Abstract
We investigate a discrete-time Chen system. First, we give the topological classifications of the fixed points of this system. Then, we analytically show that the discrete Chen system underlies a Neimark–Sacker (NS) bifurcation and period doubling (PD) under specific parametric circumstances. We confirm the existence of a PD and NS bifurcation via the explicit PD-NS bifurcation criterion and determine the direction of both bifurcations with the help of center manifold theory. We performed numerical simulations to confirm our analytical results. Furthermore, we use the 0-1 chaos test to quantify whether there is chaos in the system or not. At the end, the hybrid control strategy and the OGY (Ott, Grebogi, and Yorke) method are applied to eliminate chaotic trajectories of the system.
1. Introduction
A dynamical system is referred to as a system that changes over time. Today, a mathematical model linked to dynamical systems is used in fields such as weather prediction, ecosystem regulation, heart rate control, power system collapse prevention, and biological applications to human psychiatry, among others. A dynamical system can be divided into continuous dynamical systems and discrete dynamical systems. While many academics have focused on and conducted in-depth investigations into systems bifurcation in continuous dynamical systems, only a few studies have focused on systems bifurcation in discrete dynamical systems. In the continuous dynamical system, many renowned researchers [1–5] have extensively investigated three or higher dimensional. In 1963, Lorenz [2] made the discovery of a 3-D chaotic system while studying the weather model for atmospheric convection. His classical pioneering work inspired scientists to investigate several 3-D chaotic systems. Lü et al. [3] and Ueta and Chen [5] constructed a new critical chaotic system by anticontrol technique in the Lorenz system [2]. These systems are known as Lü system and Chen’s system, respectively. Qualitative analysis of these empirical works found many dynamical properties, including local bifurcations, chaotic, periodic, quasiperiodic orbits, and route to chaos. They also obtained super-critical and subcritical bifurcation conditions around positive equilibriums. A novel 3-D chaotic system [4] is investigated to increase the complexity of the chaotic system and the precision of weak signal detection. A 3-D jerk dynamical system examined in [1], which can be utilized as an analog simulator for experiments made in a lab. This work investigated several dramatic and uncommon bifurcation situations, such as those with multiple attractors, symmetry-recovering crises, and basins of attraction for a variety of coexisting attractors. However, many exploratory works have suggested that discrete-time models are more suitable compared to the differential equation model, since the discrete-time model reveals rich chaotic dynamics and gives effective computational models for numerical simulations [6–23]. These researches explored unpredicted properties, including the emergence of (PD-NS) bifurcations and chaos phenomena either numerically or by the applications of normal form theory. These studies focused solely on three-dimensional discrete systems.
Recently, a very short number of contributions have been dedicated to the study of the dynamics of three-dimensional discrete systems [24–31]. For example, discrete-time epidemic models SIR, SEIR, and hypertensive or diabetic exposed to COVID-19 discussed in [24, 26, 28, 32], respectively. In [30], the authors investigated a discrete financial system. In [29], the authors studied a discrete chaotic system. In these works, the researchers concentrated their effort to determine the direction and stability of the PD and NS bifurcations by using explicit PD-NS bifurcation criterion, center manifold theory, and bifurcation theory. The studies in [27] investigated discrete population models. In [25], dynamics of the discrete three-species food chain model is studied through NS bifurcation. These studies used only explicit PD-NS bifurcations criterion and numerical simulations for the existence of PD and NS bifurcations. In nonlinear field research, chaos theory has recently attracted a lot of attention. Chaos is strongly hooked into the initial conditions for the answer trajectories, or the exponential aberration in solution trajectories for little differences within the initial conditions in a discrete system.
In this paper, we consider the Chen system [5]:
In system (1), are the state variables denoting the rate of convective overtuning, the horizontal temperature difference, and the vertical temperature difference, respectively. The parameters in the system represent the Prandtl number, the Rayleigh number, and some physical proportions of the region under study, and for a more detailed description of these parameters, we refer to ref [33].
One potential application of the Chen system is weather prediction. The chaotic behavior of the system can help to model complex atmospheric phenomena, such as the formation and movement of storm systems. By simulating the behavior of the Chen system under various initial conditions, researchers can generate forecasts for weather patterns with greater accuracy than traditional methods. This has the potential to greatly benefit industries such as agriculture, transportation, and emergency response, which rely on accurate weather predictions to make important decisions.
Another example is the application of the Chen system in biological research. The chaotic dynamics of the system can be used to model complex biological systems, such as neural networks in the brain. By simulating the behavior of the Chen system, researchers can gain insights into how these systems operate and potentially develop new treatments for neurological disorders.
