Existence of Solution and Self-Exciting Attractor in the Fractional-Order Gyrostat Dynamical System

Marwan, Muhammad; Ali, Gauhar; Khan, Ramla

doi:https://doi.org/10.1155/2022/3505634

Complexity

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Abstract Introduction Conclusion Appendix Data Availability Ethical Approval Conflicts of Interest Authors’ Contributions References Copyright Related Articles

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Chaos in Applied Sciences and Engineering 2021

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Research Article | Open Access

Volume 2022 | Article ID 3505634 | https://doi.org/10.1155/2022/3505634

Existence of Solution and Self-Exciting Attractor in the Fractional-Order Gyrostat Dynamical System

Muhammad Marwan,¹Gauhar Ali,²and Ramla Khan³

Academic Editor: Chun-Biao Li

Received25 Mar 2022

Revised05 May 2022

Accepted09 May 2022

Published18 Jul 2022

Abstract

This work identifies the influence of chaos theory on fractional calculus by providing a theorem for the existence and stability of solution in fractional-order gyrostat model with the help of a fixed-point theorem. We modified an integer order gyrostat model consisting of three rotors into fractional order by attaching rotatory fuel-filled tank and provided an iterative scheme for our proposed model as a working rule of obtained analytical results. Moreover, this iterative scheme is injected into algorithms for a system of integer order dynamical systems to observe Lyapunov exponents and a bifurcation diagram for our proposed fractional-order dynamical model. Furthermore, we obtained five equilibrium points, including four unstable spirals and one saddle node, using local dynamical analysis which acted as self-exciting attractors and a separatrix in a global domain.

1. Introduction

System of ordinary differential equations [1]

Is called the dynamical system, and a parameter in the velocity vector field is termed as bifurcation parameter if system (1) changes its topological structure with the variation in parameter values, whereas the process of changing in qualitative structures is known as bifurcation. There are several types of bifurcation including saddle node [2], Hopf [3–7], and zero-Hopf [8–11]. The bifurcation diagram [12] for the parameter makes it easy for predicting the type of bifurcation and existence of chaos in system (1). Chaos has a vital role in engineering [13–17], medical [18–20], aeronautics [14, 21] and fluid dynamics [22–24]. Apart from the above cited applications, its great influence can also be found in fractional calculus [25–27] and reference therein. Dynamical systems based on ordinary differential equations with an integer order, , describe velocity vectors, but for fractional order, , researchers aim to target velocity vectors and replace it by differential equation with order between 0 and 1. Several discretization techniques such as fractional linear multistep [28], Adam [29], predictor-corrector [30], and Adam–Bashforth/Moulton [31] are used to solve fractional-order dynamical systems since decade, but the most flexible scheme with fast convergence in solving nonlinear problems is the variation iteration method (VIM). This technique was used for integer order dynamical systems, but later on modified for fractional-order systems by introducing Lagrangian multiplier [32] into it. Many researchers have enhanced its importance by using it in several engineering-based complex problems, such as in 2006, the variation iteration scheme was utilized for fractional-order systems by Odibat and Momani [33], whereas new development in the VIM was carried by Wu and Baleanu [32] in 2013 to overcome its limitation. Recently in 2021, Kumar and Gupta [34] worked on application of the VIM in a fuzzy-based system.

It has been observed from the above-cited work and our knowledge from the literature that dynamical systems related to spacecrafts or its attached devices such as beam and gyrostat have never been considered for the existence of solution and self-exciting attractors in a fractional-order form. Therefore, we have restructured the gyrostat chaotic system [35] into a fractional order along with the addition of a rotatory liquid-filled tank to discuss its unique solution, bounds, and stability using the fixed point theory. Moreover, for bringing novelty into our work, a variation iteration scheme has been used in our proposed fractional-order system to observe chaos into it. For this purpose, several algorithms such as by Wolf et al. [36] and the bifurcation diagram [37] were modified by injecting the VIM iteration scheme into these algorithms. Finally, analyzing local dynamics of our proposed model, trajectories around five equilibrium points with four unstable spirals and a single saddle node motivated us to search for self-exciting attractors with a separatrix in a global domain.

