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Chaotic Analysis and Combination-Combination Synchronization of a Novel Hyperchaotic System without any Equilibria
Abstract
This manuscript examines a novel 4D continuous autonomous hyperchaotic system with two scroll attractor. This system does not produce any equilibrium point so, it generates hidden hyperchaotic attractor. The Lyapunov exponent, bifurcation diagram, Poincaré section, Kaplan–Yorke dimension and phase portraits are given to justify the hyperchaotic nature of the system. The novel system displays hyperchaotic orbit, chaotic orbit, periodic orbit, quasi-periodic orbit as the parameters values varies. Furthermore, the combination-combination synchronization is performed by considering four identical 4D novel hyperchaotic systems.
... The synchronization problem of four systems was studied in article [16] by using active backstepping technique. Article [17] developed a combination-combination synchronization scheme for a kind of 4-D autonomous hyperchaotic system. Article [18] coped with the CS problem for the complex chaotic systems by designing an adaptive control proposal. ...
... However, it can be found that the scaling functions given in [16][17][18][19][20][21][22] are constants or fixed functions. This means that once the synchronization scheme is given, the scaling functions are specified and can be expressed exactly. ...
... 2) The scaling functions given in [16][17][18][19][20][21][22] are constants or fixed functions, while the scaling functions in our combination synchronization are governed by the chaotic dynamic systems, which can improve the security of information transformation to some extent. ...
The concerns of this article is to investigate the fixed time function combination synchronization (FCS) among three different chaotic systems with disturbances. First, a new stability theorem is presented by virtue of the integral inequality technique and a more accurate upper estimation of the convergence time is given. Next, the fixed time FCS problem is considered for the perturbed chaotic systems in which the scaling functions are generated by the bounded dynamical systems. Besides, the synchronization time is related to the control parameters rather than the initial values. The time upper bound can be calculated out by the simple formula, which is the function of control parameters. Finally, a simulation example is provided to illustrate the correctness of the derived criteria.
... This subsection demonstrates the performance of the control law (3.2) in terms of synchronization error to zero, and time of convergence and describes the comparative analysis with the feedback controller [12]. ...
... The controller parameters ρ and γ are set as ρ = 0.01 and γ = 0.1. The control signals u i (t) , i = 1, 2, 3, 4, given in (5.6) are synthesized by the feedback control strategy [12], then these control signals in the error system (5.5) establish the synchronization of the master and slave systems (5.2) and (5.3). ...
... The feedback control technique [12]: ...
This paper proposes, designs, and analyses a novel nonlinear feedback controller that realizes fast, and oscillation free convergence of the synchronization error to the equilibrium point. Oscillation free convergence lowers the failure chances of a closed-loop system due to the reduced chattering phenomenon in the actuator motion, which is a consequence of low energy sm ooth control signal. The proposed controller has a novel structure. This controller does not cancel nonlinear terms of the plant in the closed-loop; this attribute improves the robustness of the loop. The controller consists of linear and nonlinear parts; each part executes a specific task. The linear term in the controller keeps the closed-loop stable, while the nonlinear part of the controller facilitates the fast convergence of the error signal to the vicinity of the origin. Then the linear controller synthesizes a smooth control signal that moves the error signals to zero without oscillations. The nonlinear term of the controller does not contribute to this synthesis. The collaborative combination of linear and nonlinear controllers that drive the synchronization errors to zero is innovative. The paper establishes proof of global stability and convergence behavior by describing a detailed analysis based on the Lyapunov stability theory. Computer simulation results of two numerical examples verify the performance of the proposed controller approach. The paper also provides a comparative study with state-of-the-art controllers.