Overall, the Chen system’s broad application prospects make it an exciting area of research with the potential to revolutionize a variety of fields.
In order to obtain the discrete Chen system with integral step size , the forward Euler approach is used as follows:
In a discrete system, both the PD and NS bifurcations play a significant role in the generation of critical chaotic dynamics and trigger a route to chaos. The objective of this work is to analyze systematically the conditions for the occurrence of PD and NS bifurcations using an explicit PD-NS bifurcation criterion and to determine the stability and direction of both bifurcations using applications of bifurcation theory.
The remainder of this paper is structured as follows. Section 2 explores the topological classifications of fixed points. In Section 3, we analytically discuss that under a certain parametric condition, system (2) undergoes PD or NS bifurcations. In Section 4, we present the dynamics of system (2) numerically including diagrams of bifurcations, phase portraits, and maximum Lyapunov exponents (MLEs) to validate our analytical findings. We also use the chaos test to confirm the existence of chaos in the system. In Section 5, we implement hybrid control strategy and the OGY method to stabilize chaos of the uncontrolled system. In Section 6, we present a short discussion.
2. Existence and Local Stability Analysis of Fixed Points
The following system of equations solutions represents the fixed points of system (2):
Lemma 1. (i)There is only one fixed point in the system for any variation of the parameter values (ii)If , then the system has three fixed points ,
The Jacobian matrix of system (2) evaluated at is as follows:
The eigenvalues of the matrix are the roots of the following characteristic equation:where
We begin by outlining the following Lemma regarding the prerequisites that must be met for stability near a fixed point of system (2)
Lemma 2 [34]. Suppose that , . Then, each root of the following equation has to follow the necessary and sufficient conditionsto satisfy are and .
Now, the topological classifications of system (2) around a fixed points and are given as follows.
At , the Jacobian matrix has eigenvalues and satisfies the following equation:where , . We discuss the following Lemma regarding the bifurcation behavior of system (2) around the fixed point .
Lemma 3. For every parameter value selection, the fixed point is a sink if source if nonhyperbolic if
Let
Then, system (2) experiences a PD bifurcation at when the parameters fluctuate within a narrow region of . Letthen system (2) experience a NS bifurcation at if the parameters change around the set . Now, the eigenvalues of the matrix satisfy the following equation:where
The following Lemma is provided for stability condition of the fixed point .
Lemma 4. The fixed point of system (2) is locally asymptotically stable if and only if the coefficients of of (6) satisfy and
3. Analysis of Bifurcations
In this section, we will discuss the existence, direction, and stability analysis of PD and NS bifurcations near the fixed point and by using an explicit PD-NS bifurcation criterion without computing the eigenvalues of the respective system and bifurcation theory [35–37]. We consider as the bifurcation parameter, otherwise stated.
3.1. NS Bifurcation around
When and the parameters , then the eigenvalues of the system (2) are
Let , then we have
Moreover, the transversality and nonresonance conditions yield
Theorem 5. Suppose (15) holds and , then NS bifurcation at a fixed point for system (2) if changes its value in a small neighborhood of . Moreover, if , then there is a smooth closed invariant curve that bifurcates from and the bifurcation is subcritical, either attracting or repelling (resp. super-critical).
Proof. System (2) can be written aswhere .
Then, by [35], we write system (16) aswhereare the symmetric multilinear functions of , and these functions can be defined byIn particular,Let be two eigenvectors of and , respectively, such thatthen, we obtainwhere , with .
In order to obtain , we set the normalized vector , where .
We decompose as by considering vary near and for . The precise formulation of is . So, system (16) changed to the following system as a result for the system for close to :where with and is a smooth complex-valued function. Applying the Taylor expansion to the function , we obtainBy using symmetric multilinear vector functions, the Taylor coefficients can be defined asThen, we obtain , whereTo ensure the direction of NS bifurcation, we require that the coefficient, , does not equal to zero. Hence,Therefore, , whereHence, the theorem is proved.
3.2. Bifurcation Analysis around
3.2.1. PD Bifurcation at : Existence
To investigate the existence of PD bifurcation, we will use the following lemma discussed in [37].
Lemma 6. The PD bifurcation of system (2) takes place around the fixed point at if and only ifand , where are given as in (12) and with
If the values of the system parameters vary in a small neighborhood of the setthen one of the eigenvalues of the characteristic (11) is , and the other two are , and therefore, system (2) underlies a PD bifurcation around the fixed point .
