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Chaos control of Chen chaotic dynamical system
Abstract
This paper is devoted to study the problem of controlling chaos in Chen chaotic dynamical system. Two different methods of control, feedback and nonfeedback methods are used to suppress chaos to unstable equilibria or unstable periodic orbits (UPO). The Lyapunov direct method and Routh–Hurwitz criteria are used to study the conditions of the asymptotic stability of the steady states of the controlled system. Numerical simulations are presented to show these results.
... Stabilization of chaotic systems has attracted a plethora of research in nonlinear systems control [11][12][13][14][15][16][17][18][19][20][21][22]. Traditional approaches including the Ott-Grebogi-Yorke (OGY) [15], time delay feedback control (TDFC) [16], and occasional proportional feedback (OPF) [17] use the natural characteristics of a chaotic system, hence dealing with the complicated system dynamics, to remove chaotic behavior and stabilize the system. ...
... The stabilization process involves assigning specific parameters such as time delays to the chaotic system and using phase portrait and Pioncare map -which cannot be determined in a straightforward way-to mitigate or eliminate chaotic conduct. In addition to traditional methods, different nonlinear control approaches, including sliding mode control (SMC) [18], nonlinear feedback control [19], and backstepping control [20], and Lyapunov exponent placement methods [21,22] have been suggested for stabilizing chaotic systems. ...
... (18) Proof. According to (1) and (17), the following relations yielḋ (19) and the dynamics of perturbation z, the state difference between the subsystems, is found aṡ ...
... In the study of chaotic dynamical systems, one of the interests is determining the essential system parameters given the observed data. In this section, we will investigate three chaotic dynamical systems, Lorenz63 (Lorenz 1963), Agiza and Yassen (2001) and Yassen (2003), that can be summarized as the first-order ODE:ẋ = f (x; u). We will apply the CES framework (Sect. ...
... The true parameter that we will infer is u † = (a † , b † , c † ) = (35, 3, 28). With u † , the system has three unstable equilibrium states given by (0, 0, 0), (γ , γ , 2c − a), (Yassen 2003). Figure 14 illustrates the two-scroll attractor (left), the chaotic trajectories (middle) and their marginal and pairwise distributions (right) of their coordinates viewed as random variables. ...
Inverse problems with spatiotemporal observations are ubiquitous in scientific studies and engineering applications. In these spatiotemporal inverse problems, observed multivariate time series are used to infer parameters of physical or biological interests. Traditional solutions for these problems often ignore the spatial or temporal correlations in the data (static model), or simply model the data summarized over time (time-averaged model). In either case, the data information that contains the spatiotemporal interactions is not fully utilized for parameter learning, which leads to insufficient modeling in these problems. In this paper, we apply Bayesian models based on spatiotemporal Gaussian processess (STGP) to inverse problems with spatiotemporal data and show that the spatial and temporal information provides more effective parameter estimation and uncertainty quantification (UQ). We demonstrate the merit of Bayesian spatiotemporal modeling for inverse problems compared with traditional static and time-averaged approaches using a time-dependent advection–diffusion partial different equation (PDE) and three chaotic ordinary differential equations (ODE). We also provide theoretic justification for the superiority of spatiotemporal modeling to fit the trajectories even if it appears cumbersome (e.g. for chaotic dynamics).
... In the study of chaotic dynamical systems, one of the interests is determining the essential system parameters given the observed data. In this section, we will investigate three chaotic dynamical systems, Lorenz63 [37], Rössler [2] and Chen [60], that can be summarized as the first-order ODE:ẋ = f (x; u). We will apply the CES framework (Section 2) to learn the system parameter u and quantify its associated uncertainty based on the observed trajectories. ...
... The true parameter that we will infer is u † = (a † , b † , c † ) = (35,3,28). With u † , the system has three unstable equilibrium states given by (0, 0, 0), (γ, γ, 2c − a), and (−γ, −γ, 2c − a) where γ = b(2c − a) [60]. Figure 14 illustrates the two-scroll attractor (left), the chaotic trajectories (middle) and their marginal and pairwise distributions (right) of their coordinates viewed as random variables. ...
Inverse problems with spatiotemporal observations are ubiquitous in scientific studies and engineering applications. In these spatiotemporal inverse problems, observed multivariate time series are used to infer parameters of physical or biological interests. Traditional solutions for these problems often ignore the spatial or temporal correlations in the data (static model), or simply model the data summarized over time (time-averaged model). In either case, the data information that contains the spatiotemporal interactions is not fully utilized for parameter learning, which leads to insufficient modeling in these problems. In this paper, we apply Bayesian models based on spatiotemporal Gaussian processes (STGP) to the inverse problems with spatiotemporal data and show that the spatial and temporal information provides more effective parameter estimation and uncertainty quantification (UQ). We demonstrate the merit of Bayesian spatiotemporal modeling for inverse problems compared with traditional static and time-averaged approaches using a time-dependent advection-diffusion partial different equation (PDE) and three chaotic ordinary differential equations (ODE). We also provide theoretic justification for the superiority of spatiotemporal modeling to fit the trajectories even it appears cumbersome (e.g. for chaotic dynamics).
... 10 Chaos has widely applications, such as weather forecasting, nonlinear control and so on. [11][12][13] In the fields of secure communication and multimedia encryption, it is desirable to have some chaotic systems with better chaotic characteristics. 14 Therefore, finding chaotic systems with better performance has become a hot issue in the field of nonlinear research. ...