... Similarly, using computer search algorithms, 17 chaotic systems without equilibrium were discovered . Some other diverse chaotic or hyperchaotic systems without equilibrium were reported in [Borah & Roy, 2017;Pham et al., 2017b;Wang & Chen, 2012;Khan & Shikha, 2018;Singh & Roy, 2018;Hu et al., 2016;Hu et al., 2017;Escalante-Gonzalez et al., 2017]. According to the published literatures, the chaotic or hyperchaotic systems with infinite number of equilibriums include the following types (including but not limited to): the equilibriums linearly distributed ; circularly distributed [Gotthans & Petrzela, 2015]; squarely distributed [Gotthans et al., 2016]; elliptically distributed [Pham et al., 2016a] and curvilinear distributed [Chen & Yang, 2015;Pham et al., 2016b]. ...
... The relevant literature comparison is shown in Table 2. The number of scrolls generated by the hyperchaotic systems with hidden attractors that were published in Khan & Shikha, 2018;Singh & Roy, 2018;Chen & Yang, 2015] are limited, at most, only 2-scroll hidden attractors can be generated, and no hardware circuit verifications have been carried out. The multiscroll chaotic system with hidden attractors proposed in literature can generate any number of scroll hidden attractors only in numerical simulation. ...
Based on the study on Jerk chaotic system, a multiscroll hyperchaotic system with hidden attractors is proposed in this paper, which has infinite number of equilibriums. The chaotic system can generate N + M + 2 scroll hyperchaotic hidden attractors. The dynamic characteristics of the multiscroll hyperchaotic system with hidden attractors are analyzed by means of dynamic analysis methods such as Lyapunov exponents and bifurcation diagram. In addition, we have studied the synchronization of the system by applying an adaptive control method. The hardware experiment of the proposed multiscroll hyperchaotic system with hidden attractors is carried out using discrete components. The hardware experimental results are consistent with the numerical simulation results of MATLAB and the theoretical analysis results.
... Vincent et al. developed a multiswitching combination synchronization of chaotic systems [18], and this synchronization type is further developed by Ahmad et al. as globally exponential multi-switchingcombination synchronization control for chaotic systems in the field of secure communications [21]. Based on the combination synchronization, with the consideration of four or more chaotic systems, the researchers further proposed and explored combination-combination synchronization in the cases where the numbers of drive systems and response systems are both larger than one [13,[22][23][24][25]. ...
... Time response of synchronization errors e ij , i=1, 2 and j=1, 2, 3, for the combination synchronization-II of Case 2 among systems (4), (5),(6), and(7).With this controller, the error system can be changed as the following (notice that s 1 = 1 and s 2 = 0 for the dynamics of e 11 , e 12 , and e 13 , s 1 = 0 and s 2 = 1 for the dynamics of e 21 , e22 , and e 23 ): 11 = 36 ( 12 − 11 ) , ...
Based on former combination synchronization studies, a new type of combination synchronization approach is developed in this research, with the consideration of parallel combination of drive systems. This new synchronization approach is referred to as combination synchronization-II. As a representative case, the combination synchronization-II between three drive systems and one response system is studied. Applying Lyapunov stability theorem and active backstepping design, sufficient conditions for the proposed combination synchronization approach are derived. Numerical simulations are performed to show the feasibility and effectiveness of the proposed approach. Based on the investigation in this research, the proposed approach provides an applicable method for designing universal combination synchronization among multiple chaotic systems.
... As chaos theory progresses, many new Lorenz-type systems [4][5][6] have been proposed, specially Lorenz hyperchaotic systems [7][8][9][10]. The Lorenz system is widely used in electric circuits, chemical reactions, and forward osmosis. ...
... Experiment . We consider the 3D autonomous chaotic Lorenz-type system[7] Chebyshev nodes and the number of nodes = 60 and the parameters = 10, = 15 and the initial conditions (0) = 10, (0) = −0.2, (0) = 0.75. ...
Although some numerical methods of the Lorenz system have been announced, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, this paper introduces a novel numerical method to solve the Lorenz-type chaotic system which is based on barycentric Lagrange interpolation collocation method (BLICM). The system (1) is adopted as an example to elucidate the solution process. Numerical simulations are used to verify the effectiveness of the present method.
