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Due to its chaotic carrier function and pseudorandom nature, chaotic systems are frequently applied in the area of secure communication. To render chaotic signals more complicated, a novel multi-wing chaotic system is developed in this study. Initially, a double-wing attractor generated by a novel 3D Lorenz-like chaotic system that comprises two quadratic terms and four linear terms. By inserting a nonlinear feedback controller, the complex system gets transformed into a multi-wing chaotic system. It is noteworthy that the topological shape of the attractor and the number of equilibrium points can both be altered in the particular multi-wing chaotic system. Secondly, the dynamic features of the multi-wing chaotic system are discussed via employing the bifurcation diagram and the Lyapunov exponential spectrum. According to results from simulation, the system displays widely dispersed chaotic characteristics over a large range of parameters. In addition, the complexity analysis findings of chaotic sequences testify that the chaotic sequences have strong complexity. Therefore, the multi-wing chaotic system will have an excellent effect in image encryption. Finally, a scrambling-diffusion encryption scheme is built using the rich chaotic sequences discussed above. The scheme includes two times scrambling of pixel position and once transform of pixel value by the virtue of Galois field (GF). The proposed algorithm has superior efficiency and excellent security according to the results of the feasibility and security analysis. Moreover, the investigation of the multi-wing chaotic system property and its usage in image encryption additionally broadens the applicability of chaos systems and offers some standards for confidential communication. [ABSTRACT FROM AUTHOR]

Abstract In this contribution, the effects of two key internal parameters, i.e. the linewidth-enhancement factor (α) and the gain nonlinearity (ε), on time-delay signature (TDS) concealment of the optical injection chaos system with phase-conjugate feedback (PCF-OICS) are numerically investigated. The results show that, as the injection and feedback parameters (injection and feedback strength, frequency detuning) are varied, higher linewidth-enhancement factor and lower gain nonlinearity are useful for concealing the TDS, and our simulations uncover that combining the mechanism of PCF and chaotic optical injection is beneficial for TDS concealment in the slave semiconductor lasers (SSL). [ABSTRACT FROM AUTHOR]

We studied two methods that quantify the orbital instability of a laser diode (LD) in two typical chaotic systems. In one method, a coefficient named the orbital expansion exponent (OEE) is calculated from only the time series of the laser intensity. In the other method, the Lyapunov exponent is calculated from the rate equations. Then we compared the OEE and the Lyapunov exponent to discuss the appropriateness of the OEE. In an optical feedback LD system, the OEE cannot represent the dynamics of the chaotic LD satisfactorily for the large feedback ratio. However, when the LD has a larger drive current and the inherent parameters of the LD are dominant in the dynamics, the OEE represents the dynamics of the chaotic LD as well as the Lyapunov exponent. On the other hand, in an optical injection LD system, the correlation coefficient between the OEE and the Lyapunov exponent is large even if the injection ratio is large. To find the reason for this, we replaced the optical feedback system with a multistage optical injection system, and showed that the feedback terms prevent the OEE from accurately representing the orbital instability of a chaotic laser. [ABSTRACT FROM AUTHOR]

In this paper, digital watermark based on a new memritive chaotic system is studied. Firstly a new memristive chaotic system is constructed, and has complicated dynamical chaotic behavior; then the memristive chaotic sequence extraction and watermark encryption algorithm are presented to obtain digital watermark; finally concealment analysis and anti-attack capability analysis are carried out to show the effectiveness of our digital watermarking scheme. [ABSTRACT FROM AUTHOR]

The present study investigates the generalized Biswas–Milovic model using Kudryshov's law of refractive index by employing the G ′ / (b G ′ + G + a) -Expansion method. This technique facilitates the extraction of singular exponential, trigonometric and kink soliton solutions for the model under examination. The outcomes are subsequently presented as 2D, 3D, and contour plots. Additionally, the model is transformed into a planner dynamical system, and the bifurcation theory is utilized to explore all potential parameter dependencies of the governing model. The chaotic behavior of dynamical systems is also studied by adding an external periodic force. The phase portraits of the obtained findings are also displayed. The results demonstrate the rich and sophisticated behavior of dynamical systems and emphasized the significance of exact solutions for determining their behavior. [ABSTRACT FROM AUTHOR]

