A Numerical Study of the Caputo Fractional Nonlinear Rössler Attractor Model via Ultraspherical Wavelets Approach

1  Introduction

In today’s world, accuracy is paramount across all fields, making the application of advanced mathematical techniques essential [1–4]. Fractional calculus (FC), an extension of traditional calculus, plays a crucial role in this regard [5–8]. Incorporating fractional derivatives and integrals, FC provides a strong framework for modeling complex systems with memory and inherited features, capturing phenomena that traditional models often overlook [9]. Its applications are increasingly widespread across various fields of science and engineering, offering new insights into real-world challenges. The reach of FC now extends into the areas such as robotics [10], stock market analysis [11], signal processing [12], chaotic dynamical systems [13], image processing [14], viscoelasticity [15], life sciences [16], neural networks [17], pharmacokinetic [18], and earthquake modeling [19]. This concept enhances the understanding and prediction of systems exhibiting non-local and time-dependent behavior.

Fractional operators [20–22] can be applied in any real or complex order, providing deeper analysis with the flexibility to approach dynamic systems. Fractional integrals generalize traditional integration to non-integer orders, using the Riemann-Liouville [23] or Caputo definitions [24,25]. Integral and differential operators are used in the Liouville and Riemann framework to investigate FC. The Caputo operator is a novel fractional operator that Caputo created, utilizing the framework of FC to study and analyze the models [26,27]. The Caputo fractional derivative is highly regarded as an efficient derivative for real-world applications, as it effectively incorporates initial and boundary conditions. Compared to other fractional derivatives like the Riemann-Liouville derivative, the Caputo derivative avoids certain mathematical complexities, such as needing non-local initial conditions that are more computationally challenging. The use of the Caputo derivative in real-world scenarios often leads to more accurate solutions than other derivatives, when combined with orthogonal polynomials and wavelet-based methods. By incorporating kernel functions that account for historical effects, they effectively model systems with long-range dependencies, such as anomalous diffusion and viscoelastic behavior [28]. Their ability to capture non-local interactions makes them invaluable in fields like physics [29], engineering [30,31], finance [32], and control theory [33] and provide essential tools for understanding complex phenomena. One such tool is a wavelet transform.

Wavelet Transform is a mathematical technique that deals with an expansion of functions as a basis function. Operations on wavelets form wavelets theory [34–36] which can be employed in various fields such as image analysis, control systems, communication systems, stock market analysis, a meteorological model, solutions to differential equations [37,38], and many others [39–41]. In recent years, the wavelets approach has become more and more popular in the areas of numerical methods. Several types of wavelets and function approximation were utilized in these references [42–44]. Wavelet-based approaches are employed for solving differential equations, particularly non-linear differential equations, providing highly accurate solutions by using integration to transform differential equations into integral equations [45–48]. The functions or signals associated with the equations are then compared using truncated orthogonal series expansions. The integral operations within these equations are eliminated by applying an operational matrix of integration, thereby effectively reducing the considered problem to a series of algebraic equations, and simplifying the calculation process. One of its applications includes analyzing chaotic dynamic systems.

Chaotic dynamical systems [49,50] are sensitive to the state of initial conditions applied to the systems. Its characteristics such as unpredictability, sensitivity to initial conditions, and complex dynamics play vital roles in solving real-world scenarios. They are crucial in understanding weather patterns [51], predicting population dynamics [52], encryption in cryptocurrency [53] encryption in image [54], and other fields. In the realm of chaos and dynamical systems, particularly for low-dimensional models, the Lorenz system [55] and Rössler model [56] are two seminal examples that have been widely examined. These models are fundamental in exploring and understanding chaotic dynamics and have contributed a significant role in the advancement of the related fields. The Lorenz system is more complex and closely associated with physical phenomena, generating a double-lobed butterfly attractor representing more intricate chaos. On the other hand, the Rössler system is simpler and serves as an abstract model of chaos, making it ideal for theoretical studies and easier visualizations. It produces a single spiral attractor, which reflects smoother chaos. The coexistence of both asymmetric and symmetric features in the Rössler system allows researchers to explore a broader spectrum of dynamical behaviors as compared to the asymmetric Lorenz system, making it a versatile and insightful model for studying chaotic dynamics. Due to these characteristics, a system of non-linear equation of Rössler attractor model is analyzed numerically in this study by using a wavelet-based technique.