The following pattern can be followed for understanding the rest of the paper. In Section 2, the gyrostat chaotic system is remodeled by adding rotatory liquid-filled tank and modified into fractional order. Several theorems have been proved in Section 3 for the existence of solution and stability. An iteration scheme for our proposed model has been introduced in Section 4, while several applications of this scheme related to dynamical analysis are discussed in Section 5. Finally, Section 6 comprises concluding remarks and future target.

2. Modeling of Gyro Chaotic System Attached with Fuel-Filled Tank

Gyrostat is a device consisting of rotors, used as an attachment in larger objects for bringing stability in their dynamics with the passage of time. The system of three-dimensional ordinary differential equations for the gyrostat model is designed by Qi et al. [35]:where is the angular velocity vector, are the principal moments of inertia of the gyrostat in the body axis frame, are constants of total angular momentum, whereas and are external and disturbed torques applied on the gyrostat, respectively.

A tank, rotating about an angle at desired point (shown in Figure 1), is attached with an originally disturbed gyrostat system given in (2). It is observed that is a vector of external forces applied on the gyrostat. Therefore, we have attached a tank containing fuel which exert external forces on the gyrostat due to rotation of the attached tank with respect to axis about angle at desired point . Hence, we replaced withwhere , is the rotation matrix for axis and is a desired point about which one can rotate the attached tank. Therefore, using vector in ((2) [38]) and given in (3) into (2), we obtain the following equation:

System (4) is a fractional-order mathematical representation of the model given in Figure 1 in which is the fractional number between 0 and 1 exclusive, = is a damping constant vector, while , , and are defined in equation (2). Moreover, system (4) shows chaotic behavior for , , , , and . The phase portrait of system (4) with given initial and parameter values can be seen in region of Figure 2.

3. Existence and Stability of Solution

In this part of our paper, we determined results based on the existence theory for system (4) using the fixed point theorem with Banach space. Therefore, basic definitions and important lemmas are considered for the understanding of this work.

Definition 1 (see [39]). The integral of fractional order for a function is given by

Definition 2 (see [39]). The Caputo fractional derivative of order of a continuous function is given bywhere .
Following two lemmas have importance in achieving the solutions of the systems consisting of fractional differential equations.

Lemma 1 (see [39]). Assume , then the solution of fractional differential equationOf order is

Lemma 2 (see [39]). Let us consider , with a derivative of fractional order , then

We begin our work by introducing , , and on the right side of equation (4) and for convenience, we use the following notions:

and

System (4) can be rewritten, using (5) as

According to Lemma 2, problem (12) can be converted into an integral equation

Definition 3. Let us consider a Banach space under the suitable normand the operator is defined aswhere and . Then, the following assumptions are true:
There exists a positive constant such that The following inequality holds for positive constants :

Theorem 1. Let us consider that and assumption is satisfied. Then, there exists a unique solution of system (4) with the contraction of operator .

Proof. Let , then one hasThis shows that is a contraction. Hence, our desired result is obtained, that system (4) has a unique solution.

Theorem 2. The integral equation (8) has at least one solution if under the assumptions of and .

Proof. For existence of a solution for operator , it is enough to show that is completely continuous, and there exists an element such that for . Therefore, our proof will pass through three steps for achieving our desired results.

Step 1. Let us consider a sequence in and for each , we have Hence, approaches as time tends to infinityEquation (20) identifies continuity of an operator .

Step 2. Let us consider a bounded set , where is a positive real number. Then, for any , we have Hence, maps a bounded set into a bounded set.

Step 3. The image of a bounded set under is equicontinuous in .
Let in and , we have As , then , and thus, is continuous and bounded. Hence, shows uniform continuity of . Therefore, steps show that is completely continuous.

Step 4. Finally, we have to show that for some , is bounded. Let and for any , we haveSimplifying inequality (19) yieldsThis shows that the defined set is bounded. Hence using Schaefer’s theorem [40], system (4) has at least one solution.
For achieving stability, a negligible perturbation parameter can be included in such that(i)(ii) for

Lemma 3. Solution of the perturbed problemSatisfies the following relation:

Theorem 3. Gyrostat system (4) achieves Ullam–Hyers stability if and assumption , together with Lemma 3 is satisfied.