3.2.2. PD Bifurcation around : Direction and Stability
Here, the direction of PD bifurcation will be determined by the applications of center manifold theory [35]. We consider the fixed point of system (2) with arbitrary parameters .
Theorem 7. Suppose equation (32) holds well and . If changes its value around the bifurcation point, then a PD bifurcation will occur for system (2) at the fixed point . Furthermore, if , then there exists unstable (resp. stable) period-2 points that bifurcate from .
Proof. Let , then the eigenvalues of areNext, we use the transformation , where and set . Next, we transfer the fixed point of system (2) to the origin since the symmetric multilinear functions are not associated with a fixed point. Therefore, these bilinear and trilinear functions for flip bifurcation will remain unchanged as defined in (18).
Consider two eigenvectors of for eigenvalue such thatwhere and must satisfy the inner product property . Therefore, the sign of can be used to determine the direction of PD bifurcation, which is calculated byHence, the theorem is proved.
3.2.3. NS Bifurcation around : Existence
We will introduce the following Lemma in [36, 37] for the existence of NS bifurcation with the help of explicit Flip-NS bifurcation criterion.
Lemma 8. The NS bifurcation of system (2) occurs around the fixed point at if and only ifwhere are given as in (12) with
For parameter perturbation in a small neighborhood oftwo roots (eigenvalues) of the characteristic (11) are complex conjugates that have modulus one and the magnitude of the other root is not equal to one, then system (2) experiences NS bifurcation around .
3.2.4. NS Bifurcation around : Direction and Stability
This section will present the direction of the NS bifurcation with the help of the center manifold theory.
Next, we choose the fixed point of system (2) with arbitrary parameters .
Theorem 9. Suppose (38) holds and , then NS bifurcation at a fixed point for system (2) if changes its value in a small neighborhood of . Moreover, if , then there is a smooth closed invariant curve that bifurcates from and the bifurcation is subcritical, either attracting or repelling (resp. super-critical).
Proof. Let , then the matrix has the eigenvalues satisfyingFor eigenvalues and , let be two eigenvectors of and , respectively, such that the following conditions hold:and the eigenvector must satisfy .
The sign of the first Lyapunov coefficient determines the direction of NS bifurcation and is defined in statement (27).
Hence, the theorem is proved.
4. Numerical Simulations
In this section, we will perform numerical simulations to support our theoretical results for system (2) which includes diagrams of bifurcation, phase portraits, and MLEs. We take parameter values given in Table 1 to investigate bifurcation analysis.
Example 10. We take the values of the parameters as in case (i). We find a fixed point and bifurcation point for system (2) is evaluated at .
Now, at , the Jacobian matrix of the system (2) takes the form ofand the eigenvalues of are and with .
Also,Let be two complex eigenvectors of and corresponding to , respectively. Therefore,For , then we can take normalized vector as where . Then, we getAlso, by (25), the Taylor coefficients are .
From (39), we obtain the Lyapunov coefficient . Therefore, the NS bifurcation is super-critical, and the requirements of Theorem 5 are established.
The NS bifurcation diagrams are displayed in Figures 1(a)–1(c) which reveal that the condition of stability for the fixed point occurs when and loses its stability at , and there appears an attracting closed invariant curve when . The nonstability of dynamical system is justified with the sign of MLEs (1 (d)).
The phase portraits of system (2) corresponding to the bifurcation diagram shown in Figure 1 are plotted in Figure 2. This figure explicitly illustrates the mechanism of how an invariant smooth closed curve bifurcates from a stable fixed point when changes near its critical value. We noticed that NS bifurcations occur at . When , there appears an invariant closed curve, and further increasing of , the NS bifurcation instigates a route to chaos.
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Example 11. We take the values of the parameters as in case (ii). We find a fixed point and bifurcation point for the system (2) is evaluated at .
Now, at , the Jacobian matrix of system (2) takes the formand the eigenvalues of are and with . Moreover,This shows that all requirements of Lemma 6 are validated with , and so, PD bifurcation of system (2) exists around at . Next, let the two eigenvectors of corresponding to , be , respectively. Then, we obtainTo set , we can choose normalized vector as where . Therefore,Then, from statement (34), the Lyapunov coefficient is obtained. This guarantees the appropriateness of Theorem 7.
The bifurcation diagrams shown in Figures 3(a)–3(c) express the stability of the fixed point when , lose its stability at , and when , a period-doubling phenomena trigger chaotic dynamics. The sign of MLEs illustrates instability of system (2) (see Figure 3(d)). The phase portrait in Figure 4 shows the existence of period orbits and chaos in system (2)
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Example 12. We select the parameters as in case (iii). By calculation, we find a fixed point of system (2), and the bifurcation point is obtained as .
