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Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (i...
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Multistablity analysis and formation of spiral wave in the fractional-order nonlinear systems is a recent hot topic. In this paper, dynamics, coexisting attractors, complexity, and synchronization of the fractional-order memristor-based hyperchaotic Lü system are investigated numerically by means of bifurcation diagram, Lyapunov exponents (LEs), ch...
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... Next, we will test with a system of equations known to produce multiple attractors. Equation (3) is the system of differential equations for Yu's chaotic system [27] using a value of 2.5 for a. e region for initial states is defined below in Equation (5) Figure 4 shows the full-generated SSCC network in 3D Euclidean space and in the network-embedding space. Both visuals show two attractors. ...
The concept of attractors is considered critical in the study of dynamical systems as they represent the set of states that a system gravitates toward. However, it is generally difficult to analyze attractors in complex systems due to multiple reasons including chaos, high-dimensionality, and stochasticity. This paper explores a novel approach to analyzing attractors in complex systems by utilizing networks to represent phase spaces. We accomplish this by discretizing phase space and defining node associations with attractors by finding sink strongly connected components (SSCCs) within these networks. Moreover, the network representation of phase space facilitates the use of well-established techniques of network analysis to study the phase space of a complex system. We show the latter by introducing a new node-based metric called attractivity which can be used in conjunction with the SSCC as they are highly correlated. We demonstrate the proposed method by applying it to several chaotic dynamical systems and a large-scale agent-based social simulation model.
... In 1963 Lorenz discovered the chaotic attractor and proposed the Lorenz system [21], and then Chen and Liu successively proposed the Chen system and the Liu system [22,23]. These three typical systems are widely used [24][25][26][27][28][29]. Gao et al. adds a new state feedback controller on the basis of the three-dimensional Chen system to obtain the four-dimensional Chen system [24]. ...
... Gao et al. adds a new state feedback controller on the basis of the three-dimensional Chen system to obtain the four-dimensional Chen system [24]. Zhou et al. proposed a model with two independent attractors based on the Lü system model [26]. Liu et al. proposed a simple chaotic system using hyperbolic sinusoidal nonlinear functions and applied this system to medical images [28]. ...
... Step 9: XOR the pixels after replacement according to Eqs. (25)(26)(27). In this process, every l i pixels, m i is updated (16) to a new number, and then continue to diffuse. ...
A new four-dimensional autonomous dynamic system is proposed and the dynamic characteristics are analyzed. First calculate the equilibrium point and dissipation of the system, and then discuss the non-dynamic behavior of the system through the bifurcation graph, Lyapunov exponential spectrum, and spectral entropy (SE) complexity. The experiment found that the existence of a large range of parameter values keeps the system in a hyper-chaotic state. Therefore, we propose a color image encryption algorithm based on a four-dimensional hyperchaotic system and multiple S-boxes. First, the color image is scrambled globally, then three sets of S-boxes are constructed and replaced, and finally chaotic sequences are used to further diffuse the image. Through the experimental analysis, this algorithm has a good encryption effect on the image.
... c e-mail: fahimenazarimehr@yahoo.com d e-mail: haydernatiq86@gmail.com e e-mail: sajadjafari83@gmail.com f e-mail: iqtadarqau@gmail.com (corresponding author) try [14], multi-scroll attractors [15], hidden attractors [16], amplitude control [17], algebraically plainest equations [1], hyper-chaos [18], fractional-order formation [19,20], topological horseshoes [21], multi-stability [22], and extreme multi-stability [23]. Another top class of nonlinear chaotic systems contains periodically forced nonlinear oscillators, that the van der Pol oscillator is one of the most antecedent [24,25]. ...