Proof. Let be any solution and is a unique solution, thenThis implies thatHence, solution of the proposed system (4) is Ullam–Hyers stable.

4. Variational Iterative Scheme for System (4)

An iterative scheme, variational iterative method (VIM), is introduced in this section using successive approximations of the solution for rapid convergence and analytical results discussed in Section 2.

4.1. Working Rule

To express the VIM, we consider general nonlinear differential equation aswhere , , and are linear, nonlinear, and source functions, while the corrector function for (29) is considered as

in (30) is defined as , whereas is used as a restricted value with . Then, the exact solution can be obtained as

System (4) can be discretized using VIM aswhere , and .

For where

For , we have

The values of , and , in (35) are given in Appendix . In the next section, we have discussed system (4) analytically and qualitatively. For numerical simulations, our designed algorithm is used to plot Lyapunov exponents and bifurcation diagram in integer order as well as fractional-order chaotic systems.

5. Dynamical Analysis

The fractional-order dynamical system exhibits chaos for some values of fractional term, , but using a hit and trial method for such purpose is difficult to investigate chaos in dynamical systems. Therefore, we plotted the bifurcation diagram for system (4) with respect to fractional term, . A noisy dense area is observed in Figure 3 that illustrates occurrence of chaos in the fractional-order gyrostat system beginning with .

In section 2, we used the concept of the fixed point theory to obtain at least one solution of system (4). Hence, for fixed points, we consider the following function , equals to zero:

After fixing all parameters given in section (1), then we solve equation (36) to get the following five equilibrium points:

In Theorem 4 local dynamical analysis of system (4) is used for observing trajectories around equilibrium points (27).

Theorem 4. A gyrostat chaotic system (4) is composed of five equilibrium points, in which is the saddle node and are all unstable saddle spirals. Moreover, these spirals lead to four attractors and one saddle node that act as a separatrix as t extends.

Proof. Five equilibrium points are calculated in equation (27). The Jacobian matrix plays a vital role in the system of differential equations for local dynamical analysis. Therefore, the Jacobian matrix of system (4) isAnd the Jacobian matrix for fixed parameter values at isThe characteristic equations of the Jacobian matrix (29) isSolution of (40) results into single positive and two negative eigenvalues:Equation (41) illustrates that two states will move away from , while a single state will move inward towards equilibria: , and such information shows occurrence of the saddle. The Jacobian matrix at isAnd the corresponding characteristic equation isSolution of (43) gives three eigenvalues with one negative real and two complex numbers with positive real part:Equation (44) describes occurrence of the unstable spiral. In a similar fashion, one can achieveEigenvalues of , , and . In view of (45), equilibrium points, are also unstable spirals.
Analytical results (29–35) are explained in Figure 4, which illustrate trajectories of system (4) around their equilibrium points. Five different colors are used for each equilibrium point, which are also highlighted as a legend in Figure 4. It is observed that are unstable spirals plotted in green, brown, blue, and black colors, while red color shows the saddle node. In detail, we can see that the red trajectory starts from and passes through the regions of , , , and with the passage of time. The trajectory for shows a spiral emerging from its equilibrium point and is moving away from it. After some time it has been observed that the green trajectory is acting as a heteroclinic orbit: . The same theory can be observed between and , when a brown colored orbit starts with high unstable oscillations and approaches to a region occupied by . Apart from these four unstable spirals, one can also locate saddle node equilibria in the red color, in which its trajectory passes through regions acquired by unstable spirals and act as a separatrix between them. For further analysis, we have extended time for observing the trajectories around five equilibrium points in the greater domain. It has been analyzed that four unstable equilibrium points are self-exciting attractors and occupy four basins. Moreover, the combination of all these four regions leads to the concept of a strange attractor in system (4). Studying in more depth, it has been also observed that the saddle node in the global domain is busy in separating regions of self-exciting attractors. For getting more knowledge about chaoticity in the fractional-order gyrostat system (4), some basic results are used for the possibility and detection of chaos.