The Rössler system [57] consists of three equations, being one non-linear, three system parameters a, b, c and variables x, y, z representing state variables. The parameter a controls the strength in the y-equation, affecting the rotation in xy-plane, b parameter affects the z-equation, i.e., influences the coupling between x and z components of the system and can be thought of as an external forcing term and c influences the interaction between the x and z variables, controlling the rate at which the z component grows and interacts with the x component. Here a,b,c∈R, and they are assumed to be dimensionless and positive. The Rössler attractor is typically visualized in 3D space with x, y, and z axes, depicting chaotic trajectories that are sensitive and dependent on initial conditions. In this work, the following forms of Caputo fractional Rössler attractor model are considered.

Model 1: Asymmetric fractional Rössler attractor [58]:

00CDταx(τ)=−y(τ)−z(τ),00CDταy(τ)=x(τ)+ay(τ),τ∈[0,1],00CDταz(τ)=b+z(τ)(x(τ)−c),(1)

with the conditions

x(0)=x1,y(0)=y1,z(0)=z1.(2)

In contrast to the asymmetric Lorenz system, the Rössler system exhibits both symmetry and asymmetry characteristics. Asymmetric Rössler attractor is shown by the model given in Eq. (1). One can change the structure by making changes to the linear or nonlinear variables to build a symmetric system. A 180∘ rotation around the z-axis corresponds to the system that is invariant undergoing the transformation (x,y,z) → (−x,−y,z), which can be used to construct a system that is rotationally invariant. System in Eq. (1) requires that some terms be multiplied by suitable variable selections [58].

Model 2: Symmetric fractional Rössler attractor [58]:

00CDταx(τ)=−y(τ)−y(τ)z(τ),00CDταy(τ)=x(τ)+ay(τ),τ∈[0,1],00CDταz(τ)=b+z(τ)((x(τ))2−c),(3)

with the conditions

x(0)=x2,y(0)=y2,z(0)=z2.(4)

The operator 00CDτα in Eqs. (1) and (3) represents the fractional derivative in the Caputo sense of order α∈(0,1]. The above Caputo fractional Rössler attractor model is an extension of classical Rössler system.

We will utilize the above two systems for our study under the conditions mentioned. Several researchers have studied the above models, their applications, and their nature by utilizing various methods and tools. In [59], Kontorovich et al. used the degenerated cumulant equations method for analysis and provided an application through Rössler attractor output signals to model radio frequency interferences provided by Peripheral Component Interface express. Rysak et al. [60] implemented the Grunwald-Letnikov method for numerical solutions of Caputo fractional Rössler attractor model and did a recurrence quantification analysis. In [61], Elbadri et al. presented a fractional Laplace decomposition technique with an adaptive predictor-corrector algorithm for solving rotationally symmetric Rössler attractor. Kekana et al. [62] conducted the analysis of Rössler attractor by residual and Joubert-Greeff methods. In [63], Barrio et al. studied the regions of parameters for chaotic behavior by using different chaos indicators and conducted a thorough analysis of the global and local bifurcations of co-dimension one and two of limit cycles. Santra et al. [64] provided the simulation of Rössler attractor through the power series approach. In [65], Boulehmi solved the Caputo fractional Rössler attractor model under the Atangana-Baleanu-Caputo fractional derivative and compared the results with the method.

In this study, we are interested in adapting a novel approach based on ultraspherical Wavelets (USWs) as basis functions for simulating Caputo fractional Rössler attractor model. These wavelets play a significant role in numerical analysis as well as in approximation theory. The importance of studying the fractional Rössler system lies in its potential for better analyzing the applications of such models. To the best of the author’s knowledge, this is the first time that the considered model under the Caputo derivative is numerically simulated using USWs. Therefore, motivated by the existing literature, the present work illustrates an application of USWs with collocation nodes to analyze the Caputo fractional Rössler attractor model under the Caputo derivative. An additional motivation is that the Legendre wavelets and Chebyshev wavelets can be inferred as particular instances of the USWs. The presented scheme does not appear to have any significant flaws. However, the suggested scheme works effectively in a limited domain, and processing a large number of USWs could lead to high computational costs. The novelty of this work is presented as:

•   The Caputo fractional derivatives have been employed to get more accurate solutions to the Caputo fractional Rössler attractor model.