... , the iteration point is located on a finite cuboid region, so the iteration sequence of Eqs. (12) and (13) will not diverge. ...
Chaos, as an important subject of nonlinear science, plays an important role in solving problems in both natural sciences and social sciences such as the fields of secure communications, fluid motion, particle motion and so on. Aiming at this problem, this paper proposes a nonlinear dynamic system composed of product trigonometric functions and studies its chaotic characteristics. Through the mathematical derivation of the system’s period, the analysis of the necessary conditions at the fixed point, the experimental drawing of the Lyapunov exponential graph and the branch graph of the system, it is proved that the system has larger chaotic interval and stronger chaotic characteristics. The parameters of the proposed dynamic system are generated randomly, and then the chaotic sequence can be generated. The chaotic sequence is used to encrypt the digital image, a good encryption effect is obtained, and there is a large key space. At the same time, the motion of the particles in the space magnetic field is simulated, which further proves that the trigonometric system has strong chaotic characteristics.
... Chen Kaotik Sistemi (CKS), Eşitlik 7'de tanımlanmıştır (Liang & Qi, 2017;Yassen, 2003): ...
... a=35, b=3 ve c=28 değerlerini aldığında sistem eşsiz bir kaotik çekiciye sahip olur. Nümerik çözümler, 5. Dereceden Runge-Kutta metodu (RK5B) kullanılarak gerçekleştirilmiştir. Adım aralığı dt = 0.001 ve başlangıç koşulları x 0 = 5; y 0 = -15; z 0 = 40 olarak alınmıştır (Liang & Qi, 2017;Yassen, 2003). Şekil 3 ve Şekil 4'te sırasıyla Chen Sistemine ait zaman serileri ve Chen Sistemine ait kaotik çekiciler gösterilmektedir. ...
Activation functions is an important parameter that affects the performance of the network in the process of training of Artificial Neural Network (ANN) structures. This paper presents the modelling of Aizawa Chaotic System (ACS) using the structures of Feed Forward Neural Network (NN) and Feedback NN. Runge Kutta 5 Butcher (RK-5-B) algorithm has been used for the numeric solution describing ACS. Nonlinear activation functions like Radial Basis (RadBas), Logarithmic Sigmoid (LogSig) and Tangent Sigmoid (TanSig) have been used in the modelling process and the analysis study has been performed related to the modelling performance of ACS by using these functions in the created network structures. It has been observed that the TanSig activation function which is one of the actication functions used in the modelling with FFNN structure has produced more sensitive results than others and the LogSig activation function which is one of the actication functions used in the modelling with RNN structure. has produced more sensitive results than others.
... Therefore, the investigation of controlling chaos is of great significance. Many schemes have been presented to carry out chaos control [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] of which using time-delayed controlling forces proves to be a simple and viable method for a continuous dynamical system. ...
... The characteristic equation (8) with the weak kernel case takes the form where 1 ( ) = 2 + − − , 2 ( ) = ( + ) + 2 ( − + 1) + ( * − ) , ...
We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable when
α
∈
(
0
,
α
0
)
and unstable when
α
∈
(
α
0
,
∞
)
; Hopf bifurcation occurs when
α
crosses a critical value
α
0
by choosing
α
as a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regarding
α
as a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter value
α
.
... Thus the equilibrium E(10, 12.85, 13) is asymptotically stable for τ < τ0 ≈ 0.6 and unstable for τ > τ0 ≈ 0.6 which is shown in Fig. 2. When τ = τ0 ≈ 0.6, (13) undergoes a Hopf bifurcation at the equilibrium E(10, 12.85, 13), i.e., a periodic solution of small amplitude occurs around E (10, 12.85, 13) when τ is close to τ0 = 0.6 which can be seen in Fig. 3. In addition, we find that the varying initial values have no effect on the stability and Hopf bifurcation behavior of system (13). Chaos vanishes when τ = 0.8 > τ0 ≈ 0.6. ...
... Hopf bifurcation occurs from the equilibrium E (10, 12.85, 13). The initial value is (10,10,13) ...
In this paper, the control of chemical chaotic dynamical system is investigated by time-delayed feedback control technique. The controllability and the stability of the equilibriums and local Hopf bifurcation of the system are verified. Some numerical simulations which show the effectiveness of the time-delayed feedback control method are provided.
... On the other side, random motion of a system in a chaotic situation was considered to be unfavorable in engineering fields. Therefore, many efforts were given to eliminate or control chaos in the systems, which led to launching the work on chaos control in 1990 such as the OGY (Ott, Gerbogi and York) control method [8], linear feedback control [9], [10], time delay feedback control [4], [11], [12] and some others were reported: the control of chaos such as sliding mode control [13][14][15][16][17], fuzzy control [18][19][20][21][22], polynomial approach [23] high order sliding mode control [24] and harmonic approach [25], [26], etc. Many researchers have been addressed the control of Chua's circuit, in several control methodologies. ...
Chua?s circuit is the classic chaotic system and the most widely used in serval areas due to its potential for secure communication. However, developing an accurate chaos control strategy is one of the most challenging works for Chua?s circuit. This study proposes a new application of super twisting algorithm (STC) based on sliding mode control (SMC) to eliminate or synchronize the chaos behavior in the circuit. Therefore, the proposed control strategy is robust against uncertainty and effectively regulates the system with a good regulation tracking task. Using the Lyapunov stability, the property of asymptotical stability is verified. The whole of the system including the (control strategy, and Chua?s circuit) is implemented under a suitable test setup based on dSpace1104 to validate the effectiveness of our proposed control scheme. The experimental results show that the proposed control method can effectively eliminate or synchronize the chaos in the Chua's circuit.