We present a detailed study on the dynamics of a single Loop optoelectronic Oscillator (SLOEO), with the variation of feedback loop delay. The feedback delay can be varied by changing the length of the optical fiber delay line. From the numerical study it is observed that with the variation of the delay the oscillator produces chaotic oscillation following the sequence of Hopf bifurcation, stable periodic and period doubling oscillation. In OEO, incrementing the feedback loop delay increases the number of neighboring modes. The effect of these modes on the dynamics of the OEO is reported through simulation study. It has been seen that due to the generation of additional modes, instability grows in the system and the oscillator changes its state from periodic to chaotic oscillation. Consequently the effect of periodic external synchronizing signal is studied in the chaotic OEO. It is observed that application of an external periodic signal with suitably chosen amplitude and frequency can destroy the chaotic oscillation and produce single frequency oscillation. This mechanism has an additional advantage in the reduction of phase noise with regard to the use of larger delay in the loop. Moreover, the sync. signal increases the effective quality factor (Q) of the circuit thereby reducing the deleterious effect of phase noise. [ABSTRACT FROM AUTHOR]

A novel color image encryption algorithm is proposed in this paper which utilizes SHA-256 hash function to modify the initial conditions and control parameters of chaotic system. Three channels of a color image called red, green and blue is arranged into one dimensional vector and sort according to chaotic sequence generated by Piecewise Linear Chaotic Map. Then this permuted array split into three parts where each represents a color channel and permuted again using Lorenz’s chaotic system independently. After dual permutation, each pixel of every channel is independently encoded into Deoxyribonucleic Acid (DNA) bases chaotically. The novelty of algorithm is that the substitution of each pixel of a channel is achieved by Exclusive-OR operation with DNA complementary rules. This operation is repeated with multiple DNA rules in a sequence to some random number of times. This repetition of operation continues in a cycle. The selection of DNA rule in the beginning of this cyclic operation and continuation of operation is made reliant on Chen’s chaotic sequence. The extensive simulated experimental results such as Key Sensitivity, Correlation, entropy, Number of Pixel Chang Rate (NPCR), Unified Average Cipher Intensity (UACI) scores prove that proposed algorithm has excellent encryption results obtained in just one round. This scheme has advantage over other encryption techniques that it has nice feature to resist transmission noise and loss of a part of an encrypted image. [ABSTRACT FROM AUTHOR]

Synchronization of chaotic system is useful in information hiding and secure communication. The convergence time of chaotic synchronization is a key factor that will affect its application in message recovery. A special principle of constructing a finite time stable system was proposed to realized the finite time control and synchronization of chaotic PMSM system. After that, a kind of hybrid constructing method was proposed by integrating a common stable system and finite time stable system. Also parameter identification problem was studied, and the use of error information of parameter was avoided to construct a safe scheme for information hiding and recovery. At last, detailed numerical simulation result were done to testify the rightness of all proposed new theorems and new methods. [ABSTRACT FROM AUTHOR]

In the commented paper, the authors consider a three-dimensional system and analyze the presence of Shilnikov chaos as well as a Hopf bifurcation. On the one hand, they state that the existence of a chaotic attractor is verified via the homoclinic Shilnikov theorem. The homoclinic orbit of this system is determined by using the undetermined coefficient method, introduced by Zhou et al. in [Chen's attractor exists, Int. J. Bifurcation Chaos 14 (2004) 3167–3178], a paper that presents very serious shortcomings. However, it has been cited dozens of times and its erroneous method has been copied in lots of papers, including the commented paper where an even expression for the first component of the homoclinic connection is used. It is evident that this even expression cannot represent the first component of a Shilnikov homoclinic connection, an orbit which is necessarily non-symmetric. Consequently, the results stated in Section 3, the core of the paper, are worthless. On the other hand, the study of the Hopf bifurcation presented in Section 4 is also wrong because the first Lyapunov coefficient provided by the authors is incorrect. [ABSTRACT FROM AUTHOR]

This paper firstly discusses chaotic behaviors in a four-wing fractional-order system based on frequency-domain approach and time-domain approach, respectively. And chaotic dynamics is indeed found to exist in the system by numerical analysis. Then, in order to apply the system in secure communication, a new chaos circuit based on the four-wing fractional-order chaotic systems is designed to implement the synchronization mentioned. And the effectiveness and feasibility of the proposed synchronization scheme are verified by the new analog circuit. At last, an example of secure communication is given to show that the fractional-order four-wing system is appropriate for secure communication of some signals. [ABSTRACT FROM AUTHOR]