•   The design of the computational framework based on USWs is presented for the first time to solve the Caputo fractional Rössler attractor model numerically.

•   The relative representations have been presented through relative errors and L2 error, which improve the strength of the computing structure USWs to solve the Caputo fractional Rössler attractor model.

•   A detailed error and equilibria analysis of the proposed model is provided in this study.

This work is organized as follows: Section 2 presents the fundamental ideas about fractional Caputo operators employed in this work. The USWs basis and approximation of function by USWs are described in Section 3. The USWs technique for simulating Caputo fractional Rössler attractor model is presented in Section 4. Section 5 provides the error and equilibria analysis. The nonlinear Caputo fractional Rössler attractor model is numerically analyzed in Section 6, showcasing results in tables and graphs that demonstrate the effectiveness of the suggested scheme. Section 7 contains the concluding remarks and the future scope.

2  Mathematical Preliminaries

Some preliminaries and notations about FC are included in this section. This work employs the following fractional operators.

Definition 2.1. The fractional Caputo derivative of a function h(τ) defined on [0, 1] with order α∈(0,1] is given by [66]

0CDταh(τ)=1Γ(1−α)∫0τh′(t)(τ−t)−αdt,0<α<1.(5)

Definition 2.2. For a function h(τ), the fractional integral with α order is expressed by [66]

0CIταh(τ)=1Γ(α)∫0τ(τ−t)α−1h(t)dt,0<τ,α∈(0,1].(6)

•   For 0<α≤1, the relationship between the operators in Eqs. (5) and (6) is described as

00CIτα(00CDταh(τ))=h(τ)−∑j=0⌈α⌉−1τjΓ(j)h(j)(0),τ>0.(7)

3  Wavelets and Function Approximation

3.1 Brief Overview of Ultraspherical Wavelets

The USWs ψn,mγ(τ)=ψ(k,n,m,γ,τ) contain five arguments, where k∈Z0+, n=1,2,3,...,2k, m be the degree of Ultraspherical polynomials such as m=0,1,2,...,M−1, M∈N, τ be the normalized time, and γ is known as Ultraspherical parameter such that γ>−12.

The USWs are defined on [0, 1] as [67–69]

ψn,mγ(τ)={2k+12μm,γUmγ(2k+1τ-2n+1),(n-1)2k≤τ≤n2k,0,elsewhere(8)

where

μm,γ=2γΓ(γ)Γ(m+1)(m+γ)2πΓ(m+2γ),(9)

and Γ(.) denotes the Gamma function [70].

Here, Umγ(τ) is the Ultraspherical polynomial [67,68] defined on [−1,1] having degree m and satisfies following recurrence relations:

U0γ(τ)=1,

U1γ(τ)=2γτ,

Un+1γ(τ)=2(n+γ)τUnγ(τ)−(n−1+2γ)Un−1γ(τ)n+1,n=1,2,...

With respect to the weighted function w(τ)=(1−τ2)γ−12, the aforementioned polynomials are orthogonal and the set of USWs is orthogonal on [0,1] under the weight function wn,k(τ), where

wn,k(τ)=w(2k+1τ-2n+1).

3.2 Function Approximation

Let {ψ1,0γ(τ),...,ψ1,M−1γ(τ),ψ2,0γ(τ),...,ψ2,M−1γ(τ),...,ψ2k−1,0γ(τ),...,ψ2k−1,(M−1)γ(τ)}∈L2[0,1] is the set of USWs,

S=Span{ψ1,0γ(τ),...,ψ1,M−1γ(τ),ψ2,0γ(τ),...,ψ2,M−1γ(τ),...,ψ2k−1,0γ(τ),...,ψ2k−1,(M−1)γ(τ)}, and h(τ)∈L2[0,1] is any element. Then, from the finite-dimensional vector space S, h(τ) has the best approximation such that

‖h(τ)−h0(τ)‖<‖h(τ)−f(τ)‖;h0(τ),f(τ)∈S.