... In the case of nonautonomous differential equations, i.e. nonlinear damped systems driven by external periodic signal, the minimal order of the oscillator can be reduced to two. Chaos was found useful with great potential in many fields, including liquid mixing with low power consumption, human brain and heartbeat regulation, and secure communications [5][6][7][8][9][10][11][12][13][14][15]. However, the external periodic signal required to generate chaotic behaviors are not always easy to obtain because frequency generator is expensive. ...
... From (8), it follows that the position of the Hopf bifurcation does not depend on the parameter β. This is confirmed by the numerical simulations of system (4) which show in Fig. 2, the bifurcation diagrams of x(t) versus the parameter k for = 0.5 and three different values of the parameterβ. ...
In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.
... We also prove that this is a system with three lines of equilibrium, and no other equilibrium. This is a very interesting case, as it belongs to the class of systems with hidden attractors, compared to other new works [15][16]. ...
Based on our previous work investigating chaos theory, in this article we suggest a hyperchaotic system with an equilibrium line and its synchronization. The paper introduces a new hyperchaotic system with highly complex dynamic behavior, so it is interesting to note that this system has three fixed point lines. Finally, we propose a method for synchronizing the proposed hyperchaotic system.
... Zhou et al. (2014) proposed synchronisation between two combinations of chaotic systems. Combination-combination synchronisation of hyperchaotic systems was developed by Khan and Singh (2018). Combination synchronisation of Lotka-Volterra chaotic systems was proposed by Jahanzaib et al. (2020). ...
... Zhou et al. (2014) proposed synchronisation between two combinations of chaotic systems. Combination-combination synchronisation of hyperchaotic systems was developed by Khan and Singh (2018). Combination synchronisation of Lotka-Volterra chaotic systems was proposed by Jahanzaib et al. (2020). ...
... A comparison analysis between the proposed finite-time combination-combination (C-C) synchronization (FTCCS) scheme and the earlier published work is as follows. In Ref. [62], the author applied the adaptive control method to achieve C-C synchronization among four identical hyper-chaotic systems where it noted that the synchronization states happened at t = 5 (approx). In Ref. [61], the author used the sliding mode control scheme to address multiple chaotic systems with unknown parameters and disturbances in which the synchronization happened at t = 5 (approx). ...
This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the finite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the finite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB.
... Designing a slave system that imitates the behavior of the master system is called chaos synchronization. Introduced the concept of chaos later on several fast-growing interest research areas has reported such as complete synchronization [3], anti-synchronization [4], hybrid synchronization [5], projective synchronization [6], lag synchronization [7] , modulus synchronization [8,9], combination synchronization [10], dual combination synchronization, combination-combination synchronization (CCS) [11,12], etc. Moreover, unlike other nonlinear analytical techniques, such as perturbations, this method does not depend on small parameters used to give an approximate solution. ...
The present paper tries to study a new adaptive modulus hybrid projective combination synchronization (AMHPCS) with time-delay hyperchaotic (HC) systems containing uncertainty and disturbance. Time-delay, uncertainty, and disturbance are the three basic occurrences in nonlinear systems, which gravely affect the control and synchronization of chaotic systems. The controlling and parameter updating equations are designed through the adaptive controllers’ method and the Lyapunov function. This AMHPCS approach gives a systematic procedure for an automated arrangement of controllers in real-time, to perform or to sustain the wanted level of control system execution when the parameters of the chaotic systems are strange and/or vary in time. The time-delay complex HC Lü system and Lorenz systems have been considered as the master and slave systems respectively. This is the first problem in which modulus hybrid projective combination synchronization of time-delay HC with uncertainty and disturbance has been achieved for unknown parameters. Time-delay chaotic/HC exhibits multi-stability and is useful to construct more significant complex dynamics that improve the transmitted data security. Based on the AMHPCS, a secure communication scheme is described. The information message has been recovered accurately by using the chaotic signal masking method. Numerical simulations are also performed to establish the effectiveness of the proposed method, numerical simulations have been performed by using the Runge–Kutta method of the delay-differential equations. This ensures that the designed controllers and adaptive parameter laws are effective to secure and synchronize chaotic time-delay systems and our results demonstrate the novelty over the existing results.