... In addition, the controlling of chaotic systems is still one of the most worthwhile research directions of chaos theory. Feedback control is one of the most commonly used methods for stabilization, synchronization, and deformation in chaotic systems [Yassen, 2003]. Furthermore, the coexisting attractors of parameters as a way of offsetting control have also attracted people's attention in recent years since it can realize the arbitrary movement of the attractor without changing the dynamic characteristics of the chaotic system Li et al., 2022]. ...
Electrolysis is an important way to produce manganese metal, but the low current efficiency and random growth of dendrites have always been challenging problems for enterprises. The lack of understanding of the dynamic system during the electrolysis process is the main reason for the accurate control of the electrolysis process. Based on this consideration, a new four-dimensional continuous hyperchaotic system with high Lyapunov exponents is designed. The amplitude control, frequency modulation, and offset boosting of the hyperchaotic system are obtained through the selection of feedback term. A circuit simulation and corresponding simplified circuit are established. In addition, the actual hyperchaotic circuit is applied to the manganese electrolysis process through the self-designed current amplification module (the amplification of w signal is realized by the offset boosting control). The experimental results of the hyperchaotic electrolysis of metal manganese showed that the hyperchaotic current can delay the occurrence time of electrochemical oscillation, and reduce the generation of cathode metal manganese dendrites. Furthermore, the results show that the hyperchaotic current can enhance the current efficiency and reduce the energy consumption. Based on the new experiment, it is suggested that the formation of anodic porous structures, whose primary phase compositions were PbSO4, MnO2, and Mn2O3, is one factor for the occurrence of electrochemical oscillations, while the conversion between Mn3+ and Mn4+ is another main factor for the mutation of electrochemical signal (manganese autocatalysis).
... The control of complex characteristics consists of designing a regulatable input capable of reshaping the dynamics of a complex system and hence transforming the system to a stable form around some selected states or positions. Chaos is currently controlled in dynamic systems using feedback and non-feedback control methods (Yassen 2000;Yassen 2006;Kemnang Tsafack et al. 2020). ...
This paper reports concerning the microcontroller validation of a self-governing jerk oscillator with quadratic nonlinearities (AJOQN) and operation investigations based on chaos control and difference synchronization. AJOQN displays self-excited chaotic attractors with different shapes. The total amplitude control of AJOQN is achieved by tuning one of its parameters. The dynamical characteristics reported in AJOQN are vindicated via the microcontrollerprobing. A single controller is delineated to quash the complex characteristics of AJOQN. The validity of the designed single controller is confirmed by the numerical simulations. In the bargain, controllers are formulated to establish difference synchronization in the triple similar coupled chaotic AJOQNs advancing from different incipient states. In closing, simulations numerically of the triple alike coupled chaotic AJOQNs manifest the efficacy of difference synchronization.
... Kaos kontrolde temel amaç kaotik yörüngeleri bir denge noktasına taşımak veya kararlı bir yörüngede tutmaktır. Bu amacı gerçekleştirmeye yönelik bazı çalışmalar söyle sıralanabilir: [12]'de zaman gecikmeli geri besleme kullanılarak, [13]'de kaotik davranış gösteren ekonomik bir modelin durum değişkenleri ve sistem parametreleri kullanılarak, [14]'de birden fazla geri besleme kullanılarak, [15]'de kayma kipli kontrol yöntemi kullanılarak, [16]'da meta sezgisel bir optimizasyon yöntem kullanılarak ve [17]'de kesir dereceli PI kontrolör kullanılarak kaos kontrol işlemi gerçekleştirilmiştir. Yapılan çalışmalara bakıldığında, eğer kaotik sistemin sistem parametrelerine ve durum değişkenlerine pratikte erişim mümkün değilse PI kontrolör kullanımı kaos kontrolü mümkün kılabilir. Bu makalede, literatürde ilk defa doğrusal sistemler için önerilen kararlılık sınır eğrileri yöntemi kullanılarak doğrusal olmayan yapıdaki kaotik davranış gösteren Genesio-Tesi sistemin PI kontrolör kullanarak bu davranıştan kurtarılması amaçlanmıştır. ...
Bu makalede, kararlılık sınır eğrileri yöntemi kullanılarak kaotik davranış gösteren Genesio-Tesi sisteme PI kontrolör eklenerek sistemi kaotik davranıştan kurtarmak amaçlanmıştır. Bu amaçla, kaotik davranış gösteren Genesio-Tesi sistemin denge noktasındaki doğrusal modeli elde edilmiştir. Elde edilen model kullanılarak PI kontrolör eklenmiş ve sistemin denge noktasını kararlı hale getirecek kontrolör parametre seçimi gerçekleştirilmiştir. Denge noktasını kararlı hale getirecek kontrolör parametrelerinin belirlenmesi için kararlılık sınır eğrileri yöntemi kullanılmıştır. Son olarak, denge noktasını kararlı ve kararsız hale getirecek kontrolör parametreleri için MATLAB/Simulink ortamında sistemin benzetim sonuçları elde edilmiş ve böylece sonuçların doğruluğu gösterilmiştir.
... These methods involve assigning parameters such as time delays to the chaotic system and using phase portrait and Pioncare map, which cannot be determined in a straightforward way, to mitigate or eliminate the chaotic conduct. Also, various nonlinear control approaches, including sliding mode control (SMC) [5], nonlinear feedback control [6], and backstepping control [7] have been also used for the control of chaotic behavior. ...