An arbitrary function h(τ)∈L2[0,1] may be described as a combination of USWs as

h(τ)≈∑n=1∞∑m=0∞cn,mψn,mγ(τ),(10)

where cn,m be the unknown coefficient corresponding to ψn,mγ(τ).

If the given series in Eq. (10) is truncated, we obtain

h(τ)≈∑n=12k−1∑m=0M−1cn,mψn,mγ(τ)=CTΨσ~γ(τ),(11)

where Ψσ~γ(τ) and C are 2k−1M×1 order matrices provided by

CT=[c1,0,c1,1,...,c1,M−1,c2,0,c2,1,...,c2,M−1,...,c2k−1,0,c2k−1,1,...,c2k−1,M−1],(12)

Ψσ~γ(τ)=[ψ1,0γ(τ),...,ψ1,M−1γ(τ),ψ2,0γ(τ),...,ψ2,M−1γ(τ),...,ψ2k−1,0γ(τ),...,ψ2k−1,M−1γ(τ)]T.(13)

In this study, we use 2k−1M=σ~ to represent the total USWs basis in the calculating process. Any unknown function can be computed with ease utilizing the concept of function approximation corresponding to known USWs and the computational complexity is minimized in comparison to other available techniques.

4  Solution of Caputo Fractional Rössler Attractor Model

In the present section, the Caputo fractional Rössler attractor model is solved by the USWs scheme with the collocation points. In this study, we selected the specific collocation points given in Eq. (25) because they are uniformly distributed over the interval, simplifying implementation and ensuring that the computational load is evenly distributed across the domain. Uniformly distributed points are particularly well-suited for our method, as they align well with the USWs framework, preserving the accuracy of the numerical solution for the fractional Rössler attractor model.

To evaluate the solutions of the Caputo fractional Rössler attractor model, the procedure of the described wavelets scheme is given as:

4.1 For Model 1 (Asymmetric Caputo Fractional Rössler Attractor Model)

Consider the asymmetric Caputo fractional Rössler attractor model given in Eq. (1) and provide the approximation for the unknown function 00CDταx(τ) of Eq. (1) as combination of USWs using Eq. (11) as

00CDταx(τ)≈∑n=12k−1∑m=0M−1cn,m(1)ψn,mγ(τ)=C10TΨσ~γ(τ),(14)

where Ψσ~γ(τ) is given in Eqs. (8) and (13) and wavelet coefficients C1T is defined as

C1T=[c1,0(1),c1,1(1),...,c1,M−1(1),c2,0(1),c2,1(1),...,c2,M−1(1),...,c2k−1,0(1),c2k−1,1(1),...,c2k−1,M−1(1)].(15)

Taking the integral of Eq. (6) on Eq. (14), and using Eqs. (2) and (7), we get

00CIτα(00CDταx(τ))=C10T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dtx(τ)−x(0)=C10T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dtx(τ)=x1+C10T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dtx(τ)=x1+C10T1Γ(α)G(τ,α,γ),(16)

where G(τ,α,γ)=∫0τ(τ−t)α−1Ψσ~γ(t)dt is computed directly by performing integration in Mathematica.

Similarly, the unknown function 00CDταy(τ) in Eq. (1) is expressed through USWs as

00CDταy(τ)≈∑n=12k−1∑m=0M−1cn,m(2)ψn,mγ(τ)=C20TΨσ~γ(τ),(17)

where wavelet coefficient C2T is defined as

C2T=[c1,0(2),c1,1(2),...,c1,M−1(2),c2,0(2),c2,1(2),...,c2,M−1(2),...,c2k−1,0(2),c2k−1,1(2),...,c2k−1,M−1(2)].(18)

Taking integral of Eq. (6) on Eq. (17) and using Eqs. (2) and (7), we get

00CIτα(00CDταy(τ))=C20T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dty(τ)−y(0)=C20T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dty(τ)=y1+C20T1Γ(α)G(τ,α,γ).(19)