... When k = 0, by solving Eq. (8), it is easy to get x = y = 0, dz = k by taking it into the third equation we get −ck/d = 0, but the parameters c and k are all non-zero constants, obviously this is a contradiction Therefore, in the case of k = 0, there is no solution to the equilibrium equation, so no equilibrium point exists in the system. According to the definition of hidden attractors, [29] no matter what the value of parameter k is, the system has hidden attractor generation. Let the initial values of the state variables be (1, 1, 1, 10). ...
A five-value memristor model is proposed, it is proved that the model has a typical hysteresis loop by analyzing the relationship between voltage and current. Then, based on the classical Liu–Chen system, a new memristor-based four-dimensional (4D) chaotic system is designed by using the five-value memristor. The trajectory phase diagram, Poincare mapping, bifurcation diagram, and Lyapunov exponent spectrum are drawn by numerical simulation. It is found that, in addition to the general chaos characteristics, the system has some special phenomena, such as hidden homogenous multistabilities, hidden heterogeneous multistabilities, and hidden super-multistabilities. Finally, according to the dimensionless equation of the system, the circuit model of the system is built and simulated. The results are consistent with the numerical simulation results, which proves the physical realizability of the five-value memristor-based chaotic system proposed in this paper.
... In addition, these variables are set to zero except three of them, an extreme approximation. The following set of equations as in (1), (2) and (3) represents the model, which represents Lorenz equations [10][11][12][13][14]: ...
In spite of, several mathematical approaches of the Lorenz solver system have been declared, fast and effective approaches has always been the direction in which scientists are trying to achieve. Based on this challenge, this paper initiates to boost the processing of Lorenz ordinary differential equations (ODE) system by applied hardware accelerator to take advantage of parallel architecture. A chaotic solution is found for various parameters and primary conditions. Specifically, Lorenz attractor was a group of chaotic outputs of the Lorenz equation. The 3D plotted the shape of Lorenz attractor was like “‘butterfly wings” which depend on initial conditions of the nonlinear system. Lorenz has more application in the real world as in biomechanical, physical, Control systems. As noted earlier, the Lorenz system is a scheme of ordinary differential equations exhibiting chaotic behavior for certain parameter values and initial conditions. The result presents the different Lorenz waveform of various parameter values to calculate the time of execution for each case study on a different platform.
... The more the number of the positive lyapunov exponents in system of higher dimension, the more chaotic the system is. For instance in system of dimension four if we obtain one positive L.E. we say the system is chaotic, but if we have two positive L.E. we say that the system is hyper-chaotic [46]. The maximal lyapunov exponent for system and its first, second and third approximation respectively are 0.4583, 0.26892, 0.47238, 0.45998. ...
Journal of Vibration Testing and System Dynamics is an interdisciplinary journal as a platform for facilitating the synergy of dynamics, experimentation, design, and education. The journal publishes high-quality, original articles that explore the theory, modelling, and application of dynamical systems and data-driven dynamics for high-impact engineering solutions. Manuscripts exploring data science, machine learning, and artificial intelligence to the design and control of complex dynamical systems including cell and neuro networks are solicited. Articles on data mining, deep learning and big data applicable to physical sciences and large-scale dynamic systems are equally encouraged. Progress made in the following topics, but not limited to, are of interest to the journal:
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... Further, in Ref. [42], author studied C-C synchronization among four complex nonlinear chaotic systems where it has been recorded that the error synchronization is realized at t = 5 (approx.). Also in Ref. [45], author applied ACM to achieve C-C synchronization between four identical HC systems where it noted that the synchronization state is attained at t = 5.1 (approx.). Moreover, in Ref. [46], author utilized robust adaptive sliding mode control to generalize the C-C synchronization of n-dimensional chaotic systems with time delay in which the synchronization is obtained at t = 4.9 (approx.). ...