5.1. Lyapunov Exponents

The Lyapunov exponent is one of the fundamental results, which help researchers in pointing out existence of unpredictability in trajectories of their corresponding systems. Moreover, in a three-dimensional autonomous system of ordinary differential equations, there exist three Lyapunov exponents , . Now, if = , then it shows existence of chaos, whereas illustrates the existence of periodic solutions. In Figure 5, three Lyapunov exponents can be observed, emerging from and leading to , which motivated us to work further on it and find out chaotic trajectories in it. For further investigation, we have used the concept of the bifurcation diagram [12].

5.2. Bifurcation Leading to Chaos

For confirmation of existence of chaos in our proposed model (4), we fixed all other parameter values except for . For damping coefficient, , it is observed in Figure 6 that there seems no bifurcation in system (4) for . The single bifurcation emerges at and continues till , which changes into period doubling bifurcation (PDB) for . Trajectories of our proposed system jump into the chaotic region for lying in interval . One can observe symmetric behavior in sense of bifurcation leading to chaos in Figure 6. If we start from , two lines can be observed that are converted into period doubling bifurcations, then period period chaotic region. This concept is also explained with the aid of a series of phase portraits, which confirms chaotic behavior in our proposed system. Therefore, we have divided bifurcation diagram 5 in nine regions and plotted phase portraits to their corresponding values.

Figure 2 is validation of Figure 6, which explains existence of chaos in detail by moving clockwise or anticlockwise. Therefore, we have an indexed sequence of phase portraits for . If we start from region , a spiral trajectory can be observed and is expanding in regions and . This trajectory is converted into period doubling and period 4 bifurcations in region and for to 5.1, respectively. In region , chaotic movement of trajectories can be observed, which gradually declines to period period period 2 bifurcations by moving in the anticlockwise direction from region to . Similarly, if we begin in the clockwise direction, region to , one can see trajectory starts with period doubling bifurcation for is gradually increasing to period period chaos from region to , then decline in a symmetric way is observed from chaos to the period doubling bifurcation till region which finally shrinks into spiral and bifurcation disappearing in region to .

Figure 7 is the series of Lyapunov exponents corresponding to each subregion of the bifurcation diagram (plotted in Figure 6). In Figure 2, the existence of chaos in a symmetrical way is thoroughly discussed, but in Figures 7(a)–7(h) the same concept is explained in more detail where for each value of the damping coefficient ; there exist different values of Lyapunov exponents. Moreover, it is also observed that the Lyapunov exponent of system (4) tends to as the damping coefficient approaches to 6.43.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

6. Conclusion and Future Work

An integer ordered dynamical system of the gyrostat was considered by researchers since decade, and a variety of work related to chaos was achieved with the help of sensitivity in its initial conditions. But we have analyzed the gyrostat model with modification by attaching a rotatory cylinder and conversion into fractional order for the first time. Several theorems were proved in this work for the existence of solution and Ullam–Hyers stability. Moreover, dealing with the fractional-order system does not work on ; therefore, an iterative scheme was designed for system (4) to attain chaos in the fractional order. Studying local dynamics of system (4) leads to five solutions with four unstable spirals and one saddle node, but observing trajectories around these equilibrium points in global domain acted as a self-exciting attractor and separatrix. In future, we aim to target fractional-order dynamical systems for codimension 2 bifurcations, which itself is a tedious task due to a large number of involved parameters. Apart from bifurcation, our future aim also involves application of (integer and fractional) ordered chaotic systems in strategy-based mobile gaming.

Appendix

Our discretization scheme is based on an iterative technique; therefore, for and 1, analytical work is presented in Section 3. But increasing the number of leads to tedious analytic. Hence, for , the leftover calculation in equation (25) is done here:where

In a similar way, the values of are calculated as

With

Finally, the values of arewhere

For further iterations, things were very tedious; therefore, we used MATLAB for further numerical calculations.

Data Availability

MATLAB is used for graphs and tedious calculations in this work, and codes are available upon reasonable request from the corresponding author.

Ethical Approval

This paper does not contain any studies with human participants or animals performed by any of the author.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Muhammad Marwan conducted formal analysis, methodology, investigation, and writing of the original draft. Gauhar Ali conducted formal analysis, methodology, and conceptualization. Ramla Khan conducted methodology, software, conceptualization, and writing.

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