Similarly, the unknown term 00CDταz(τ) in Eq. (1) is expressed via UVWs as

00CDταz(τ)≈∑n=12k−1∑m=0M−1cn,m(3)ψn,mγ(τ)=C30TΨσ~γ(τ),(20)

where wavelet coefficient C3T is defined as

C3T=[c1,0(3),c1,1(3),...,c1,M−1(3),c2,0(3),c2,1(3),...,c2,M−1(3),...,c2k−1,0(3),c2k−1,1(3),...,c2k−1,M−1(3)].(21)

Taking integral of Eq. (6) on Eq. (20) and using Eqs. (2) and (7), we get

00CIτα(00CDταz(τ))=C30T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dtz(τ)−z(0)=C30T1Γ(α)∫0τ(τ−t)α−1Ψσ~γ(t)dtz(τ)=z1+C30T1Γ(α)G(τ,α,γ).(22)

Substituting Eqs. (14)–(22) in Eq. (1), we get the wavelet approximate form of Eq. (1) as

C1TΨσ~γ(τ)=−(y1+C20T1Γ(α)G(τ,α,γ))−(z1+C30T1Γ(α)G(τ,α,γ))C2TΨσ~γ(τ)=(x1+C10T1Γ(α)G(τ,α,γ))+a(y1+C20T1Γ(α)G(τ,α,γ))C3TΨσ~γ(τ)=b+(z1+C30T1Γ(α)G(τ,α,γ))((x1+C10T1Γ(α)G(τ,α,γ))−c),(23)

Collocating Eq. (23) at the collocation point τj, (3σ~) algebraic equations are obtained as

C1TΨσ~γ(τj)=−(y1+C20T1Γ(α)G(τj,α,γ))−(z1+C30T1Γ(α)G(τj,α,γ))C2TΨσ~γ(τj)=(x1+C10T1Γ(α)G(τj,α,γ))+a(y1+C20T1Γ(α)G(τj,α,γ))C3TΨσ~γ(τj)=b+(z1+C30T1Γ(α)G(τj,α,γ))((x1+C10T1Γ(α)G(τj,α,γ))−c),(24)

where τj is given by

τj=((2j−1)2σ~);j=1,2,...,σ~.(25)

Solve the system of algebraic equations in Eq. (24) by Newton iteration method, we can readily obtain the wavelet coefficient vectors C1, C2 and C3. Using the value of C1, C2 and C3 in Eqs. (16), (19) and (22), we estimate the wavelets approximation for x(τ), y(τ) and z(τ) of Model 1.

The algorithm of the suggested approach for Model 1 is described in Table 1.

4.2 For Model 2 (Symmetric Caputo Fractional Rössler Attractor Model):

Express the unknown terms 00CDταx(τ),00CDταy(τ),00CDταz(τ) in Eq. (3) as a combination of USWs and obtain the same Eqs. (14)–(22) using Eq. (4). Now substituting Eqs. (14)–(22) in Eq. (3), we obtain the wavelet approximate form of Eq. (3) as

C1TΨσ~γ(τ)=−(y2+C20T1Γ(α)G(τ,α,γ))−(y2+C20T1Γ(α)G(τ,α,γ))(z2+C30T1Γ(α)G(τ,α,γ))C2TΨσ~γ(τ)=(x2+C10T1Γ(α)G(τ,α,γ))+a(y2+C20T1Γ(α)G(τ,α,γ))C3TΨσ~γ(τ)=b+(z2+C30T1Γ(α)G(τ,α,γ))((x2+C10T1Γ(α)G(τ,α,γ))2−c).(26)

Collocating Eq. (26) at the collocation point τj, (3σ~) algebraic equations are obtained as

C1TΨσ~γ(τj)=−(y2+C20T1Γ(α)G(τj,α,γ))−(y2+C20T1Γ(α)G(τj,α,γ))(z2+C30T1Γ(α)G(τj,α,γ))C2TΨσ~γ(τj)=(x2+C10T1Γ(α)G(τj,α,γ))+a(y2+C20T1Γ(α)G(τj,α,γ))C3TΨσ~γ(τj)=b+(z2+C30T1Γ(α)G(τj,α,γ))((x2+C10T1Γ(α)G(τj,α,γ))2−c),(27)

where τj is provided in Eq. (25).

Evaluate the algebraic system in Eq. (27) by Newton iteration method, we can readily obtain the wavelet coefficient vectors C1, C2 & C3. Using the value of C1, C2 & C3, we estimate the wavelets approximation of x(τ), y(τ) & z(τ) of Model 2.

