Chaotic dynamics is widely used to design pseudo-random number generators and for other applications such as secure communications and encryption. This paper aims to study the dynamics of discrete-time chaotic maps in the digital (i.e., finite-precision) domain. Differing from the traditional approaches treating a digital chaotic map as a black box with different explanations according to the test results of the output, the dynamical properties of such chaotic maps are first explored with a fixed-point arithmetic, using the Logistic map and the Tent map as two representative examples, from a new perspective with the corresponding state-mapping networks (SMNs). In an SMN, every possible value in the digital domain is considered as a node and the mapping relationship between any pair of nodes is a directed edge. The scale-free properties of the Logistic map's SMN are proved. The analytic results are further extended to the scenario of floating-point arithmetic and for other chaotic maps. Understanding the network structure of a chaotic map's SMN in digital computers can facilitate counteracting the undesirable degeneration of chaotic dynamics in finite-precision domains, helping also classify and improve the randomness of pseudo-random number sequences generated by iterating chaotic maps.

Dynamic Analysis of Digital Chaotic Maps via State-Mapping Networks

The definition of fractional calculus is introduced into a 3D multi-attribute chaotic system in this paper. The fractional multi-attribute chaotic system (FMACS) numerical solution is obtained based on the Adomian decomposition method (ADM). The balance points and dynamical behaviors of self-excited and hidden attractors in FMACS are compared and analyzed through the Lyapunov spectrum, bifurcation model, and complexity. It is worth noting that some hidden coexistence attractors with different shapes are affected by the order. Besides, a novel chaotic system without equilibrium points is constructed, in which the nonlinear function term in FMACS is replaced with a rare nonlinear function e x . Meanwhile, its degradation phenomenon and state transition phenomenon are analyzed in detail. Finally, the digital circuit of the system is realized on the DSP board. The research result shows that FMACS has richer dynamical behaviors and higher complexity. This research provides a theoretical basis and guidance for the application of fractional chaotic systems.

A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation

A fast and efficient multiple images encryption based on single-channel encryption and chaotic system

In this paper, a new four-dimensional dissipative chaotic system which can produce multiple asymmetric attractors is designed and its dynamical behaviors are analyzed. The basin of attraction reveals the asymmetric multistability of the system. In addition, it is very interesting to observe different types of asymmetric coexisting attractors as the bifurcation parameters change. The spectral entropy complexity chaotic diagrams are used to observe the changes in the sequence complexity when the two bifurcation parameters change simultaneously. Moreover, the difference of the system complexity between the two different initial values is analyzed. In order to facilitate engineering applications, the offset boosting control is introduced to the state variable, and the numerical simulation shows that the offset boosting control scheme can flexibly change the polarity of the chaotic signal. Finally, an analog circuit and a digital circuit were designed to verify the new chaotic system. The new research results will enrich the theoretical basis of multistability, offset boosting control and circuit implementation of chaos.

Dynamical analysis of a new chaotic system: asymmetric multistability, offset boosting control and circuit realization

In this paper, a novel no-equilibrium Hindmarsh-Rose (HR) neuron model with memristive electromagnetic induction is proposed. This memristive HR neuron model exhibits complex memristor initial offset boosting dynamics, from which infinitely many coexisting hidden attractors sharing the same shape but with different positions can be generated, therefore breeding the interesting phenomenon of hidden homogeneous extreme multistability. The complicated dynamical behaviors are detailedly investigated via bifurcation diagrams, Lyapunov exponents, time series, attraction basins and spectral entropy (SE) complexity. Moreover, PSIM circuit simulations and DSP hardware experiments are carried out to demonstrate the theoretical analyses and numerical simulations. Finally, a pseudorandom number generator is also designed by using the memristor initial-controlled chaotic sequences extracted from the memristive HR neuron model. The performance analysis results show that these chaotic sequences can yield pseudorandom numbers with excellent randomness, which are more suitable for chaos-based engineering applications.

A novel no-equilibrium HR neuron model with hidden homogeneous extreme multistability

In this paper, we focus on a novel simple chaotic circuit with a memristor, a memcapacitor and a linear inductor in parallel. Then we establish the circuit’s dimensionless mathematical model. Nineteen types of different chaotic attractors are found in the circuit. The chaotic system’s equilibrium point and stability are analyzed by using the traditional dynamic analysis methods, and the dynamical behaviors with three varying parameters of this circuit are analyzed in detail. Furthermore, some special phenomena such as state transition, chaos degradation and the multiple coexisting attractors are discovered. Finally, we implement this circuit through the DSP platform and the results illustrate the validity of the theoretical analysis. Theoretical analysis and simulation results indicate that the simple chaotic circuit has very rich dynamical characteristics.

A novel simple chaotic circuit based on memristor–memcapacitor

In this paper, a new 4-D incommensurate fractional-order chaotic system based on adomian decomposition method is investigated. The equilibrium points of the system with incommensurate orders are analyzed. The coexisting attractors of the system with different parameters and orders are unveiled through coexisting attractors phase diagram, coexisting bifurcation model, coexisting Lyapunov exponent spectrum. The effects of the incommensurate orders of the system are also analyzed. Meanwhile, some special phenomena such as hidden attractors and chaos degradation are found. The results show that the new 4-D incommensurate fractional-order chaotic system has rich dynamic characteristics. In addition, using the DSP platform to realize the system, the results are in good agreement with those on simulation. The research results provide a theoretical basis for the application of incommensurate order chaotic systems in secure communication.

Coexistence of multiple attractors for an incommensurate fractional-order chaotic system

Chaotic dynamics play a vital role in real secure communication and image encryption This paper focuses on the characteristic analysis and complex self-synchronization (CSS) of a laser system in complex field Its dynamical features are described by Lyapunov exponent spectrum, bifurcation diagram, phase portrait and the basin of attraction The result of the investigation shows some attractive dynamical behaviors such as two different bifurcation routes, coexistence of axis-symmetric attractors, intermittent chaos and infinite transition of period and sink In particular, two chaotic attractors with different topological structures are implemented on digital signal processor platform Finally, the CSS is defined, and a novel scheme is proposed to achieve homogeneous and heterogeneous CSS of the laser complex-variable chaotic systems in complex field The researches provide basis for the laser system in real applications

Characteristics of a laser system in complex field and its complex self-synchronization

Generalized synchronization is a typical dynamical phenomenon in nonlinear systems, for which the real-valued setting has been widely investigated. The complex-valued functions relationship in gene...

Generalized Synchronization and Parameters Identification of Different-Dimensional Chaotic Systems in the Complex Field

Chaotic dynamics analysis of complex-variable chaotic systems (CVCSs) is an important problem in real secure communication and encryption. In this paper, a simple one-parameter chaotic system in complex field is proposed, whose nonlinear terms are the same as Lorenz system but the linear terms are much simpler. The proposed system has circular equilibria and therefore multi-stability can be measured by phase portraits, bifurcation diagrams and Lyapunov exponent spectrum. Its basin of attraction is filled with initial points leading to chaotic behaviors. The coexistence of infinitely many attractors is found in the proposed system, which is not reported in the existing complex-variable Lorenz system. Finally, two complexity indexes are used to measure dynamic characteristic with respect to parameter.

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Xiu Zhao

Papers

A novel simple chaotic circuit based on memristor–memcapacitor

Coexistence of multiple attractors for an incommensurate fractional-order chaotic system

Characteristics of a laser system in complex field and its complex self-synchronization

Generalized Synchronization and Parameters Identification of Different-Dimensional Chaotic Systems in the Complex Field

Dynamic Analysis of a One-Parameter Chaotic System in Complex Field