In this paper, we investigate a hybrid projective combination–combination synchronization scheme among four non-identical hyperchaotic systems via adaptive control method. Based on Lyapunov stability theory, the considered approach identifies the unknown parameters and determines the asymptotic stability globally. It is observed that various synchronization techniques, for instance, chaos control problem, combination synchronization, projective synchronization, etc. turn into particular cases of combination–combination synchronization. The proposed scheme is applicable to secure communication and information processing. Finally, numerical simulations are performed to demonstrate the effectivity and correctness of the considered technique by using MATLAB.
... Firstly, we compared our results with previously published results for combination-combination synchronization of chaotic systems using a different technique. In [32], the author investigated combination-combination synchronization of a novel HC system without equilibria using active control, and they achieved the synchronization error at t = 5(approx). In [48], author studied Multi-switching combination-combination synchronization of non-identical fractional-order CS and they achieved the synchronization error at t = 1.5(approx). ...
In this paper, we propose an adaptive hybrid complex projective combination–combination synchronization method to synchronize the hyperchaotic (HC) complex Lorenz system and HC complex Lu system. The adaptive control laws and parameter update laws are derived from making the state of HC complex systems asymptotically stable by using Lyapunov stability theory. During these studies, the coupled HC complex systems (master and slave systems) evolve in a distinct direction with a constant intersection phase angle. Numerical simulations are performed to illustrate the validity and effectiveness of the proposed scheme using MATLAB.
... Many hyperchaotic systems were constructed by modifying an already existed 3D chaotic system. A continuous autonomous hyperchaotic system without equilibrium points was constructed by adding a feedback controller to a 3D autonomous chaotic Lorenz-type system [13]. Bonyah [14] proposed a new hyperchaotic system with four wings by changing the non-local and non-singular fractional derivatives. ...
This paper concerns the application of adaptive control method in a four-dimensional hyperchaoticsystem. Firstly, we carry out a systematic dynamic analysis including the properties of equilibriumpoint, stability, dissipation, Lyapunov exponent spectrum, and bifurcation. Both the existenceof two positive Lyapunov exponents and the Lyapunov dimension value show the hyperchaotic property of the system. Based on Lyapunov stability theorem, we then construct an adaptive controller and the adaptive law to suppress hyperchaos to the origin, which is an unstable equilibrium point under a certain parameter set. The effectiveness of the adaptive control is veried by theoretical analysis and numerical simulation. We nally brie y demonstrate the control eciency of self-linear feedback control and misaligned feedback control. For the fourdimensional hyperchaotic system, the adaptive control outperforms them from the view of control speed.
... The rising requirements for reasonably secure and extra trustworthy encryption algorithms has shaped innovative study in secure communication and has got cryptographers' attention to propose better cryptosystems based on new concepts. Recently, it has been proved that the chaos [1][2][3][4][5] based schemes demonstrate the majority of valid properties that are suitable for encryption systems [6][7][8] thus chaotic [9,10] encryption schemes captured good ground in cryptography together with steganography and watermarking [11][12][13][14][15][16][17][18][19][20][21][22]. It was 1960s when the concept of chaotic dynamics was initiated by E. Lorenz [23]. ...