The algorithm of the proposed approach for Model 2 is described in Table 2.

5  Error and Equilibria Analysis

This section first presents a detailed error analysis, followed by a comprehensive stability analysis of the Caputo fractional Rössler attractor model.

5.1 Error Analysis

The residual error formula is presented for comparison purposes and to examine the efficacy of the described approach. The analytical solution of the considered model is not available for integer and non-integer order, therefore we provide a residual error function to assess the precision of the given scheme.

•   For asymmetric fractional Rössler attractor, the residual error function is given as

Rx(τ)=|00CDταxσ~(τ)+yσ~(τ)+zσ~(τ)|,Ry(τ)=|00CDταyσ~(τ)−xσ~(τ)−ayσ~(τ)|,Rz(τ)=|00CDταzσ~(τ)−b−zσ~(τ)(xσ~(τ)−c)|,

where xσ~(τ),yσ~(τ)andzσ~(τ) are wavelets approximated solutions of the considered Model 1.

•   For symmetric fractional Rössler attractor, the residual error function is given as

Rx(τ)=|00CDταxσ~(τ)+yσ~(τ)+yσ~(τ)zσ~(τ)|,Ry(τ)=|00CDταy(τ)−xσ~(τ)−ayσ~(τ)|,Rz(τ)=|00CDταzσ~(τ)−b−zσ~(τ)((xσ~(τ))2−c)|.

•   The maximum residual error (REmax) is calculated as

RE(x(τ))max=maxτ∈[0,1]|Rx(τ)|,RE(y(τ))max=maxτ∈[0,1]|Ry(τ)|,RE(z(τ))max=maxτ∈[0,1]|Rz(τ)|.

•   The minimum residual error (REmin) is calculated as

RE(x(τ))min=minτ∈[0,1]|Rx(τ)|,RE(y(τ))min=minτ∈[0,1]|Ry(τ)|,RE(z(τ))min=minτ∈[0,1]|Rz(τ)|.

•   The L2-Residual error is determined by

L2error=||R(τ)||0L2[0,1]=∫01(R(τ))2dτ.

The fundamental findings corresponding to the approximation of Ultraspherical polynomials serve as the foundation for exploring the convergence of USWs approximations. The convergence of the series expansion of h(τ)∈L2[0,1] is given by the following theorem.

Theorem 5.1. Consider the USWs expansion ∑n=12k−1∑m=0M−1cn,mψn,mγ(τ)=∑j=0σ~cjψjγ(τ)=CTΨσ~γ(τ) of a function h(τ)∈L2[0,1], the error estimate is obtained as

‖h(τ)−∑j=0σ~cjψjγ(τ)‖2≤ℵ(1+γ)2(M+γ)(M−3)7/24k,

where ℵ=maxτ∈[0,1]|hm+1(τ)|.

Also,

limσ~→∞‖h(τ)−∑j=0σ~cjψjγ(τ)‖=0,

where σ~=2k−1M.

Proof. For proof, see [68]. ▪

5.2 Existence and Uniqueness of Solution

Theorem 5.2 Every solution of the system (1) and (3) with positive initial values x(0),y(0),z(0) exists and is unique in the interval [0,∞).

Proof. For Model 1:

Let X∗=(x,y,z)T and G(X∗)=(G1(X∗),G2(X∗),G3(X∗))T such that

G1(X∗)=−y(τ)−z(τ)=0,G2(X∗)=x(τ)+ay(τ)=0,G3(X∗)=b+z(τ)(x(τ)−c)=0.(28)

Therefore, the system in (28) can be written in the form X^∗=G(X∗) where G(X∗):R+→R+3 with X∗(0)=X0∗∈R+3,Gr∈R∞(R+3) for r=1,2,3. Consequently, the vector function G is a fully continuous and locally Lipschitzian function of the variables x,y & z in the positive quadrant W={(x(τ),y(τ),z(τ));x>0,y>0,z>0}. As a result, we can easily state that any solution (x,y,z) to system (1) with positive initial values exists and is unique across the interval [0,∞) for all τ≥0.