The Jacobian matrix is evaluated at isand the eigenvalues of are and with .
Furthermore,From the resonance condition , we get .
So, the criterion for the existence of NS bifurcation is fulfilled with . This confirms the correctness of Lemma 8. Therefore, a NS bifurcation occurs around fixed point if crosses its critical value .
Let be two complex eigenvectors of and corresponding to , respectively. Therefore,For , we can take normalized vector as where . Then,Also by statement (25), the Taylor coefficients are .
From statement (27), we obtain the Lyapunov coefficient . As a result, it is established that the NS bifurcation is super-critical, and the conditions of Theorem 9 are met.
The NS bifurcation diagrams are displayed in Figures 5(a)–5(c) which reveal that the stability condition for the positive fixed point occurs when , loses its stability at , and there appears an attracting closed invariant curve when . The MLEs related to Figure 5(a) are shown in Figure 5(d). The nonstability of system dynamics is justified with the sign of MLEs.
The phase portraits of system (2) corresponding to the bifurcation diagram shown in Figure 5 are plotted in Figure 6. We noticed that the appearance of invariant closed curve occurs when and further changes of , and NS bifurcation triggers a route to chaos.
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Example 13. For the selection of parameter values in the case (iv) and considering as bifurcation parameter, we find a fixed point of system (2), and bifurcation parameter is calculated at . Moreover,From the resonance condition , we get .
The correctness of Lemma 8 shows the existence of NS bifurcation at . We find the eigenvalue values and with and the Lyapunov coefficient . This makes the NS bifurcation super-critical and supports the validity of Theorem 9.
From the bifurcation diagram Figures 7(a)–7(c) and the phase portrait Figure 8, we noticed that for a small value of , the system first enters into a chaotic dynamics, and with an increase in , the chaotic state certainly disappears through an NS bifurcation occurring at . The dynamics of the system then abruptly transitions to a stable state, after which the system experiences a flip bifurcation occurring at , and the period-doubling phenomenon leads to chaos. The sign of MLEs (Figure 7(d)) guarantees that both bifurcations take place in system (2).
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Example 14. Two-dimensional parametric space is plotted in Figure 9(a) by taking parameter values as in case (v). The critical values curves of NS and PD bifurcations are shown in that Figure. We observe that for the parameter values of (approximately), only NS bifurcations occur. Further increase value of , the system (2) experiences two bifurcations, firstly NS bifurcation and later flip bifurcation. Thereafter, we notice that both bifurcations take place simultaneously at the same point . In two-dimensional parameter space, the diagrams of bifurcation are presented in Figure 9(b). The instability conditions of system (2) are conformable from the sign of MLEs presented in Figure 9(c).
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4.1. 0-1 Chaos Test Algorithm
For chaos test algorithm [38–40], suppose is a measured discrete set of data where , and the entire amount of the data is . We pick a number at random and form new coordinates as follows:where .
Now, the mean square displacement is now defined as follows:
The modified mean square displacement is something else we define as follows:
The median correlation coefficient value is then described as follows:wherein which , and covariance and variance of vectors of length are defined as follows:
Now, we can illustrate the output as follows:(i)The dynamics are consistent (i.e., periodic or quasi-periodic) when , whereas suggests that the dynamics are chaotic.(ii)As opposed to Brownian-like (unbounded) trajectories, which show chaotic dynamics, bounded trajectories on the plane show regular dynamics (i.e., periodic or quasiperiodic dynamics).
Example 15. Taking parameters as in case (iii) with , the system dynamics is chaotic (see in Figure 6), which is consistent with Brownian-like (unbounded) trajectories in new coordinates plane displaying in Figure 10(a) with the value of . The correlation coefficient value curve versus is plotted in Figure 10(b). We find a very nice coincidence between the bifurcation diagram (see in Figure 5) and diagram showing that with the increase value of , the topological properties of system (2) from nonchaotic to chaotic.
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5. Chaos Control
It can be difficult to keep chaos under control. We introduce a hybrid control [41] and OGY [42] approach to control chaos in the Chen system.The effectiveness, cost, and difficulty of implementing a hybrid control strategy and OGY method to eliminate the chaotic behavior of the Chen system will depend on various factors such as the specific control algorithm used, the complexity of the system, and the hardware and software required for implementation.