When we listen regarding quantum cryptography, the first thinking that comes up to mind is, how can there be any connection between physics and cryptographic ciphers. In fact, it comes into view that is one of the hottest research in the field of cryptology to exploit physics and has been confirmed as the important objective in the protection of digital data. The topic of quantum magnets and matrix Lorenz Systems has begun to draw increasing attention in recent years. In last decade, chaotic systems have been utilized in many encryption algorithms. In this regard, Lorenz chaotic system based on differential equations achieved more recognition because of its fundamental sensitivity to initial conditions and easiness in use. This manuscript has two parts. First, we have constructed Substitution Boxes (S-Boxes) based on quantum magnets and matrix Lorenz chaotic system and permutation of the symmetric group S256 and then designated an image encryption algorithm based on proposed S-boxes and Lorenz chaotic system. In cryptography, it is essential to construct secure S-Boxes to propose cryptographically strong systems. Recently, some image encryption techniques that are weak against robustness have been proposed based on chaotic S-boxes. In the first part of the manuscript, an S-Box design is proposed. Continuous-time Lorenz system is chosen as the chaotic system. The presented image encryption algorithm is based on two rounds of a novel chaotic substitution-permutation network uses proposed S-boxes for substitution purpose by iterations of the Lorenz chaotic system and special kind of permutation process to get the full cipher image. The proposed algorithms for S-box construction and image encryption give very good and coherent performance results when we analyze them with renowned analyses.
This paper introduces a new hyperchaotic oscillator base on a new boundary-restricted Hewlett- Packard memristor model. Firstly, the complex system is designed based on a memristor-based hyperchaotic real system, and its properties are analyzed by means of Lyapunov exponents, Lyapunov dimension and phase portraits diagrams. Secondly, a simple feedback control based on the minimum variance control technique is designed to stabilize the hyperchaotic oscillator system, which is one of the new developed approaches for controlling the chaos in high-dimensional hyperchaotic systems. In this method, the time series variance is considered for designing and calculating the state feedback control gain. Furthermore, the state feedback control is designed so that to minimize the variance as a cost function, followed by developing an online optimization technique using the particle swarm optimization method in order to calculate the state feedback control based on the minimum variance strategy. Then, the application of this method is examined on a hyperchaotic memristor-based oscillator. Finally, the sensitivity of the proposed method is evaluated in different initial conditions that greatly influence the hyperchaotic dynamics. Considering that the optimization is online, simulation results show highly good effectiveness of the presented technique in controlling the chaos in high-dimensional hyperchaotic oscillators.
Substitution boxes are used in different security techniques and cryptosystems to ensure the secure communication of data. To enhance the randomness and perplexity of data, chaos theory has utmost importance in encryption schemes and multimedia security. In this paper, Substitution boxes are developed by using linear fractional transformation and combination of chaotic systems with the increased chaotic range as compared to their seed maps. The Substitution boxes are assessed by using various analyses which include nonlinearity, strict Avalanche criterion, bit independence criterion, linear and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.
The memristor-based FitzHugh-Nagumo neuron system shows special dynamical behaviors, which make it have a good application in the fields of image processing and signal transmission. The accurate dimensionality reduction model of the system is established, which facilitates the internal interpretation and physical control of the multistability that depends on the initial values. The dynamical behavior analyses of the dimensionality reduction memristive system with different original initial states are developed through bifurcation diagram and Lyapunov index spectrum. Multistability and antimonotonicity with periodic-chaotic bubbles are investigated. The sliding mode control method with a smooth function is designed to achieve the finite-time synchronization of the multistable memristive neuron systems, which make three different behaviors of systems synchronized. The sliding mode controller can enable the system to quickly and smoothly implement combination synchronization within finite time. Numerical simulations show the effectiveness of the sliding mode controller designed.
Purpose
This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems.
Design/methodology/approach
A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C 0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components.
Findings
The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q . Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system.
Originality/value
The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.
Chaos is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. This article examines the controlling of a chaotic system via optimal control technique which is based on the Pontryagin minimum principle. A 3D chaotic system is considered to apply this scheme which have 5 equilibrium points. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed method. The simulation results illustrated the stabilized behaviour of states and control functions for different equilibrium points.