In this paper, the problem on a memristor-based chaotic system and its application in image encryption is discussed. A new memristive chaotic system is presented, corresponding dynamical behaviors are analyzed. Then encryption method based on the new memristive chaotic system is proposed to achieve the image encryption, Finally encrypted image analysis is carried out to demonstrate the effectiveness of the image encryption method. [ABSTRACT FROM AUTHOR]

This paper reports a new 4-D dissipative chaotic system. When compared with the available 4-D chaotic/hyperchaotic systems, the unique properties of the proposed system are (i) the system exhibits coexistence of different kind of asymmetric hidden attractors, (ii) the system has a curve of equilibria and (ii) the system is simple. This system does not satisfy Shilnikov criterion for the existence of chaos. The system has a total of eight terms including only one nonlinear term. The coexistence of rich chaotic dynamics in the system is investigated through Lyapunov spectrum, bifurcation diagram, Poincaré map, frequency spectrum and attractors plot. The coexistence of the asymmetric hidden dynamics is shown using chaotic attractors, 3- torus and 2-torus plots. The system is developed from the type-IV 3-D Rossler chaotic system. [ABSTRACT FROM AUTHOR]

In this article, based on tracking control strategy and the stability theory of the fractional differential equation, a new controller has been designed to realize the dual projective synchronization (DPS) between integer-order and fractional-order chaotic systems. The proposed method is respectively applied to dual projective synchronization of the following two pairs of chaotic systems: Sprott-Lü and Liu-Chen chaotic systems. The numerical simulations verify the correctness and effectiveness of the strategy. [ABSTRACT FROM AUTHOR]

This paper investigates the control problems of the Bose–Einstein condensate (BEC) system. Firstly, a new linear feedback control method for the control problems of a class of chaotic systems in any dimension based on triangle system structure. Then, based on the obtained method, the stabilization, synchronization, and anti-synchronization of BEC system are investigated extensively, and the corresponding controllers are designed. It should be pointed out that the obtained controllers are single input linear controllers, which are simpler than the existing results. Finally, numerical simulations are given to show the effectiveness and validity of the proposed results. [ABSTRACT FROM AUTHOR]

A new method for generating M × N -grid double-scroll Rucklidge chaotic attractors is presented. By designing arctangent function series to construct a nonlinear function, which replace the state variables x , y of the Rucklidge system, a new M × N -grid double-scroll chaotic system is created. The formation mechanism of this M × N -grid double-scroll chaotic model is further investigated. Moreover, some basic properties are theoretically analyzed and numerically simulated, including equilibria and their stability, Lyapunov exponents and bifurcation diagram. This system can be widely utilized in secure communication and data encryption. [ABSTRACT FROM AUTHOR]

Referring to fractional-order systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions and different orders . By taking a master system of dimension n and a slave system of dimension m , the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach presents some useful features: i) it enables commensurate and incommensurate non-identical fractional-order systems with different dimension n < m or n > m to be synchronized; ii) it can be applied to a wide class of chaotic (hyperchaotic) fractional-order systems for any differentiable scaling function; iii) it is rigorous, being based on two theorems, one for the case n < m and the other for the case n > m . Two different numerical examples are reported, involving chaotic/hyperchaotic fractional-order Lorenz systems (three-dimensional and four-dimensional master/slave, respectively) and hyperchaotic/chaotic fractional-order Chen systems (four-dimensional and three-dimensional master/slave, respectively). The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function. [ABSTRACT FROM AUTHOR]

The paper cryptanalyses the symmetric key image encryption scheme proposed by Mrinal et al. [1] . The encryption scheme proposed by Mrinal et al. uses Rossler chaotic system for generating three chaotic sequences which are used for creating the cipher image. A secret key of 128 bits is used to generate the initial conditions for the Rossler chaotic system. The equations used for generating the initial conditions in Mrinal et al. encryption scheme have got a loophole. The loophole is utilized to carry out the cryptanalysis on the cipher image generated using Mrinal et al. encryption scheme. Simulation results show that the encryption scheme can be easily cryptanalysed to revealed the exact plain image. An improved version is proposed with statistical and security analyses to ascertain the practicability of the encryption scheme. [ABSTRACT FROM AUTHOR]