... Therefore, lots of efforts were given to eliminate or control chaos in the systems, which led to the seminal work of chaos control in 1990, namely, the OGY control [2]. After the OGY method, time-delay feedback [3], linear feedback [4], [5], impulse control [6], [7], and others, were reported to control chaos. ...
Chaos has been successfully applied in many fields to improve the performance of engineering systems, such as communication, vibration compact, and mixing. Generating chaos from originally non-chaotic systems is a relevant topic because of potential applications. In this work, the impulse control is shown to generate chaos from non-chaotic system. Using non-chaotic Chen system as an example, we prove by analytical and numerical methods that chaos is indeed generated. The features of the chaos generated by impulse control are analysed using Lyapunov exponents, bifurcation diagram, power spectrum, Poincaré mapping and Kaplan-Yorke dimension. Furthermore, we demonstrate the chaotic attractor generation by impulse control using a circuit experiment. The last but not minor point is that the existence of topological horseshoe is given by rigorous computer-aided proof.
... To overcome this limitation of low-dimensional schemes, researchers have proposed high-dimensional systems having complex dynamical features such as Chen system, Lorenz system and Bao system Lee and Singh (2011);Yassen (2003); Khellat (2015). Increasing the dimension of chaos systems can effectively negate decryption attacks such as phase space reconstruction, however, these algorithms are prone to plaintext attacks. ...
https://dergipark.org.tr/en/pub/chaos
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https://www.researchgate.net/project/Chaos-Theory-and-Applications-CHTA
... To overcome this limitation of low-dimensional schemes, researchers have proposed high-dimensional systems having complex dynamical features such as Chen system, Lorenz system and Bao system Lee and Singh (2011);Yassen (2003); Khellat (2015). Increasing the dimension of chaos systems can effectively negate decryption attacks such as phase space reconstruction, however, these algorithms are prone to plaintext attacks. ...
With the advancement in digital technologies and the demand for secure communication, there is an increased interest in the design and implementation of reliable image encryption schemes. This paper presents a thorough review of chaos theory and its application in image encryption schemes. Due to ergodicity and initial key sensitivity, chaos-based image encryption schemes have several advantages over traditional encryption schemes. The paper discusses the major applications of chaos theory, particularly in image encryption area. The use of different chaotic maps such as one and multi-dimensional chaos, hyper and composite chaos in image encryption have been presented. The paper also discusses current trends and future research directions in the field of chaotic image encryption. This work provides a foundation for future research work along with providing basic understanding to new researchers. Several recommendations have been suggested that can improve a chaos-based cryptosystem.
... The behaviors of this system are uncertain, unrepeatable, and unpredictable. Therefore, the chaotic sequences generated by the hyperchaotic Chen system are more pseudorandom [43]. The hyperchaotic Chen system is described in formula (2): ...
To improve the abilities of image encryption systems to resist plaintext attacks and differential attacks, a novel plaintext-related image encryption scheme based on Josephus traversing and pixel permutation is proposed. In this scheme, the step sizes of the Josephus traversing are associated with the pixel values, and the Josephus traversing method is improved, increasing the dependence of the cipher image on the plaintext image, further enhancing the plaintext sensitivity of the algorithm, and reducing the number of iterations of the index sequence generated by the chaotic system. By segmenting the image and combining a chaotic system, bit XOR and crossover operations between the modules are performed to achieve the effects of confusion and diffusion, and the randomness of the cipher image is improved. Finally, the confusing and spreading characteristics of the algorithm are further enhanced by cipher feedback. The experimental results and security analysis show that the proposed algorithm is sensitive to keys and can effectively resist attacks such as statistical attacks, selective-plaintext attacks and exhaustive attacks. The algorithm has high potential for real-time and secure image applications.
... The engineering applications of fractional-order chaotic systems have been reviewed in [13]. Many fractional-order chaotic systems have been introduced up to now such as Chen [14], Rossler [15], Lorenz [16], Liu [17], Genesio-Tesi [18], Novel [19] and some others. ...
In this paper, a fuzzy control method is proposed for a class of fractional-order chaotic systems. The dynamics of the system are unknown and are perturbed by the external disturbances. Also, the value of the fractional-order is assumed to be unknown. The type-2 fuzzy systems (T2FSs) are employed to estimate the unknown functions in the dynamics of the system. The parameters of T2FS and the value of fractional-order are estimated by unscented Kalman filter. The upper bound of the approximation error is online estimated, and a new fractional-order compensator is designed to eliminate the effect of the uncertainties and to guarantee the closed-loop stability. The effectiveness of the proposed method is shown by simulations, and the results are compared with some other techniques. It is shown that the proposed method results in better performance in the presence of unknown fractional-order and unknown perturbed dynamics.
... The control of chaos involves eliminating and restraining the chaos phenomenon when it is unavailable and harmful. It has been noticed that purposeful control of chaos can be a key issue in many technological applications [12][13][14][15]. Generally, the existing chaos control methods can be divided into two categories, feedback control and non-feedback control, according to their characteristics. ...
Abstract This paper is concerned with chaos control and bifurcations of the Leslie–Gower type generalist predator model in a tri-trophic food web system with the time-delayed feedback control. First, the distribution of the roots of the related characteristic equations is analyzed by the polynomial theorem, the conditions to guarantee the existence of Hopf bifurcation are given by choosing the time delay as a bifurcation parameter. Then, the explicit formula for direction of Hopf bifurcation and stability of periodic solutions bifurcating are determined by using the normal form theory and center manifold theorem. Finally, the correctness of our theoretical analysis is verified by some numerical simulation.