Designing novel mega-stable chaotic oscillators has been a hot topic of research lately. In the current paper, a two-dimensional mega-stable oscillator is presented. The oscillator has a vast amount of coexisting limit cycles that spread on a surface. The forced version of this system is a novel nonlinear oscillator with an immense number of coexisting limit cycles and chaotic attractors. The dynamical features of this oscillator are examined by the assistance of equilibria study, the plot of the basin of attraction, bifurcation scheme, Lyapunov Exponents (LEs) perspective, and Kaplan-Yorke dimension.
... Besides, the Lü system has three equilibrium points, the present system has six. In addition, a chaotic system that is similar but not equivalent, to system (1) was presented in [42]; they have unequal numbers of equilibriums. Therefore, due to the highlighted differences among all the systems, there is no nonsingular transformation by which any of them can be converted to system (1). ...
This paper presents a new chaotic system that has four attractors including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial condition, Lyapunov spectrum, measure of strangeness, basin of attraction including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation are rigorously treated. It is found by numerical computations that the system has a far-reaching composite basin of attraction, which is important for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding values of electronic components such as the circuital resistances while ensuring the dynamic ranges are well contained. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to innate properties of the system such as sensitivity on initial condition and topological mixing, the robot’s path becomes unpredictable and guaranteed to scan the workspace, respectively.
... While the Lü system has three equilibrium points, the present system has six. Moreover, a chaotic system, similar but not equivalent to the current system, was presented in [183]. The system and the current system have unequal numbers of equilibriums. ...
This work presents a new chaotic system having four attractors, including two fixed point attractors and two symmetrical chaotic strange attractors. It also discusses dynamical properties of the system viz. sensitive dependence on initial conditions, Lyapunov spectrum, the measure of strangeness, basin of attraction, classification and quantification of the basin of attraction, the existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation. It is found by numerical computations that the system has far-reaching basins of attraction that encompass roughly $95$ percent of the state space. Moreover, sensitive dependence on initial conditions implies that long-term prediction of the system is impossible and any two such systems with slightly different initial conditions will become increasingly uncorrelated as time tends to infinity. However, despite this chaotic and unpredictable behavior, we realize various control schemes, including nonlinear control, active control, adaptive control, and robust adaptive control to synchronize two such systems in a master-slave topology irrespective of their initial conditions and parametric differences. The Lyapunov stability theory is used to ensure the boundedness of the closed-loop error dynamics. A Lyapunov-like analysis, involving the use of Barbalat lemma, is used to ensure asymptotic or exponential convergence of the error signals. Besides, the Lyapunov functions used are radially unbounded. Hence, global asymptotic or exponential stability of the closed-loop error dynamics follows. Besides, a secure communication application based on adaptive synchronization of a transmitter and a receiver based on the new chaotic system is realized. This situation is nontrivial as the adaptation mechanism tends to react against the information signals to suppress its effect on the equilibrium of the closed-loop system. Despite this situation, we can retrieve the information signals at the receiver. Depending on the nature of the modulating information signal, a detection mechanism may be used to reconstruct the original information signal at the receiver. Furthermore, the communication system based on adaptive synchronization of a transmitter-receiver system is subjected to a more realistic situation by transmitting the modulated signal via an additive white Gaussian noise (AWGN) channel of various signal-to-noise-ratios (SNRs) and noise power levels. A parametric sweep of various SNRs gives margins for continued useful communication.
... A chaotic system is a dynamical system with the following properties: (1) high sensitivity to initial conditions, (2) dense periodic orbits, and (3) topological mixing. Consequently, it is impossible to carry out accurate predictions about its long-term dynamic behavior [20][21][22][23]. In spite of that, the boundedness of its states can be guaranteed. ...
The objective of this paper is to estimate the unmeasurable variables of a multistable chaotic system using a Luenberger-like observer. First, the observability of the chaotic system is analyzed. Next, a Lipschitz constant is determined on the attractor of this system. Then, the methodology proposed by Raghavan and the result proposed by Thau are used to try to find an observer. Both attempts are unsuccessful. In spite of this, a Luenberger-like observer can still be used based on a proposed gain. The performance of this observer is tested by numerical simulation showing the convergence to zero of the estimation error. Finally, the chaotic system and its observer are implemented using 32-bit microcontrollers. The experimental results confirm good agreement between the responses of the implemented and simulated observers.