Recently, a novel image fusion encryption algorithm based on DNA sequence operation and hyper-chaotic system were proposed. It was reported that the encryption algorithm can be broken by chosen-plaintext attack. This paper re-evaluates the security of the encryption algorithm, and finds that the encryption algorithm can be broken by known-plaintext attack as well. Besides, it also gives another method of chosen-plaintext attack which is more effective. The effectiveness of the proposed know-plaintext attack and chosen-plaintext attack is supported both by rigorous theoretical analysis and experimental results. [ABSTRACT FROM AUTHOR]

In this article, the synchronization between fractional order complex chaotic systems with uncertainty is considered using Active control method. Synchronization of the fractional order chaotic systems with uncertainty have already been studied in real variable, but for the case of complex variable it is first of its kind. Due to complexity and the presence of additional variables, it is more interesting and important for the researchers working on the real life problem. The fractional order complex Lorenz system and complex T-system are chosen to address the theoretical result in the presence of uncertain parameters. Numerical simulation is carried out using Adams-Bashforth-Moulton method for different particular cases, which reveals that the method is reliable and effective, even for fractional order complex chaotic systems, and in the presence of uncertain terms. [ABSTRACT FROM AUTHOR]

The study of nonlinear system with coexisting chaotic attractors is of recent interest. In this letter, we introduce a new four-dimensional chaotic system with coexisting attractors. The system has three quadratic nonlinearities and only one unstable fixed point. Basic dynamics of the system are analyzed, including dissipativity, equilibria, etc. The coexisting attractors in the system is investigated by numerical simulation. The control problem of the system is also considered. By using an ordinary feedback controller and a speed feedback controller, the chaotic behavior of the system can be well suppressed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the synchronization problem of chaotic systems with unknown control direction under input saturation nonlinearity. By utilizing backstepping technique, an adaptive neural synchronization control scheme is developed via neural approximation technique and Nussbaum-type function method. To deal with the problem of sharp corner of saturation, a tangent function based smooth function is used to approximate input saturation. To facilitate controller design, an auxiliary signal is constructed to augment the plant. Two Nussbaum functions are respectively used to deal with unknown control direction and compensate for the time-varying gain arising from the partial differentiation of the smooth function of input saturation. Tracking-differentiator is introduced to handle the problem of “explosion of computation” in the backstepping design. Adaptive robust effects are designed to deal with the “disturbance-like” term and tracking-differentiator estimation error. Lyapunov based analysis gives the transient performance and convergence of synchronization errors. Finally, a simulation example is presented to verify the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Genesio system was taken as an example to show the design process of synchronization of chaotic system with reduced number of active inputs which can be transferred into a control problem of special second order system. A kind of adaptive transfer function method is integrated with sliding mode control method to solve the control problem of second order system with double control directions. And sliding mode method can reduce system order and make the design process more simple. But double control direction will caused a derivative of control problem, which is difficult to solve with traditional method. So a kind an equivalent transfer function method is proposed to cope it. Also, the proper parameters of transfer function are estimated by adaptive methods. At last, detailed simulation result is done to testify the rightness of the proposed method and simulation shows some problems which is caused by the adaptive method that the stable condition for transfer function can not be guaranteed. Also, the simulation of the complex chaotic synchronization was realized with the proposed adaptive transfer function method, so the rightness of this novel method was totally verified. [ABSTRACT FROM AUTHOR]

For the past five years there has been much attention to systems with uncountable equilibria. This work introduces a new system with an infinite number of equilibrium points. It is worth noting that equilibrium points, which located on a heart-shaped curve, lead the presence of hidden attractors in the system. By using phase portraits, bifurcation diagram, maximal Lyapunov exponents, dynamics of the system have been investigated. Moreover, an electronic implementation is proposed to illustrate the feasibility of the theoretical system. [ABSTRACT FROM AUTHOR]

Chaotic systems generating multi-wing attractors have received considerable attention in the literature. In this work, we propose a novel three-dimensional chaotic system with hyperbolic sine nonlinearity. It is worth noting that the system is elegant and includes only one parameter. Despite its simple structure, the new system displays double-wing and four-wing chaotic attractors. By studying dynamics of the system, coexistence of limit cycles or chaotic attractors is discovered. The capable of the synchronization of new chaotic system is verified by using an adaptive control. Furthermore, an electronic circuit for implementing the system is reported to indicate its feasibility. [ABSTRACT FROM AUTHOR]