However, in general, both the hybrid control strategy and OGY method have been shown to be effective in stabilizing the chaotic behavior of the Chen system in simulation studies and experimental validations. These methods are based on the idea of applying a feedback control signal to the system that counteracts the chaotic behavior and stabilizes the system.
For the cost of implementing these control strategies, it depends on the specific hardware and software requirements, as well as the complexity of the control algorithm used. However, in general, the cost of implementing these methods is not prohibitive, and it is usually reasonable for most practical applications.
The difficulty of implementing these control techniques can vary depending on the level of expertise and experience of the control engineer. The hybrid control strategy and OGY method require a good understanding of control theory, nonlinear dynamics, and feedback control. However, with proper training and experience, these methods can be implemented effectively.
In practical applications, the cost of implementing a control strategy must be balanced against the benefits it provides. If the cost of implementing the control strategy is too high, it may reduce the practical significance of eliminating chaotic behavior. Therefore, it is essential to carefully evaluate the specific system, and the control method is used to determine its practical significance.
Hybrid control strategy is applied to system (2) controlling chaos. We rewrite our uncontrolled system (2) aswhere , is nonlinear vector function, and is bifurcation parameter. Applying hybrid control strategy, the controlled system of (59) becomeswhere is the control parameter. Now, if we implement the above mentioned control strategy to system (2), then we get the following controlled system:
For the controlled system (61), the Jacobian matrix at fixed point (which is a fixed point of system (2)) takes the following form:
Then, the zeroes of (eigenvalues of J) satisfy the equationwhere
Lemma 16. If the fixed point of the uncontrolled system (2) is unstable, then it is a sink (stable) for the controlled system (61) if the roots of (63) lie inside open disk satisfying conditions in Lemma 1.
Example 17. To determine whether a hybrid control method is effective at containing chaotic (unstable) system dynamics, we fix with . This therefore demonstrates that the fixed point of system (2) is unstable (see Figure 6); however, the controlled system (61) is stable at this set position iff . Taking , the unstable system dynamics around are eliminated showing that is a sink for the controlled system (61) which have been displayed in Figure 11(b), and time series of the controlled system (61) is displayed in Figure 11(a).
Next, we apply OGY approach in [42] to control the Chen system’s chaotic motion on the stable period −1 orbit. Assume that is the control parameter which somewhat perturbed over time and that system (2) is stated as in (59):Let indicate the unstable equilibrium point. Using the Taylor first-order expansion, the system (2) can be written aswhere is the Jacobian matrix of with respect to the set of variables ,and is the derivative matrix of to the variable ,Put the parameter values , and the fixed point in the matrices and :We define the control parameter in the following form:Then, (67) becomesThe fixed point will be a sink is the eigenvalues of lie in the unit disc. We can determine the control matrix as follows:To stabilize the chaotic trajectories into stable periodic orbit and to get the solution of the pole placement, we set the matrixand , where is a matrix of order 3In (73), are the coefficients of the characteristic equation of :and are the coefficients of the characteristic equation of the matrixwhich is defined asNow, the matrix becomesTherefore, from (73), the matrix is given by . Figure 12 shows how the aforementioned chaos management strategy works in practice.
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6. Conclusion
We present a qualitative analysis of the Chen system in discrete time. We explicitly find the conditions and directions of the NS bifurcation of system (2) in the vicinity of the fixed point . Then, we find the existence criteria of the PD and NS bifurcations of system (2) around a fixed point . In addition, the directions of both bifurcations are given by the center manifold theory. More specifically, we notice that system (2) emerges with a PD bifurcation around and a PD or NS bifurcation around when the parameters and vary in a small vicinity of their critical values. For the mechanism of both bifurcations, the dynamics of system (2) switches from stable state to unstable state and triggers a route to chaos and vice versa. When the topological properties of system (2) change through a PD bifurcation, we obtain period orbits and two coexisting attracting chaotic sets. When the topological properties of system (2) change through a NS bifurcation, we find a closed invariant curve that attracts chaotic sets. From Figure 9, we find that system (2) experiences both bifurcations when (approx.). We present the 3D bifurcation diagrams to see how the PD and NS bifurcation advance or delay when two parameters change its value. In all cases, the dynamic complexities of system (2) are justified with the sign of MLEs. We also use the test algorithm for chaos to detect whether there is chaos in the system or not. Finally, we are able to eliminate chaotic dynamics by applying a hybrid control strategy and the OGY method. For this discrete system, it is open to study the other properties such as synchronization and codimension-2 bifurcation.
Data Availability
The data presented in this study are intended solely for illustrative purposes and are not based on actual observations.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to carry out the proof of the main results and approved the final manuscript.