For Model 2: Same proof as above. ▪

5.3 Equilibria Analysis

The solution of the Caputo fractional Rössler attractor model is overall assessed on the groundwork of steadiness points [64,65]. The steadiness points of the regarded model are attained by resolving the model under uniform state conditions. The examination of equilibrium points is split up into the eigenvalues signs. The Jacobian matrix of the model is constructed to analyze the signs of the eigenvalues to determine stability. Then, the behavior of the considered fractional model for each point of equilibrium can be subjectively evaluated by inspecting the nature of eigenvalues. When the Jacobian matrix has complex eigenvalues and the real part of the eigenvalues corresponding to the equilibrium points is positive, then the equilibrium points are the saddle spiral points. Also, if the Jacobian matrix has complex eigenvalues and positive real eigenvalues corresponding to the equilibrium points, then the equilibrium points are also saddle spiral points. The equilibrium points of the considered system will be asymptotically stable if all eigenvalues of the associated Jacobian matrix fulfill the Matignon condition [71]. The particular values of the parameters involved in the considered model are taken from [58].

For Model 1: The equilibrium points for asymmetric Caputo fractional Rössler attractor model are obtained as

00CDταx(τ)=0,00CDταy(τ)=0,00CDταz(τ)=0,

i.e.,

−y(τ)−z(τ)=0,x(τ)+ay(τ)=0,b+z(τ)(x(τ)−c)=0.(29)

The equilibrium points (Ei) of Eq. (1) are determined by solving Eq. (29) as

(E1,E2)=((aβ1,−β1,β1),(aβ2,−β2,β2)),

where

β1=12(c−c2−4aba),β2=12(c+c2−4aba).

Computing Jacobian MatrixJ for the considered Model 1 as

J=(0−1−11a0z0−c+x).(30)

•   The Jacobian Matrix at E1 using Eq. (30) is given as

J(E1)=(0−1−11a0(12(c−c2−4aba))0−c+a(12(c−c2−4aba))).

The eigenvalues of J(E1) are computed by finding the roots of its characteristic equation as the form of (−Υ1−η1i,−Υ1+η1i,κ1), where Υ1 represents real part and η1 represents imaginary part, and κ1 represents the real number.

For a=0.2,b=0.2,c=5.7, the eigenvalues are obtained as:

(−4.59607×10−6−5.42803i,−4.59607×10−6+5.42803i,0.192983).

•   The Jacobian Matrix at E2 using Eq. (30) is given as

J(E2)=(0−1−11a012(c+c2−4aba)0−c+a(12(c+c2−4aba))).

The eigenvalues of J(E2) are computed by finding the roots of its characteristic equation as the form of (Υ2−η2i,Υ2+η2i,−κ2), where Υ2 represents real part and η2 represents imaginary part, and κ2 represents the real number.

For a=0.2,b=0.2,c=5.7, the eigenvalues are obtained as:

(9.70009×10−2−9.95193×10−1i,9.70009×10−2+9.95193×10−1i,−5.6869).

The discriminant of the characteristic equation of the Jacobian matrix corresponding to the equilibrium points E1 & E2 for Model 1 is less than zero under the assumed value of the model’s parameters, therefore we obtained two imaginary roots and one real root for J(E1) & J(E2). It is observed from general and particular values of eigenvalues, E1 & E2 are spiral saddle points for Model 1.

For Model 2: The equilibrium points for symmetric Caputo fractional Rössler attractor model are obtained as

00CDταx(τ)=0,00CDταy(τ)=0,00CDταz(τ)=0,

i.e.,

−y(τ)−y(τ)z(τ)=0,x(τ)+ay(τ)=0,b+z(τ)((x(τ))2−c)=0.(31)

By solving Eq. (31), we get the model’s equilibrium points.

The equilibrium points (Ei) of Eq. (3) are obtained by solving Eq. (31) as

(E1,E2,E3)=((0,0,bc),(−aρ,ρ,−1),(aρ,−ρ,−1)),

where ρ=b+ca2.

Computing Jacobian Matrix J for the considered model as

J=(0−1−z−y1a02xz0−c+x2).(32)

•   The Jacobian Matrix at E1 using Eq. (32) is given as

J(E1)=(0−1−bc01a000−c).