This paper centers on a novel 4-D hyperchaotic system with butterfly phenomenon. The properties of the system are analyzed through invariability, dissipativity, equilibriums, Poincaré section, bifurcation diagram and the max Lyapunov exponent. The existences of the attractor are verified by an electronic circuit. [ABSTRACT FROM AUTHOR]

This paper present Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Firstly, regarding the time delay τ as a bifurcation parameter, we investigate the stability and bifurcation behaviors of this fractional order delayed memristor-based chaotic circuit system. Some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. Secondly, corresponding to different system parameters, the complex dynamics behaviors of this system are discussed by using the bifurcation diagrams, the Max Lyapunov exponents (MLEs) diagram, the time domain waveforms, the phase portraits and the power spectrums. Thirdly, we study the influence of the two parameters (time delay τ and fractional order q ) on the chaotic behavior, and it is found when time delay τ and fractional order q increases, the transitions from period one to period two and period four to chaos are observed in this memristor–based system. Meanwhile, corresponding critical values of time delay τ and fractional order q , the lowest orders q and the minimum time delay τ for generating chaos in the fractional order delayed memristor–based system are determined, respectively. Also, when the system occurs period one, the corresponding frequency is verified theoretically and experimentally. Finally, numerical simulations are provided to demonstrate the validity of theoretical analysis using the improved Adams–Bashforth–Moulton algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with the synchronization problem of two chaotic systems in a master-slave configuration. The two systems are based on a new chaotic system recently reported by Akgul and Pehlivan in [1]. The global and exponential convergence to zero of the synchronization error is guaranteed by means of a nonlinear controller designed from the Lyapunov stability theory. In order to find the appropriate parameters of this controller, the compromise between magnitude of control signal and convergence speed is quantified by using a quadratic performance index. Next, such index is minimized via four different intelligent optimization algorithms: differential evolution, brain storm, cuckoo search, and harmony search. These four algorithms are exhaustively tested and the results are systematically compared. Finally, the performance of the optimized nonlinear controller is showed graphically. [ABSTRACT FROM AUTHOR]

Chaotic systems with an infinite number of equilibrium points and chaotic ones without equilibrium have received a significant attention in the last years because they belong to a class of systems with “hidden attractor”. In this work, we introduce a three-dimensional chaotic system displaying both hidden attractors with infinite equilibria and hidden attractors without equilibrium. Surprisingly, when the system exhibits hidden attractors with infinite equilibria, it has a rounded square curve of equilibrium points. Dynamical properties of the new system are analyzed through equilibrium points, phase portraits, bifurcation diagram, and maximal Lyapunov exponents. Furthermore, circuit implementation of the system is presented showing another approach to study such system as well as its feasibility. [ABSTRACT FROM AUTHOR]

This paper is devoted to adaptive synchronization and anti-synchronization between the TSUCS and the Lü, two unified chaotic systems. It is assumed that the parameters of the both systems are unknown. A designed controller based on Lyapanov candidate theorem is introduced for adaptive synchronization/anti-synchronization of two different unified chaotic systems and is used for adjusting estimation parameters until the measurable errors converge to zero. The stability evaluation and convergence analysis is carried out by Lyapanov stability theorem. Finally, some numerical simulations are presented to assess the behavior of the synchronization procedure and to verify the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Substitution-boxes, or simply S-boxes, are used to increase confidentiality in the substitution stage of most cryptosystem approaches. In this paper, an efficient method for the construction of an S-box based on a sine map is proposed. The proposed S-box is able to generate random integer sequences with highly efficient nonlinearity in the generated values. The cryptographic analyses show that the proposed S-box method is of high performance and can be used with great potential for prominent prevalence in designing symmetric cryptosystems and copy right protection. [ABSTRACT FROM AUTHOR]

In this paper, a switched system that switches between two Chen systems based on the parameters altered periodically is established. To investigate the mechanism of switching oscillation of the whole system, Poincaré map of the whole system is defined by the switching scheme and the solutions of the subsystems. Different types of periodic switching oscillations revolved around the unstable equilibrium points and their transitions to chaotic oscillations can be observed. We conclude that bifurcation scenarios, such as period-doubling bifurcation, pitchfork bifurcation and saddle-note bifurcation, play an important role in the generation of various (symmetric or asymmetric) periodic oscillations and chaos in the switched system. [ABSTRACT FROM AUTHOR]

