A chaotic system with attractor coexistence and its synchronization circuit implementation

In this paper, a novel four-dimensional fractional-order chaotic system is proposed, and its dynamic characteristics are analyzed by equilibrium points, phase diagram, Lyapunov exponent spectrum and bifurcation diagram. Then, the electronic circuit of the chaotic system is designed and implemented by PSpice. The results of numerical simulation are in good agreement with the analog circuit simulation. In addition, the four-dimensional fractional-order system is implemented with field programmable gate array (FPGA). Finally, the synchronization of two four-dimensional chaotic systems with different initial values is realized by finite time control method, and it is also realized by FPGA. The design of fractional-order system based on hardware concerned in this paper will provide a certain theoretical basis for its application in the fields of secure communication, image encryption and so on.

In this paper, we propose a fractional-order active memristor based on fractional-order calculus, and its characteristics are analyzed. Combined with the new memristor, a fractional-order chaotic system is constructed. We study the complex dynamic behavior of the chaotic system by phase diagram, bifurcation diagram and Lyapunov exponential spectrum. The results show that with the change of fractional order and system parameters, the system will exhibit complex dynamic behaviors such as period, double period and chaos. With the change of initial values, the fractional system also exhibits the multistable characteristics of coexistence of different attractors. The research results show the complex dynamic behavior of fractional-order chaotic system.

In recent years, there are numerous studies on chaotic systems with special equilibrium curves having various shapes such as circle, butterfly, heart and apple. This paper describes a new 3-D chaotic dynamical system with a capsule-shaped equilibrium curve. The proposed chaotic system has two quadratic, two cubic and two quartic nonlinear terms. It is noted that the proposed chaotic system has a hidden attractor since it has an infinite number of equilibrium points. It is also established that the proposed chaotic system exhibits multi-stability with two coexisting chaotic attractors for the same parameter values but differential initial states. A detailed bifurcation analysis with respect to variations in the system parameters is portrayed for the new chaotic system with capsule equilibrium curve. We have shown MATLAB plots to illustrate the capsule equilibrium curve, phase orbits of the new chaotic system, bifurcation diagrams and multi-stability. As an engineering application, we have proposed a speech cryptosystem with a numerical algorithm, which is based on our novel 3-D chaotic system with a capsule-shaped equilibrium curve. The proposed speech cryptosystem follows its security evolution and implementation on Field Programmable Gate Array (FPGA) platform. Experimental results show that the proposed encryption system utilizes 33% of the FPGA, while the maximum clock frequency is 178.28 MHz.

Dynamics analysis, FPGA realization and image encryption application of a 5D memristive exponential hyperchaotic system

The problem of analyzing the parameters of a chaotic system of fractional order is considered. As an example, a chaotic system based on a Chua circuit of an integer order was chosen. To obtain a Chua circuit of fractional order, it is proposed to replace the capacitive elements in the original circuit with elements with fractal impedance. The purpose of the work is to create a mathematical model of a fractional order Chua circuit with real values of the circuit elements, on the basis of which to evaluate existing software products in the Matlab environment suitable for modeling chaotic fractional order systems, as well as to search for software products suitable for calculating the spectrum of Lyapunov exponents. An analysis of ready-made solutions in the Matlab environment, from the literature known to us, showed that the most suitable option for modeling a system of differential equations of a fractional order is the «fde_pi12_pc» program. The selected program allows you to set all the system parameters of interest and has a relatively high accuracy due to the use of the predictor-corrector algorithm. Based on this program, a domain for determining fractional indicators of the Chua system was constructed in which the presence of a characteristic attractor is visually observed. Existing software products in the Matlab environment are analyzed, allowing the calculation of Lyapunov exponents for chaotic systems described by fractional order differential equations. Of the solutions considered, the best, in our opinion, is the «FO_NC_Lyapunov» program. Using the selected program, a spatial graph of the spectrum of Lyapunov exponents was constructed when changing the fractional exponents of the Chua system. Based on the spatial graph of the spectrum of Lyapunov exponents and the criteria for the chaotic behavior of the system based on the values of Lyapunov exponents, a definition area was constructed with visualization of zones of chaotic behavior of the Chua system. The domains of definition obtained by visual analysis of the presence of an attractor and calculating the spectrum of Lyapunov exponents for the same mathematical model did not coincide; therefore, there are currently no software products for reliably assessing the chaos of fractional order systems by calculating the spectrum of Lyapunov exponents.

A hyperchaotic map with grid sinusoidal cavity

The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.