Resistor-based voltage coupling is often used to realize complete synchronization between identical nonlinear circuits while phase synchronization is investigated between non-identical nonlinear circuits (periodic or chaotic oscillation). Indeed, the coupling resistor used to consume certain Joule heat and energy before reaching the synchronization target when continuous current passed across the coupling device. In this paper, capacitor and inductor is paralleled with one coupling resistor, respectively, and the coupling devices are used bridge connection between two LC hyperchaotic circuits for investigating synchronization problems. As a result, the coupling channel can be activated to propagate energy and balance the outputs voltage from the two circuits. The dimensionless dynamical equations are obtained by applying scale transformation on the circuit equations when field coupling is switched on. It is found that the threshold of coupling intensity for reaching synchronization and the power consumption of controller can be decreased when the coupling resistor is paralleled with on capacitor or inductor. The mechanism could be that involvement of coupling capacitor(or inductor) can trigger time-varying electric field (or magnetic field), and the energy flow of field coupling via coupling capacitor (or inductor) can contribute the exchange of energy in the coupled nonlinear circuits. It can give insights to investigate synchronization on chaotic systems, neural circuits and neural networks including synapse coupling and field coupling. Finally, the experimental results on circuits are also supplied for further verification.
This paper presents a novel chaos-based technique of steganography in spatial domain. In the last decade, chaos theory has gained utmost importance in multimedia security applications. Generally, 1-D chaotic maps are employed because of computational ease and structural simplicity but their limited chaotic range is an obstacle. In the proposed work, we model the nonlinear combinations of 1-D chaotic maps. These chaotic systems possess chaotic behavior throughout the domain. We, for the first time, propose an effective application of these improved chaotic systems in steganography. These newly synthesized systems are used to embed secret information in the least significant bits (LSBs) of the host image. By comparing with some recent models, we prove that involving improved chaotic systems in steganographic approach really produces extraordinary outcomes. We determine the strength of our steganographic algorithm through the most significant statistical analyses such as information entropy, correlation, contrast, energy, homogeneity, peak signal to noise ratio (PSNR) and mean square error (MSE). We further prove the robustness of the anticipated technique against several image processing attacks. The upshot of these analysis techniques shows that our algorithm is highly reliable and produces coherent results.
In this manuscript, we investigate the methodology of combination-combination anti-synchronization of four identical fractional order hyperchaotic system. The methodology is implemented by considering a 4D fractional order hyperchaotic system. The controllers are constructed using adaptive control technique to ensure the combination-combination anti - synchronization. The synchronization schemes such as chaos control problem, projective anti-synchronization, combination anti-synchronization becomes the special cases of combination-combination anti-synchronization. The combination - combination scheme can additionally enhances the security of transmission of message signals. The theoretical results and numerical simulations are given to justify the validity and feasibility of the proposed control technique.
This manuscript investigates the synchronization between a novel integer order hyperchaotic system and a fractional order hyperchaotic system. The controllers are constructed using the technique of tracking controller and the stability theory of the linear fractional order system. Chaotic analysis of the introduced novel integer order hyperchaotic system is also investigated. The Lyapunov exponent, bifurcation diagram, Poincare section, Kaplan-Yorke dimension, equilibria and phase portraits are given to justify the chaotic nature of the system. Theoretical results are supported with the numerical simulations.
In the last decades, many studies have been done about nervous system from the mathematical and computational point of view. Many mathematical models have been proposed to describe neuron. Most of them have described the membrane potential of a neuron in terms of the leakage current and the synaptic inputs. Very recently, according to the Maxwell electromagnetic induction theorem, researchers proposed a new neuron model which considers inner magnetic fluctuation and external electromagnetic radiation as a significant missing part that can participate in neural activity. In this paper, dynamical properties of this new model, such as equilibria and its stability, phase portraits, bifurcation diagram, Lyapunov exponents, and basin of attraction, are investigated. Moreover, an implementation of electronic circuit and FPGA is also done to verify the feasibility of the theoretical model.