Some fractional-order hyperchaotic complex systems are introduced in this paper. These hyperchaotic systems appear in several fields of applied sciences, e.g., secure communication and laser physics. The values of the fractional-order and the parameters at which these systems have hyperchaotic attractors are calculated based on the sign of their Lyapunov exponents. The fractional Lyapunov dimension is computed for these hyperchaotic attractors. On the other hand, we introduced the new definition of what is called generalized function projective combination synchronization (GFPCS). This new kind of synchronization may be considered as a generalization of many kinds of synchronization in the literature. To study this kind of synchronization we state a scheme based on the tracking control technique. An example is considered for the scaling functions which is the nonlinear one. The active control method cannot be used in this scheme since its error is large. [ABSTRACT FROM AUTHOR]

This paper focuses on the dislocated projective synchronization (DPS) between the fractional-order and the integer-order chaotic systems. Based on the small-gain theorem, nonlinear controllers are designed to reach the DPS between the fractional-order and the integer-order chaotic systems. Numerical simulation illustrates the availability of the proposed scheme. [ABSTRACT FROM AUTHOR]

Genesio system was taken as an example to show the design process of synchronization of chaotic system with reduced number of active inputs which can be transferred into a control problem of special second order system. A kind of adaptive transfer function method is integrated with sliding mode control method to solve the control problem of second order system with double control directions. And sliding mode method can reduce system order and make the design process more simple. But double control direction will caused a derivative of control problem, which is difficult to solve with traditional method. So a kind an equivalent transfer function method is proposed to cope it. Also, the proper parameters of transfer function are estimated by adaptive methods. At last, detailed simulation result is done to testify the rightness of the proposed method and simulation shows some problems which are caused by the adaptive method that the stable condition for transfer function can not be guaranteed. Also, the simulation of the complex chaotic synchronization was realized with the proposed adaptive transfer function method, so the rightness of this novel method was totally verified. [ABSTRACT FROM AUTHOR]

In order to solve the synchronization problem of chaotic systems with reduced number of active control inputs, special second order system was characterized and researched. A novel kind of equivalent transfer function method was integrated with backstepping method to solve the difficulty caused by double control coefficients of a kind of minimum phase second order system without uncertainties. And by constructing a Lyapunov function, the stability of the close loop system is proved. And also an additional stable condition for transfer function should be satisfied. At last, the proper chosen of control gain interval is calculated by an inequality equation, which is testified by detailed simulation to be very useful in the controller design. Also, detailed simulation on the synchronization of chaotic system was done the testify the rightness of the proposed method. [ABSTRACT FROM AUTHOR]

A new intermittent control way, namely sinusoidal state error feedback, is adopted to synchronize two non-autonomous chaotic systems. Some synchronization criteria are deduced by the Lyapunov stability theory and the Gerschgorin disc theorem. The relationships among the control gain, control period and control width are revealed, and their influences on synchronization are also discussed. The Mathieu-Duffing equation and a gyrostat system are chosen as numerical examples to show that synchronization can be achieved under the proposed criteria. [ABSTRACT FROM AUTHOR]

In recent years, a growing interest has been devoted to the design and analysis of chaotic systems without equilibrium point. In the present contribution, we further investigate the dynamics of an equilibrium free 3D chaotic system with quadratic nonlinearities recently introduced by Yan et al. [Optik 127 (2016) 1363–1367]. Standard nonlinear diagnostic tools such as bifurcation diagrams, graphs of largest Lyapunov exponent, phase portraits, frequency spectra and Poincaré sections are plotted to characterize the dynamics of the model in terms of its parameters. It is found that the system experiences a large number of coexisting attractors for some suitable sets of its parameters, depending solely on the choice of initial conditions. Up to twelve coexisting stable attractors are revealed. An electronic analogue of the system is designed and implemented in Pspice. A very good agreement is observed between Spice based simulation results and the theoretical analysis. To the best of the authors’ knowledge, this interesting and singular behavior (i.e. the coexistence of a large number of stable attractors, including periodic, quasi-periodic and chaotic attractors) has not yet been reported both in a third order system and thus deserves dissemination. [ABSTRACT FROM AUTHOR]

A chaotic model for generating multi-directional multi-scroll attractors via hyperbolic tangent function series is proposed in this paper. The dynamical mechanisms of this chaotic model are further investigated, including M -scroll, M×N -grid scroll, M × N × L -grid scroll chaotic attractors. Moreover, the dynamical behaviors of this system are theoretically analyzed and numerically simulated, such as equilibria and their stability, Lyapunov exponents and bifurcation diagrams. This system can be widely used in data encryption and secure communication. [ABSTRACT FROM AUTHOR]