... The Chen system [45] is considered as ...
In this article, the authors have studied the triple compound synchronization among eight chaotic systems with external disturbances. The nonlinear approach is used to achieve triple compound synchronization among eight chaotic systems. The control functions are designed using Lyapunov stability theory. Numerical simulation and graphical results are carried out using the Runge–Kutta method, which shows that the designing of control functions are very effective and reliable and can be applied for triple compound synchronization among chaotic systems. The salient feature of the article is the exhibition of complexity in the error function in triple compound synchronization for which the communication via signals will be more secured through this type of synchronization process.
... Some hyperchaotic systems have also been introduced in literature. For example, one can refer to hyperchaotic complex Lorenz system [5], Wang and Chen systems [6] and integrated hyperchaotic system. The most underlying feature of such chaotic systems is their sensitiveness to change in initial conditions. ...
In this paper, a full-order sliding mode controller, based on adaptive neuro-fuzzy inference system is proposed as approximator, for controlling nonlinear chaotic systems in presence of uncertainty. At first, the full-order sliding mode controller is designed for the system in the absence of uncertainty such that the system states are converged to the sliding surface. Then, adding uncertainty to system equations, convergence of the method is illustrated using simulations. By assuming that a part of the system dynamics is uncertain and only input-output data is partly available, adaptive neuro-fuzzy inference system is used in off-line mode to approximate the uncertain dynamics of the system based on input-output data. The proposed method can effectively solve the problems of the sliding-based methods, such as chattering phenomenon and singularity. The simulation results, applied to the well-known nonlinear systems namely PMSM and plasma torch systems when they behave in chaotic mode, demonstrate effectiveness and fidelity of the proposed control method.
... It is a consequence of the theoretical properties of the decomposition method proved in Hashim. I (see [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] ...
... Chaos control has attracted a great deal of attention since Hübler published the first paper on chaos control in 1989 [Hübler, 1989]. Over the last decades, many other methods are applied for chaos control, such as the OGY method [Ott et al., 1990], PC method [Pecora & Carroll, 1990;Carroll & Pecora, 1994], adaptive method, time-delay feedback approach and backstepping design technique [Chen & Dong, 1998;Yassen, 2003;Lü & Lu, 2003;Xie & Chen, 1996;Agiza, 2004;Jiang et al., 2011;Agiza, 2002]. In this section, the time-delay feedback method is proposed to control chaos induced by dichotomous noise in the FK model. ...
Chaos and chaos control of the Frenkel–Kontorova (FK) model with dichotomous noise are studied theoretically and numerically. The threshold conditions for the onset of chaos in the FK model are firstly derived by applying the random Melnikov method with a mean-square criterion to the soliton equation, which is a fundamental topological mode of the FK model and accounts for its nonlinear phenomena. We found that dichotomous noise can induce stochastic chaos in the FK model, and the threshold of noise amplitude for the onset of chaos increases with the increase of its transition rate. Then the analytical criterion of chaos control is obtained by means of the time-delay feedback method. Since the time-delay feedback control raises the threshold of noise amplitude for the onset of chaos, chaos in the FK model is effectively suppressed. Through numerical simulations including the mean top Lyapunov exponent and the safe basin, we demonstrate the validity of the analytical predictions of chaos. Furthermore, time histories and phase portraits are utilized to verify the effectiveness of the proposed control.
... 2) Control of Chaos Using Nonlinear Control Method: Let the fractional order Vallis system is taken as a controlled system with control functions u i (t), i = 1, 2, 3 to stabilize unstable periodic orbit or fixed point as given in (10). ...
In this article the authors have studied the stability analysis and chaos control of the fractional order Vallis and El-Nino systems. The chaos control of these systems is studied using nonlinear control method with the help of a new lemma for Caputo derivative and Lyapunov stability theory. The synchronization between the systems for different fractional order cases and numerical simulation through graphical plots for different particular cases clearly exhibit that the method is easy to implement and reliable for synchronization of fractional order chaotic systems. The comparison of time of synchronization when the systems pair approaches from standard order to fractional order is the key feature of the article.
... where x 1 , x 2 , and x 3 are the state variables. When the parameters of (11) are set to a = 35, b = 3, and c = 28, this system shows a chaotic behavior [2,5,23,31]. ...
This paper presents an optimal chaos synchronization technique using particle swarm optimization (PSO) and time-delay estimation (TDE). Time-delay control (TDC), which uses the TDE to estimate and cancel the nonlinear terms in dynamics, is simple and robust, and has been recognized as a promising technique for chaos synchronization. The synchronization technique with TDC consists of three terms: a slave dynamics cancelation term using the TDE, a desired chaotic dynamics injection term, and a synchronization error dynamics describing term. In this paper, we propose the PSO algorithm for the gain of the last term, the slope of sliding surface of the synchronization error dynamics. An objective function is constructed using a sum of the absolute value of errors, and a set of particles is used to seek the optimal solution. We show that too small value of a gain results in weak robustness, and too large value of a gain worsens noise sensitivity. Our data suggest that an optimal gain set for minimizing synchronization errors of the TDC can be obtained through the proposed technique. The proposed technique can automatically find the optimized gain for the TDC in noisy environments although the estimation time-delay is altered significantly. The effectiveness of the proposed technique is verified through chaos synchronization simulations for secure communications.