... ere are many nonlinear systems known to obtain coexistence of multiple attractors [1][2][3][4][5][6][7][8][9][10]. e coexistence of multiple attractors indicates that the attractor depends crucially on the initial condition (IC). ...
... Based on the 3D multistability chaotic system [1] reported by Zhou and Ke, in which there are two coexisting conditional symmetric chaotic attractors with different initial conditions, the chaos synchronization achieved by linear resistor and capacitor coupling is studied in this paper. First, the 3D multistability chaotic system [1] is studied by using a block diagram, and its electronic circuit is realized. ...
... Based on the 3D multistability chaotic system [1] reported by Zhou and Ke, in which there are two coexisting conditional symmetric chaotic attractors with different initial conditions, the chaos synchronization achieved by linear resistor and capacitor coupling is studied in this paper. First, the 3D multistability chaotic system [1] is studied by using a block diagram, and its electronic circuit is realized. e circuit simulation results are given. ...
In this paper, a 3D multistability chaotic system with two coexisting conditional symmetric attractors is studied by using a circuit block diagram and realized by using an electronic circuit. The simulation results show that two coexisting conditional symmetric attractors are emerged in this electronic circuit. Furthermore, synchronization of this 3D multistability chaotic system and its electronic circuit is studied. It shows that linear resistor and linear capacitor in parallel coupling can achieve synchronization in this chaotic electronic circuit. That is, the output voltage of chaotic electronic circuit is coupled via one linear resistor and one linear capacitor in parallel coupling. The simulation results verify that synchronization of the chaotic electronic circuit can be achieved.
... Recently there has been growing attention in finding chaotic systems with special qualities. Systems with no equilibrium [3], [4], with stable equilibria [5], [6], with curves of equilibria [7][8][9], with surface of equilibria [10][11][12], with multi-scroll attractors [13], with hidden attractors [14], [15], with amplitude control [16], [17], with simplest form , having hyperchaos [18][19][20], having fractional order form [21][22][23], with topological horseshoes [24], [25], and with extreme multistability [26][27][28][29], are examples of them. Another major category of chaotic systems includes periodically-forced nonlinear oscillators [30]. ...
In recent years designing new multistable chaotic oscillators has been of noticeable interest. A multistable system is a double-edged sword which can have many benefits in some applications while in some other situations they can be even dangerous. In this paper, we introduce a new multistable two-dimensional oscillator. The forced version of this new oscillator can exhibit chaotic solutions which makes it much more exciting. Also, another scarce feature of this system is the complex basins of attraction for the infinite coexisting attractors. Some initial conditions can escape the whirlpools of nearby attractors and settle down in faraway destinations. The dynamical properties of this new system are investigated by the help of equilibria analysis, bifurcation diagram, Lyapunov exponents' spectrum, and the plot of basins of attraction. The feasibility of the proposed system is also verified through circuit implementation.
... Due to the typical characteristics of high irregularity, unpredictability, and complexity of chaotic systems, chaotic systems and its applications have been attracted more and more attentions in the last few decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], e.g., information processing [11], secure communication [12,13], image encryption [14,15], machine learning [16], and so on. Memristor-the missing circuit element-has been discovered by Leon Chua in 1971 [17], and it has been successfully realized in 2008 [18]. ...
Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-orders q . Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.
... Li and Sprott have proposed chaotic systems with free control and amplitude control [Li et al., 2015b;Li et al., 2017c]. Chaotic systems with topological horseshoe [Zhou & Ke, 2017;, with time delay [Li & Fu, 2011;Li & Rakkiyappan, 2013], with Hamiltonian control, and fractionalorder chaotic systems Peng et al., 2019] are other examples. ...
In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.