In this study, the control of hyperchaotic Rabinovich system is applied by using three different single state controllers, namely linear feedback, sliding mode, and passive control. Routh–Hurwitz criterions are used for constructing the sufficient conditions of the linear feedback controller. An appropriate switching surface is designed based on the sliding mode control theory. In order to ensure the global asymptotic stability of the system with the passive controller, a Lyapunov function is used. Simulation results are demonstrated not only to validate all the theoretical analyses but also to compare the effectiveness of proposed control methods. By adding the controllers to the hyperchaotic Rabinovich system, it asymptotically stabilizes towards its zero equilibrium point in the state space. Numerical simulations have also shown that the proposed single state sliding mode controller is more effective than linear feedback and passive controllers on the control of hyperchaotic Rabinovich system. [ABSTRACT FROM AUTHOR]

A new system with an exponential nonlinear term, which can exhibit ten-ring attractors, is introduced in this paper. The proposed chaotic system is totally different from the existing systems in previous works. Some basic dynamical properties of the system are analyzed theoretically and numerically. Existence and uniqueness of the solution of the presented system are proved. Next, the tracking control for the proposed system with exponential convergence rate is addressed. Finally, numerical simulations are also performed to verify the effectiveness of presented schemes. [ABSTRACT FROM AUTHOR]

The paper proposes a color image encryption scheme based on coupled hyper chaotic Lorenz systems. The novelty is to randomly inject impulse signals into coupled Lorenz system during iterations to enhance the complexity of trajectory. Six state variables sequences are generated to encrypt the red, green and blue components by bitwise operations of exclusive or (XOR) and left or right cyclic shift. Six initial values and indeterminate multiple impulse signals can make the cryptosystem own larger variable key space to resist against exhaustive attack, even the attack from quantum computer. Simulation results show that the mean encryption speed is stable, which means the speed depends solely upon hardware equipment and algorithm. Statistical analysis demonstrates the high effectiveness of the proposed image encryption algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of high order stabilization in a class of complex dynamical networks (CDNs). This class of CDNs is represented by scale-free networks with chaotic nodes. The main idea is to apply a nonlinear control method that offers the possibility of stabilizing the desired unstable periodic orbits of continuous-time chaotic systems for each node in the network. Numerical examples are presented to demonstrate the effectiveness of the proposed control strategy. [ABSTRACT FROM AUTHOR]

This paper constructs a three-dimensional autonomous chaotic system with simple algebraic structure of only five terms. Interestingly, this system is found to possess two non-hyperbolic equilibria, thus it is not under the class of Šil’nikov sense chaos for owning neither heteroclinic orbit nor homoclinic orbit. Dynamical properties have been revealed with the help of bifurcation route, Poincaré map, frequency spectrum, amplitude modulation and topological horseshoe. We hope that our work can motivate a stimulus for the further study on non-Šil’nikov chaotic system with non-hyperbolic equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, a color image encryption algorithm is presented, the new algorithm uses double-layer Joseph's scramble, XOR diffusion, and is built on a laser chaos system. Firstly, a laser chaotic system is chosen in the scheme and it has the advantages of extended bandwidth and rapid propagation speed. Secondly, the structure of the image encryption scheme follows the Fridrich structure. In the scrambling part, a modified Joseph's ring is adapted to process the chaotic sequence and the picture pixel sequence to achieve the scrambling of the plaintext image pixel positions. In the diffusion part, the XOR diffusion technique is employed to reduce the statistical characteristics of the plaintext picture, which can resist external attacks better. Finally, the presented encryption algorithm is demonstrated utilizing simulation results and security analysis. The simulation performance demonstrates that the new proposed method is practical. The security test performance indicates that the algorithm can validly encrypt the image with good security performance. In summary, the new scheme designed in this paper can offer better technical backing for the wide applications of image encryption. [ABSTRACT FROM AUTHOR]