... Therefore, the investigation of chaotic control is of great significance. Moreover, many schemes have been presented to implement chaotic control [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], of which the time-delayed feedback controlling forces is proved to be a simple and viable method for a continuous dynamical system. Fortunately, the time-delayed feedback controller can also be used to realize the control of a bifurcation. ...
This paper mainly investigates the dynamical behaviors of a chaotic system without
ilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.
... Even though the results obtained led to butterfly-shape but cannot provide good accuracy. One of the common and widely used numerical methods is RK4 for simulating the solution of chaotic system [12,15,16] and has been mostly used as comparison method [17][18][19][20]. There are other powerful methods of solving chaotic systems such as Laplace Adomian decomposition method (LADM) [21,22]. ...
... Recently, many excellent results were reported by Moon [2], Chen and Dong [3], and Kapitaniak [4]. Moreover, numerous outstanding work on chaos control was presented by EI Naschie [5] and Kapitaniak [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][30][31][32][33]. In 1998, Varriale and Gomes [23] had investigated the dynamics of the following food web consisting of three species ...
This paper is devoted to investigate the problem of controlling chaos in a 3D ratio-dependent food chain system. Time-delayed feedback control method is applied to suppress chaos to unstable equilibria or unstable periodic orbits. By local stability analysis, we theoretically prove the occurrences of Hopf bifurcation. Some numerical simulations are presented to support theoretical predictions. Finally, main conclusions are drawn.
... Since then many new chaotic systems were found, such as in [Lü & Chen, 2002;Lü et al., 2004] the transition systems that bridge the gap between the Lorenz and Chen systems were provided. Over the last twenty years, there are extensive studies on Chen system [Li et al., 2004;Zhou et al., 2004Zhou et al., , 2003Ueta & Chen, 2000;Yu & Xia, 2001;Agiza & Yassen, 2001;Lu & Zhang, 2001;Yassen, 2003;Chang & Chen, 2006]. In [Tang et al., 2012], the time-varying delays were introduced into Chen chaotic system: ...
In this paper, we study the existence of rank one strange attractor in time-delayed system. First, we try to develop rank one theory for delayed differential equations. Then, we consider Chen system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given by using the normal form method and center manifold theorem. Then, we add an external periodic force as an input and observe rank one strange attractors. Finally, several numerical simulations supporting the theoretical analysis are also given.
In this study, encrypted image transmission was carried out with a secure communication application using reversible chaotic image encryption with the synchronization of two identical chaotic systems. Encryption and decryption of the image can be done with the same algorithm using the same chaotic signal. The image can be encrypted using the same chaotic signal on the sending side and decoded with the encryption algorithm and chaotic signal on the receiving side. Experimental results show that the proposed encryption algorithm can securely encrypt and transfer images in 24-bit RGB and 8-bit grayscale images. In the study, histogram, histogram variance, salt & pepper, speckle, NPCR and UACI, four classical types of attacks, NIST, key sensitive and crop visual and statistical attacks were made on encrypted images and the robustness of the encryption method was tested. In addition, PSNR, SSIM, UIQ, BER and NCC image quality criteria were used and the results of attacks on chaotic encrypted images were measured. As a result of visual and statistical attacks, it has been seen that the proposed chaotic encryption algorithm is resistant to attacks. The NPCR and UACI analysis results of the proposed method were compared with the results of similar studies in the literature. According to similar chaotic image coding studies in the literature, the NPCR and UACI values of the proposed method gave better results than most. As a result of the tests, with the secure communication application of the proposed method, image encryption and decryption can be performed by using a common chaotic signal on the sender and receiver sides.
In recent years Parameter Estimation (PE) has attracted the attention of the scientific community.
This work considers a new generalized operator which is based on the application of Caputo-type fractional derivative is applied to model a number of nonlinear chaotic phenomena, such as the Oiseau mythique Bicéphale, Oiseau mythique and L’Oiseau du paradis maps. Numerical approximation of the generalized Caputo-type fractional derivative using the novel predictor–corrector scheme, which indeed is regarded as an extension of a well-known Adams–Bashforth-Moulton classical-order algorithm. A range of new strange chaotic wave propagation was observed for various maps with varying fractional parameters.
Here, a chaotic quadratic oscillator with only squared terms is proposed, which shows various dynamics. The oscillator has eight equilibrium points, and none of them is stable. Various bifurcation diagrams of the oscillator are investigated, and its Lyapunov exponents (LEs) are discussed. The multistability of the oscillator is discussed by plotting bifurcation diagrams with various initiation methods. The basin of attraction of the oscillator is discussed in two planes. Impulsive control is applied to the oscillator to control its chaotic dynamics. Additionally, the circuit is implemented to reveal its feasibility.
Encryption is one of the techniques that ensure the security of images used in various domains like military intelligence, secure medical imaging services, intranet and internet communication, e-banking, social networking image communication like Facebook, WhatsApp, Twitter etc. All these images travel in a free and open network either during storage or communication; hence their security turns out to be a crucial necessity in the grounds of personal privacy and confidentiality. This article reviews and summarizes various image encryption techniques so as to promote development of advanced image encryption methods that facilitate increased versatility and security.
