. Simulating the Complex Dynamics

In this Section, all the results are numerically simulated, coded and executed according to the numerical scheme represented by equation (11) using a step size of 0.01, which is shown to be adequate for capturing the intricate dynamics of the considered chaotic systems.

4.1. Numerical Simulations of the First Chaotic System

When the parameter set Α: a = 2.85, b = 200, c = 0.1, d = 100 is used, the system exhibits various types of complex dynamics. The results are outlined in Figure 5, which shows that a self-excited attractor is created when σ = 0.998; A hidden periodic attractor framing two period-one limit cycles, is shown when σ = 0.999; A hidden periodic attractor framing two period-two limit cycles, is shown when σ = 0.9992 and a hidden periodic attractor framing two O-S chaotic attractors, is shown when σ = 0.9997. The bifurcation diagrams are coded and executed to show the significance of the parameter set Α for the demonstration of the complex dynamics and hidden attractors in system (12). The results are summarized in Figures 6 and 7.

The attraction basin is illustrated in Figure 8 whose color map depicts the coexistence of hidden quasi-periodic attractors and two O-S chaotic attractors. The remaining part of the graph can be explained as follows; the riddled red area near the center of the graph refers to O-S chaotic attractor around $$; the opposite clear red area refers to O-S chaotic attractor around $$ and the blue regions at the top and bottom of the graph refer to the quasi-periodic attractors.

4.2. Numerical Simulations of the Second Chaotic System

When the parameter set Β: p₁ = 10, p₂ = 40, p₃ = 1, p₄ = 2.5, p₅ = 4 is used, system (16) displays a hidden chaotic attractor surrounding two point attractors when σ = 0.9777785 (see Figure 9). σ = 0.9997. The bifurcation diagrams are coded and executed to show the significance of the parameter set Α for the demonstration of the complex dynamics and hidden attractors in system (16). The results are summarized in Figures 10 and 11. In Figure 11, the green domain refers to the hidden chaotic attractor that is swinging between the positive and negative parts of z1 axis. This figure shows that the hidden chaotic attractor is surrounding, but not intersecting the sinks $$ and $$ that are represented by red and blue regions. The attraction basin is illustrated in Figure 12, which shows a riddled basin of attraction. This property of the Liu dynamical system was discussed in [46]. The later reference proved that any neighborhood of the attractor of the Liu dynamical system contains repelled sets with positive Lebesgue measures, which means that it has a strange attractor of Milnor’s type.

5. Chaos Control via the LFC Scheme

5.1. Chaos Control of System (12) Using LFC Scheme

The counterpart of system (12) according to the LFC scheme is

$$ ()

where k_i, i = 1, 2, 3 are the FCGs. To stabilize system (19) towards $$ we choose k₂ = k₃ = 0 and the appropriate k₁ that makes the discriminant of following polynomial equation

$$ ()

a negative real number and also satisfy

$$ ()

When the parameter set Α is selected, the inequalities (21) have the solution k₁ > 38.76. Hence, the FCGs can be chosen as k₁ = 40, k₂ = 0, k₃ = 0 and σ = 0.9998. In Figure 13, the controlled state variables are depicted.

To stabilize system (19) towards $$, we select k₂ = k₃ = 0 and the appropriate k₁ that makes following polynomial

$$ ()

has a negative real discriminant and also the following conditions hold

$$ ()

When the parameter set Α is chosen, the inequalities (23) have the solution k₁ > 9. Hence, the FCGs are chosen as k₁ = 20, k₂ = 0, k₃ = 0 and σ = 0.9998. In Figures 14 and 15, the controlled state variables are illustrated.

5.2. Chaos Control of System (16) Using LFC Scheme

The controlled version of system (16) in terms of the LFC scheme is given by

$$ ()

To stabilize system (24) towards $$ we consider following polynomial equation

$$ ()

where k_i > 0, i = 1, 2, 3, ε₁ = p₄ + k₃ and ε₂ = p₁ + k₁.

Indeed, equation (25) has negative real roots if and only if

$$ ()

When the parameter set Β with k₂ = 37 and k₃ = 1 are specified, the inequalities (26) have the solution k₁ > 0.81. Thus, the FCGs are selected as k₂ = 37, k₃ = 1, k₁ = 1 and σ = 0.998. In Figure 16, the controlled state variables are illustrated.