This paper reports a new 3-D autonomous chaotic system with novel behaviour. The proposed system has three different natures of equilibria: (i) saddle, (ii) saddle foci and (iii) stable node foci which is new in the literature. The system has invariant nature of the equilibria in the considered ranges of all five bifurcation parameters. The system has various complex dynamic behaviours like periodic (period-1, period-2, period-4), quasi-periodic, chaotic transient, stable and chaotic attractor. The system exhibits (i) inverse crises route to chaos for two bifurcation parameters and (ii) crises route to chaos for another three bifurcation parameters. It is observed that in case of normal parameter ranges, the trajectories of the system are concentrated only near an unstable equilibrium point, but in the transient chaotic parameters ranges the trajectories of the system switchover to a stable equilibrium point from the unstable equilibrium point. The transient chaotic nature of the system is analysed using the many numerical tools like Lyapunov exponents, instantaneous phase, Poincare return map, recurrence plot, 0–1 test analysis, autocorrelation plot. The findings of the numerical tools are validated using the circuit implementation. [ABSTRACT FROM AUTHOR]

Chaos-based secure communication systems are alternative of standard spread-spectrum systems that enable spreading the spectrum of the information signals and encrypting information signals with simple and inexpensive chaotic circuitry. In secure communication area, like Lorenz, Chua, Rossler, Duffing etc., classical systems are widely used. Malasoma chaotic system is topologically simple but their dynamical behaviors are very rich and its synchronization and secure communication applications has not seen in literature. This paper aims for introducing a new chaotic system which is able to use as alternative to classical chaotic systems into secure communication fields. Mentioned novel model is also supported with electronic circuit model simulations that aim for being a synchronization circuit model using P-C method as chaotic masking communication circuit model. Modelling, synchronization and secure communication applications of Malasoma are realized respectively in MATLAB-Simulink environment. Retrieved results show that this novel chaotic system is able to use in secure communication fields. [ABSTRACT FROM AUTHOR]

Synchronization of chaotic systems is very useful in secure communication. A kind of reduced number of active input method was proposed to achieve the synchronization of Genesio system and Chen system, which will not only make its application in communication more safe and more difficult to be deciphered, but also it can make the synchronization device more economical. And to make the proposed method can be applied in a more general situation of synchronization of chaotic systems, a special second order system control problem with double control coefficients was researched first. it It is very difficult and complex to compute the accurate value of the derivative of expect state of the second order system, which is caused by the double control directions. A kind of simple backstepping method is proposed to neglect the derivative of expect state. So the derivative of control will not appear and the design process of second order mini-phase system will be largely simplified. And also the stability of the system is proved by constructing a Lyapunov function. At last, detailed simulation is done to testify the rightness of proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on an important issue in the research of fractional-order system, i.e., chaos control. Unlike previous study on chaos control for fractional-order systems, for which the control target is inherently unstable equilibrium point. In our work, any desired state embedded in the chaotic attractor is the control target. For deterministic fractional-order chaotic system, a simple and general controller is designed to achieve track control. For fractional-order chaotic system with uncertain parameter and external perturbation, a novel adaptive control technique is presented to achieve tracking control. Furthermore, the new control method is extended to track the desired bounded trajectories. All the above methods have been proved rigorously according to the stability theory of fractional-order system. Numerical simulation results for fractional-order unified chaotic system are also provided to illustrate the effectiveness of proposed schemes. [ABSTRACT FROM AUTHOR]

The problem of real combination synchronization of three fractional-order complex-variable chaotic systems is investigated in this paper. Based on the stability theory of three fractional-order systems, two drive fractional-order complex-variable chaotic systems and one response fractional-order complex-variable chaotic system could be asymptotically synchronized up to two scaling matrices. Numerical simulation results are performed to verify the effectiveness and feasibility of the proposed real combination synchronization scheme. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyze dynamical behavior of a four-dimensional nonlinear tourism-based social–ecological system by using the Hopf bifurcation theory. Our purpose is using a criterion based on the Routh–Hurwitz criterion for existence of a region for Hopf bifurcation. Because existence of Hopf bifurcation and therefore bifurcated periodic solutions helps us to show an interesting type of coexistence between mass-tourists and eco-tourists through a periodic pattern, although they have different effects on environmental quality. Furthermore, numerical simulations are carried out to support our theoretical results and show sustainability of tourism. In fact by simulation we indicate a coexistence between mass-tourists and eco-tourists, the existence of chaotic behavior and sustainability of tourism industry. Moreover another criterion, Bendixson's negative criterion, which is based on the connection between compound matrices and ordinary differential equations is applied to prove the nonexistence of periodic orbits and then nonexistence of Hopf bifurcation. [ABSTRACT FROM AUTHOR]