This paper proposes a novel image block encryption algorithm based on three-dimensional Chen chaotic dynamical system. The algorithm works on 32-bit image blocks with a 192-bit secret key. The idea is that the key is employed to drive the Chen’s system to generate a chaotic sequence that is inputted to a specially designed function G, in which we use new 8x8 S-boxes generated by chaotic maps (Tang, 2005). In order to improve the robustness against differental cryptanalysis and produce desirable avalanche effect, the function G is iteratively performed several times and its last outputs serve as the keystreams to encrypt the original image block. The design of the encryption algorithm is described along with security analyses. The results from key space analysis, differential attack analysis, and information entropy analysis, correlation analysis of two adjacent pixels prove that the proposed algorithm can resist cryptanalytic, statistical and brute force attacks, and achieve a higher level of security. The algorithm can be employed to realize the security cryptosystems over the Internet.
Applications of the synchronization phenomenon of dynamical systems are important in several applications as in technology and have a wealth of science. Some applications of master–slave synchronization include the control of chaos and chaotic signal masking, where several methodologies have been considered (Martínez-Guerra, Rafael, and Wen Yu, Int J Bifurcation Chaos, 18(01), pp 235–243, 2008, [1]), (Hernault, Barbot, Ouslimani, IEEE Trans Circ Syst I: Regular Papers 55, 614, 2008, [2]). Nonetheless, it has been established that synchronization of chaotic dynamical systems is not only possible but it is believed to have potential applications in communication (Mitra, Banerjee, Int J Mod Phys B 25, 521, 2011, [3]), (Saha, Banerjee, Roy Chowdhury, Phys Lett Sect A: General, Atomic Solid State Phys 326, 133, 2004, [4]). The strategy is that when is transmitted a message; it is possible to mask it with louder chaos. An outside listener only detects the chaos, which signals like meaningless noise. But if the receiver has an adequate synchronization algorithm that perfectly reproduces the chaos, the receiver can subtract off the chaotic mask and detect the original message.
Drillability plays a crucial role in increasing the capability of blasthole drills in drilling and thereby contributing valuable input for the mining industry. The productivity of the mines and the life of the machines are directly dependent on the wear rate of the drill bit. In this light, an investigation was conducted to illustrate the relationship between the uniaxial compressive strength (UCS) of the rock, the penetration rate of the drilling machine and the bit wear-out rate at overburden benches of a surface coal mine. The wear-out rate of drilling bit estimated from digital imaging of the diametrical distribution of button bits obtained from imaging software, that was taken 'Before' and 'After' drilling. The UCS of rock mass was measured using Schmidt Hammer while wear-out rate of the drill bit was measured through computer software. The results obtained from the experimental observations and graphical plot shows that as the UCS (in N/mm2) increasing from 27.5 to 42.46 the penetration rate (in mm/s) decreases from the 30 to 14 and wear-out rate (in mm2/meter of the hole drilled) are increasing from 0.339 to 1.05.
Encryption is one of the techniques that ensure the security of images used in various domains like military intelligence, secure medical imaging services, intranet and internet communication, e-banking, social networking image communication like Facebook, WhatsApp, Twitter etc. All these images travel in a free and open network either during storage or communication; hence their security turns out to be a crucial necessity in the grounds of personal privacy and confidentiality. This article reviews and summarizes various image encryption techniques so as to promote development of advanced image encryption methods that facilitate increased versatility and security.
This paper investigates complete synchronization and stabilization of hyperchaotic systems. A new modified finite-time control method is introduced to guarantee either the stability of the closed-loop system and finite-time complete synchronization or finite-time stabilization. Since the proposed controller possesses simple construction and easy implementation, it could be utilized in synchronization and stabilization of a general class of hyperchaotic systems. Simulation results are provided to demonstrate the feasibility and the efficacy of the proposed method.
The definition of partial variables inverse sequence feedback controller of a chaotic system is proposed. We studied this kind of controller with Chen's chaotic system and get the existing condition of the controller, and extended the conclusion of the related article, so get various controllers of the chaotic system. Numerical simulations are presented to show that chaotic system control based on partial variables feedback is simple and effective.
This paper investigates the stabilization of unstable equilibrium in a new hyperchaotic system. The single state feedback control, speed feedback control and nonlinear function feedback control methods are applied to suppress hyperchaos to the unstable equilibrium and periodic orbits. Some conditions of stability for controlled system are established. Moreover some numerical examples are given to verify the theoretical results.
We study the performance of adaptive sliding mode observers in chaotic synchronization and communication in the presence of uncertainties. The proposed robust adaptive observer-based synchronization is used for cryptography based on chaotic masking modulation (CM). Uncertainties are intentionally injected into the chaotic dynamical system to achieve higher security and we use robust sliding mode observer design methods for the uncertain nonlinear dynamics. In addition, a relaxed matching condition is introduced to realize the robust observer design. Finally, a Chaotic system called Chen dynamical attractor is employed as an illustrative example to demonstrate the effectiveness and feasibility of the proposed cryptosystem.
A systematic design method for controlling chaotic systems is presented in this paper. The aggregated multiple local models are adopted to express chaotic systems. The Linear Quadratic Regulator (LQR) theory is proposed to design state feedback control system for each local model. Multi-model control strategy has come into being by combining of T-S fuzzy model and LQR. The global stability of closed loop control system can guarantee and it is illustrated with several chaotic systems as examples.
Most of scientific problems and natural phenomena can be modeled by chaotic systems of ordinary differential equations. These problems can be solved by using various methods. In this paper, we are interested to test the Runge-Kutta method of order four on the Zhou chaotic system. This system is a new three-dimensional autonomous chaotic system. Numerical comparisons are made between the Runge-Kutta of order four and the Euler's method. Comparisons were also done between the RK4 methods but with different time steps. It has been observed that the accuracy of RK4 solutions can be increased by decreasing the time step. Our work shows that RK4 method successfully can solve the Zhou system and figures are given for different number of iterations with corresponding range of time, t.