To control system (24) to $$, we set

$$ ()

Obviously, equation (27) has negative real roots if and only if

$$ ()

When the parameter set Β with k₂ = 5 and k₃ = 1 are specified, the inequalities (28) have the solution k₁ > 1.1. Thus, the FCGs are selected as k₂ = 5, k₃ = 1, k₁ = 1.2 and σ = 0.998. In Figures 17 and 18, the controlled state variables are shown.

6. Chaos Synchronization

The basic idea in this section is to pick the suitable FCGs that make the dynamic errors satisfy the stability condition (3), which make the origin equilibrium sate LAS. This means that the drive and response states coincide after a certain time. This target can easily be reached if we design the appropriate FCGs that make all the eigenvalues of the linearized part of the integer-order counterpart of the error dynamical system are negative.

6.1. Chaos Synchronization of System (12)

Firstly, we introduce the constant matrix Τ, the control law Κ and the nonlinear function Ν(ξ(t)) ∈ R³, then the dynamics of the synchronization errors ξ(t) ∈ R³ are demonstrated via the following system

$$ ()

where σ∈ and ξ ∈ R³. The vector function $$ can be selected as

$$ ()

where $$. The operator D produces the Jacobian matrix for Ν(ξ). Hence, the matrix Τ + k acts as a Jacobian of the previous error dynamical system. Therefore, the FCGs that satisfy condition (3) can be selected such that the equilibrium state ξ = 0 is LAS.

The drive and response versions of system (12) are, respectively, presented as

$$ ()

and

$$ ()

where z_i, Z_i represents the i- th component of the drive and response vectors, respectively. Then, the vector that represents the synchronization errors is given as

$$ ()

whose dynamics are governed by

$$ ()

where $$. Here, the control law Κ is presented as

$$ ()

where k_i = 0, i = 1, 2, 3. Obviously, all the eigenvalues of the linearization of the integer-order counterpart of system (34) are negative. Consequently, according to Theorem 1, the solution ξ = 0 of the fractional-order equation (34) is LAS. In Figure 19, the synchronization errors are depicted.

6.2. Chaos Synchronization of System (16)

The drive and response versions of system (16) are, respectively, presented as

$$ ()

and

$$ ()

Thus, the synchronization errors are governed by the following system

$$ ()

Now, if the FCGs are chosen to satisfy the following conditions

$$ ()

where Θ = (p₁ + p₂ + M₃p₃), Λ = (k₁ + p₁), M₁ = max(z₁, Z₁) and M₃ = max(z₃, Z₃), then the point ξ = 0 of system (38) is LAS. According to Theorem 2 of Ref. [47], all the eigenvalues of the linearization of the integer-order counterpart of system (38) are negative. Consequently, based on Theorem 1, the solution ξ = 0 of the fractional-order equation (38) is LAS. When k₁ = 0, k₂ = 10 and k₃ = 0 are selected with the parameter set Β, conditions (39) hold. In Figure 20, the synchronization errors are shown.

7. Conclusion

Two 3D systems that involve CF derivatives have been presented. A necessary and sufficient condition for achieving the local stability of an equilibrium state has been introduced. An adequate numerical scheme has been presented to integrate fractional-order equations with CF derivatives. Then, it has been used to simulate the two 3D systems. Varieties of nonlinear dynamics have been obtained such as chaotic attractors, hidden periodic attractors, self-excited and hidden chaotic attractors. This interesting multistability dynamical phenomenon has been studied in-depth and verified by the computations of attraction basins. In addition, this work reports the first examples of hidden attractors in 3D systems that involve the CF derivatives. The bifurcation diagrams have been carried out to validate the existing complex dynamics in the considered systems.

On the other hand, the systems’ chaotic states have been stabilized to their equilibrium states using the LFC scheme. The stipulations of synchronizing chaos in the proposed systems governed by the CF derivatives have been obtained when suitable selections of linear control functions are made. Thus, numerical simulations have been performed to match the theoretical conditions.

In our future studies, we will investigate the bifurcation phenomena in the considered systems when the CF derivatives are implemented. In addition, torus doubling bifurcation will be investigated in the integer-order form of the first chaotic system and its discretization. Furthermore, circuit realization of the considered systems with the CF derivatives will be designed.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

All authors contributed equally to this work.

Funding

No funding was received for this work.