Thermal images play a vital character at nuclear plants, Power stations, Forensic labs biological research, and petroleum products extraction. Safety of thermal images is very important. Image data has some unique features such as intensity, contrast, homogeneity, entropy and correlation among pixels that is why somehow image encryption is trickier as compare to other encryptions. With conventional image encryption schemes it is normally hard to handle these features. Therefore, cryptographers have paid attention to some attractive properties of the chaotic maps such as randomness and sensitivity to build up novel cryptosystems. That is why, recently proposed image encryption techniques progressively more depends on the application of chaotic maps. This paper proposed an image encryption algorithm based on Chebyshev chaotic map and S8 Symmetric group of permutation based substitution boxes. Primarily, parameters of chaotic Chebyshev map are chosen as a secret key to mystify the primary image. Then, the plaintext image is encrypted by the method generated from the substitution boxes and Chebyshev map. By this process, we can get a cipher text image that is perfectly twisted and dispersed. The outcomes of renowned experiments, key sensitivity tests and statistical analysis confirm that the proposed algorithm offers a safe and efficient approach for real-time image encryption.
In this paper, a new 4D autonomous asymmetric hyperchaotic system obtained from a three dimensional autonomous chaotic system is introduced. We analyse the hyperchaotic properties of the new system such as dissipation, equilibrium, Lyapunov exponent, stability, time series, phase portraits, Poincare map and bifurcation diagram. Furthermore, the multi-switching synchronization of the new asymmetric hyperchaotic system is analysed by using the adaptive control strategy. Based on the theory of Lyapunov stability, we sketch the input controllers and updating laws of distinct switching, and it is extended to examine the synchronization problems of response and drive systems with distinct combinations. Brief theoretical analysis and numerical results are presented to show the dynamical behaviour of the new 4D hyperchaotic system.
In this manuscript, hybrid function projective synchronization of Bhalekar–Gejji and Pehlivan chaotic systems is established by applying adaptive control technique where the system parameters are unknown. In this manuscript both the master and slave system are chosen in such a way that none of them can be derived from the member of the unified chaotic system. We construct an adaptive controller in such a manner that master and slave system attain global chaos synchronization. The results derived for the synchronization have been established using adaptive control theory and Lyapunov stability theory. Fundamental dynamical properties of both the chaotic systems are also described. The results are validated by numerical simulation which are performed by using Matlab.
A three-dimensional autonomous chaotic system with a circular equilibrium is investigated in this paper. Some dynamical properties and behaviors of this system are described in terms of equilibria, eigenvalue structures, bifurcation diagrams, Lyapunov exponents, time series and phase portraits. For specific parameters, the system displays periodic and chaotic attractors. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcad-PSpice software and experimental verification. A good qualitative agreement is shown between the experimental results, PSpice and numerical simulations. Furthermore, the commensurate fractional-order version of the system with a circular equilibrium is numerically studied. It is found that chaos exists in this system with order less than three. By tuning the commensurate fractional order, the system with a circular equilibrium displays chaotic and periodic attractors, respectively. Finally, chaos synchronization of identical fractional-order chaotic systems with a circular equilibrium is achieved by using the unidirectional linear error feedback coupling. It is shown that the fractional-order chaotic system can achieve synchronization for appropriate coupling strength.
In this paper, the methodology to achieve combination synchronization of time-delay chaotic system via robust adaptive sliding mode control is introduced. The methodology is implemented by taking identical time-delay Lorenz chaotic system. The selection of switching surface and the design of control law is also discussed, which is an important issue. By utilizing rigorous mathematical theory, sufficient condition is drawn for the stability of error dynamics based on Lyapunov stability theory. Theoretical results are supported with the numerical simulations. The complexity of this methodology is useful to strengthen the security of communication. The hidden message can be partitioned into several parts loaded in two master systems to improve the accuracy of communication.
This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.
Based on one drive system and one response system synchronization model, a new type of combination–combination synchronization is proposed for four identical or different chaotic systems. According to the Lyapunov stability theorem and adaptive control, numerical simulations for four identical or different chaotic systems with different initial conditions are discussed to show the effectiveness of the proposed method. Synchronization about combination of two drive systems and combination of two response systems is the main contribution of this paper, which can be extended to three or more chaotic systems. A universal combination of drive systems and response systems model and a universal adaptive controller may be designed to our intelligent application by our synchronization design.