In this paper, the problem of controlling chaos in a Sprott E system with distributed delay feedback is considered. By analyzing the associated characteristic transcendental equation, we focus on the local stability and Hopf bifurcation nature of the Sprott E system with distributed delay feedback. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form theory and center manifold theory. Numerical simulations for justifying the theoretical analysis are provided.
This paper is devoted to the problem of controlling chaos in the Qi system. A time-delayed feedback control method is applied to suppress chaos to unstable equilibria or unstable periodic orbits. Using a local stability analysis, we theoretically prove that the Hopf bifurcation occurs. Some numerical simulations are carried out to support the theoretical predictions. Finally, main conclusions are drawn.
In this paper, a four-dimensional (4D) autonomous hyperchaotic system is dealt with. The stability criteria of equilibria of the controlled hyperchaotic chaotic system are established. Using the dislocated feedback control, enhancing feedback control, and nonlinear function feedback control methods, the chaos of the 4D hyperchaotic system can be suppressed to unstable equilibrium. Some numerical simulations revealing the effectiveness of our control strategies are given..
This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.
Different methods are proposed to control chaotic behaviour of the Nuclear Spin Generator (NSG) and Rossler continuous dynamical systems. Linear and nonlinear feedback control techniques are used to suppress chaos. The stabilization of unstable fixed point or unstable periodic solution of chaotic behaviour is achieved. The controlled system is stable under some conditions on the parameters of the system. Stability of the controlled system is determined by the Routh–Hurwitz criterion and Lyapunov direct method. Numerical simulation results are included to show the control process.
Abstract-A continuous,feedback,controller is designed,to suppress and,eliminate the chaotic behavior of Chua’s circuit. This controller is constructed by the following two phases. First, the set of equilibrium points is extended by virtue of embedding,suitable feedback terms in the control channel. Second, a state error term is then added to steer the dynamics of the control system to a fixed point or a limit cycle. The stability region of Chua’s circuit with control is determined,via the Routh-Hurwitz criterion. Some numerical,simulations are given to demonstrate,the effect of the control design. 0 1997 Elsevier Science Ltd
A feedback controller is designed to suppress chaotic states of a modified Chua's circuit system. This controller is composed by the following two portions. One is the feedback part which constructs an equilibrium manifold by modifying the dynamics of the system. The other is the proportional feedback part which will control the system to desired states on the equilibrium manifold. The advantage of this method is that with this cllsed-loop controller the system attains faster settling time than the systems using previous controllers. Also, it will eliminate the tracking error without using other integral controllers.
Two methods of chaos control with a small time continuous perturbation are proposed. The stabilization of unstable periodic orbits of a chaotic system is achieved either by combined feedback with the use of a specially designed external oscillator, or by delayed self-controlling feedback without using of any external force. Both methods do not require an a priori analytical knowledge of the system dynamics and are applicable to experiment. The delayed feedback control does not require any computer analyses of the system and can be particularly convenient for an experimental application. On leave of absence from Institute of Semiconductor Physics, 2600 Vilnius, Lithuania.
Three methods for chaos control are briefly reviewed. Only one of them seems to be applicable to Solimans model for unemployment–inflation. © 1999 Elsevier Science Ltd. All rights reserved.
A system of two firms is studied following Puus model of Cournot duopoly. Their competing advertisements are modelled. Chaos is observed if the advertisements parameters of both firms are close to each other. Chaos control formulae are given to control the chaos due to advertisements.
A dynamical system of two nonlinear difference equations introduced by Kopel is studied (Kopel, M., Simple and complex adjustment dynamics in Cournot Duopoly model, Chaos, Solitons and Fractals, 1996, 7 (12), 2031–2048.). In this study, we determine conditions for the stability of the fixed points and we study the bifurcations and chaos (Devany, R. L., Introduction to Chaotic Dynamical Systems. Addison–Wesley, New York, 1989.) by computing the maximum Laypunov exponents (Luhta, I. and Virtanen, I., Chaos, Solitons and Fractals, 1996, 7, 2083.). The OGY chaos control formula is used to control chaos in this system.
Various control algorithms have been proposed in recent years to control chaotic systems. These methods are broadly classified into feedback and nonfeedback methods. In this paper, we make a critical analysis of nonfeedback methods such as (i) addition of constant bias, (ii) addition of second periodic force, (iii) addition of weak periodic pulse, and (iv) entrainment control. We apply these methods to a simple electronic circuit, namely, the Murali-Lakshmanan-Chua circuit system and FitzHugh-Nagumo equation. We make a comparative study of the various features associated with the algorithms.
In this paper, we present some conventional feedback controller design principles for chaos control, with mathematical controllability conditions derived via the Lyapunov function methods. The chaotic Chua's circuit and Duffing oscillator are used as examples to illustrate the fundamental concepts and basic methodology employed by this unified Lyapunov approach, in both linear and non-linear controllers design, for the control of chaotic dynamics.
The time responses of the controlled system (14) for tracking a limit cycle with f 1 ¼ 1:379; f 2 ¼ 2:418 and x ¼ 0:0176. The control is achieved
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Fig. 12. The time responses of the controlled system (14) for tracking a limit cycle with f 1 ¼ 1:379; f 2 ¼ 2:418 and x ¼ 0:0176. The control is achieved at t ¼ 24.
Nonlinear feedback for control of chaos from a piecewise linear hysteresis circuit
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