Audio encryption has gained popularity in a variety of fields including education, banking over the phone, military, and private audio conferences. Data encryption algorithms are necessary for processing and sending sensitive information in the context of secure speech conversations. In recent years, the importance of security in any communications system has increased. To transfer data securely, a variety of methods have been used. Chaotic system-based encryption is one of the most significant encryption methods used in the field of security. Chaos-based communication is a promising application of chaos theory and nonlinear dynamics. In this research, a chaotic algorithm for the new chaotic chameleon system was proposed, studied, and implemented. The chameleon chaotic system has been preferred to be employed because it has the property of changing from self-excited (SA) to hidden-attractor (HA) which increases the complexity of the system dynamics and gives strength to the encryption algorithm. A chaotic chameleon system is one in which, depending on the parameter values, the chaotic attractor alternates between being a hidden attractor and a self-excited attractor. This is an important feature, so it is preferable to use it in cryptography compared to other types of chaotic systems. This model was first implemented using a Field Programmable Gate Array (FPGA), which is the first time it has been implemented in practical applications. The chameleon system model was implemented using MATLAB Simulink and the Xilinx System Generator model. Self-excited, hidden, and coexisting attractors are shown in the proposed system. Vivado software was used to validate the designs, and Xilinx ZedBoard Zynq-7000 FPGA was used to implement them. The dynamic behavior of the proposed chaotic system was also studied and analysis methods, including phase portrait, bifurcation diagrams, and Lyapunov exponents. Assessing the quality of the suggested method by doing analyses of many quality measures, including correlation, differential signal-to-noise ratio (SNR), entropy, histogram analysis, and spectral density plot. The numerical analyses and simulation results demonstrate how well the suggested method performs in terms of security against different types of cryptographic assaults.

A four-dimensional autonomous hyperchaotic system is constructed, and the basic characteristics of the system are studied by the power spectrum, Poincaré maps, 0–1 test and Lyapunov exponents. The system has rich dynamical behaviors, such as bursting oscillations, offset boosting, transient chaos, intermittent chaos and coexistence of attractors. In addition, by studying the coexisting phenomenon and spectral entropy (SE) complexity of different initial values, an initial value that is more suitable for chaotic secure communication is selected. The circuit simulation of the system using Multisim and then the actual hardware implementation of the system by Field Programmable Gate Array (FPGA), these prove the practical existence of the system. Finally, combining the methods of backstepping control, multi-switching synchronization and synchronization of different systems, a set of controllers are proposed which can realize the backstepping multi-switching synchronization of this system with a memristive chaotic system.

A new three-dimensional chaotic system with fan-shaped Poincare maps is proposed. Based on merely six terms, the system is easy to implement. Its dynamic behaviors, such as equilibrium points, Poincare maps, power spectra, Lyapunov exponent spectra, bifurcation diagrams and forming mechanism, are analyzed theoretically and numerically. Results of theoretical analyses and numerical simulations indicate that the proposed system possesses complex chaotic attractors. Its equilibrium points are unstable, and the system can keep chaotic when its parameters vary in a wide domain. Furthermore, circuit simulations of the system are discussed. The results of numerical simulations and circuit simulations coincide very well. By virtue of its complex dynamic behaviors and wide-range parameters, the system can be adopted in some application fields where wide-range parameters and complex behaviors are usually preferred, such as secure communication, data encryption, information hiding.

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

A new three-dimensional chaotic system is reported in this paper, which is different from the Lorenz and Chen systems. This new system contains five system parameters and two quadratic cross-product terms. The basic dynamic properties of the new system are investigated via theoretical analysis, numerical simulation, Lyapunov exponent spectrum, bifurcation diagrams, Lyapunov dimension and Poincare diagrams. The different dynamic behaviors of the new system are analyzed when each system parameter is changed. Finally, the chaotic circuit is designed and realized by the Multisim software. It confirms that the chaotic system can be achieved.

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.

Memristor, as a nonlinear element, provides many advantages thanks to its superior properties to design different chaotic circuits. Thus, a novel four-dimensional double-scroll chaotic system with line equilibria as well as two unstable equilibria based on the flux-memristor model is proposed in this paper. The effects of initial values and parameters on the dynamic characteristics of the system are studied in detail by means of phase diagrams, Lyapunov exponent spectrums, bifurcation diagrams, two-parameter bifurcation diagrams and basins of attraction. Besides, a series of complex phenomena in the system, such as sustained chaos, bistability, transient chaos and coexisting attractors are observed, proving that the chaotic system has rich dynamic characteristics. Also, spectral entropy (SE) complexity algorithm and [Formula: see text] complexity algorithm based on structure complexity are adopted to analyze the complexity of the system. Additionally, PSPICE circuit simulation and Micro-Controller Unit (MCU) hardware experiment are carried out to verify the correctness of theoretical analysis and numerical simulation. Finally, the pulse chaos synchronization is achieved from the perspective of maximum Lyapunov exponent, and numerical simulations demonstrate the occurrence of the proposed system and practicability of the pulse synchronization control.

Coexisting multiple firing behaviors of fractional-order memristor-coupled HR neuron considering synaptic crosstalk and its ARM-based implementation