A hyperchaotic map with grid sinusoidal cavity

Based on a simplified model, the methods and rules for generating multi-cavity chaotic map are presented. On this basis, a new rhombic cavity hyperchaotic map is constructed. Some typical dynamic properties of the new system are investigated, including phase trace, Lyapunov exponent spectrum (LEs) and bifurcations. The results show that the new chaotic map has rich dynamical behaviors, including complicated phase space trajectory, hyperchaotic behavior, large Lyapunov exponent. By changing the system parameters, the rhombic cavity with adjustable size and number are obtained. The rhombic cavity hyperchaotic map is improved to the multi-directional mode which has more complex topological structure than the original system. The complexity of the improved system is high at the entire parameter space. To verify the validity of the model and the feasibility of the systems, the digital circuits of the rhombic cavity hyperchaotic maps are implemented based on DSP technique. This lays a foundation for the application of these hyperchaotic maps in chaotic secure communication.

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.

We construct a three-dimensional chaotic system based on TANG chaotic system and analyze the system dynamics based on the equilibrium points, phase diagram, Lyapunov exponential spectrum, bifurcation diagram, and complexity. The more complexly dynamic behaviors indicate that our designed system has different topology, bifurcation anti-monotonic property and attractor coexistence phenomenon from TANG system. The system has the coexistence of chaotic attractor and point attractor and can select the initial value that is suitable for chaotic secure communication by analyzing the coexistence attractor. The actual digital circuit is implemented with field programmable gate array (FPGA). The system is extremely sensitive to the selections of initial values and system parameters and has good and rich dynamic behavior, which is suitable for the field of chaotic confidential communication. The linear feedback method is used to control the system synchronous and the simulation of synchronous circuit is realized by Multisim software. The simulation results are consistent with the numerical analysis, which lays a theoretical foundation for the application of the system in the field of synchronized confidential communication.

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

Considering a nonlinear dynamic oscillator, a high Lyapunov exponent indicates a high degree of randomness useful in many applications, including cryptography. Most existing oscillators yield very low Lyapunov exponents. The proposed work presents a general strategy to derive an n-D hyperchaotic map with a high Lyapunov exponent. A 2D case study was analyzed using some well-known nonlinear dynamic metrics including phase portraits, bifurcation diagrams, finite time Lyapunov exponents, and dimension. These metrics indicated that the state of the novel map was more scattered in the phase plane than in the case of some traditional maps. Consequently, the novel map could produce output sequences with a high degree of randomness. Another important observation was that the first and second Lyapunov exponents of the proposed 2D map were both positive for the whole parameter space. Consequently, the attractors of the map could be classified as hyperchaotic attractors. Finally, these hyperchaotic sequences were exploited for image encryption/decryption. Various validation metrics were exploited to illustrate the security of the presented methodology against cryptanalysts. Comparative analysis indicated the superiority of the proposed encryption/decryption protocol over some recent state-of-the-art methods.

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.

A ring network with fractional-order bistable oscillators is proposed, and the relationship between synchronization and parameters, such as coupling modes and the initial structural conditions, etc., is investigated. Based on the bistable characteristics of P-R oscillator, the effects of the coupling strength and the structures in initial conditions on the dynamic behaviors of the ring network are investigated by analyzing the largest conditional Lyapunov exponents, the largest Lyapunov exponents and the bifurcation diagrams, etc. Further investigation reveals that the ring network can be controlled to form chaotic synchronization, chaotic non-synchronization, synchronous amplitude death, synchronous non-amplitude death, etc. by changing the initial conditions and the coupling strength. Furthermore, the contours of the largest conditional Lyapunov exponents and the largest Lyapunov exponents also show how the dynamic behaviors of the network are influenced by the competition between couplings along directions of y and z, strongly relies on the initial structural conditions of network.

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.

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.

Compared with integer order chaotic systems, fractional order chaotic systems can reflect natural phenomena more accurately, which are more suitable for chaotic cryptosystems. In order to explore the application of fractional order chaotic system in cryptography, a novel fractional order hyperchaotic system is constructed and implemented on DSP platform. More progressively, based on Adomian decomposition method, the dynamic behavior is studied by phase diagram, bifurcation diagram, Lyapunov exponent spectrum and spectral entropy (SE) complexity. It is found that each parameter and order have a large range of intervals that can keep the system in a hyperchaotic state. Therefore, the hyperchaotic sequences generated by the constructed fractional order hyperchaotic system have sufficient randomness and are well suited for applications in secure communications. In addition, a color image encryption scheme is designed based on the fractional order hyperchaotic system and DNA dynamic coding. Firstly, the improved Arnold algorithm is used to scramble the original image, then the column cyclic shift method is applied for secondary scrambling, and finally the pixel value is diffused by DNA sequence operation. The security analysis results indicate that the designed encryption algorithm can not only encrypt images effectively, but also has high security and can resist various common attacks.

Conservative chaotic systems have unique advantages over dissipative chaotic systems in the fields of secure communication and pseudo-random number generator because they do not have attractors but possess good traversal and pseudo-randomness. In this work, a novel five-dimensional (5D) Hamiltonian conservative hyperchaotic system is proposed based on the 5D Euler equation. The proposed system can have different types of coordinate transformations and time reversal symmetries. In this work, Hamilton energy and Casimir energy are analyzed firstly, and it is proved that the new system satisfies Hamilton energy conservation and can generate chaos. Then, the complex dynamic characteristics of the system are demonstrated and the conservatism and chaos characteristics of the system are verified through the correlation analysis methods such as phase diagram, equilibrium point, Lyapunov exponent, bifurcation diagram, and SE complexity. In addition, a detailed analysis of the multistable characteristics of the system reveals that many energy-related coexisting orbits exist. Based on the infinite number of center-type and saddle-type equilibrium points, the dynamic characteristics of the hidden multistability of the system are revealed. Then, the National Institute of Standards and Technology (NIST) test of the new system shows that the chaotic sequence generated by the system has strong pseudo-random. Finally, the circuit simulation and hardware circuit experiment of the system are carried out with Multisim simulation software and digital signal processor (DSP) respectively. The experimental results confirm that the new system has good ergodicity and realizability.

In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincare map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.

In this manuscript a chaotic system with exponential term has been considered and studied. Its basic dynamical properties such as time series, phase plots, Lyapunov exponents, bifurcation diagrams, equilibrium points, Poincare section etc. have been discussed and compared with its first, second and third approximate systems. The main aim of this study is to explore the dynamics of the approximations of the exponential chaotic systems. Also, the controllers have been designed to synchronize the chaotic system and its approximations using a technique “compound combination synchronization” and show its application in the field of secure communication. Simulations have been provided to show the efficacy of the used method.

Chaos is a typical attribute of nonlinear dynamical systems in various fields of science and technology. One of the conventional indicator of chaotic dynamics is the largest Lyapunov exponent. Chaos is implemented in a situation when in the spectrum of Lyapunov exponents for a flow there is one positive, one zero and at least one negative exponents. Using full spectrum of Lyapunov exponents it is possible to classify hyperchaos, when spectrum contains two or more positive Lyapunov exponents. In the frame of this work we describe two scenarios leading to occurrence of hyperchaos on the examples of the modified Anishchenko-Astakhov's generator and coupled generators of quasiperiodic oscillations. The first scenarios is a new scenario associated with appearance of Shilnikov's attractor, when saddle-focus with two-dimensional unstable manifold occurs via secondary Neimark-Sacker bifurcation and absorbs by chaotic attractor. For this scenario we will present cascade of secondary Neimark- Sacker bifurcations, corresponding to hierarchy of Shilnikov's attractors corresponding to hyperchaos. The second scenario was described previously, and associated with cascade of period doubling bifurcation of saddle-cycles with two-dimensional unstable manifold. Both scenarios will be presented for radiophysical generators. The reported study was funded by RFBR according to the research project № 19-31-60030.

In this paper, smooth curve models of meminductor and memcapacitor are designed, which are generalized from a memristor. Based on these models, a new five-dimensional chaotic oscillator that contains a meminductor and memcapacitor is proposed. By dimensionality reducing, this five-dimensional system can be transformed into a three-dimensional system. The main work of this paper is to give the comparisons between the five-dimensional system and its dimensionality reduction model. To investigate dynamics behaviors of the two systems, equilibrium points and stabilities are analyzed. And the bifurcation diagrams and Lyapunov exponent spectrums are used to explore their properties. In addition, digital signal processing technologies are used to realize this chaotic oscillator, and chaotic sequences are generated by the experimental device, which can be used in encryption applications.