No-equilibrium Chaotic System Research Articles

A simple 4D no-equilibrium chaotic system with only one quadratic term and its application in pseudo-random number generator

Simple-form chaotic systems with complex behaviors have always attracted the attention of researchers. This article introduces a novel 4D chaotic system that has only a quadratic nonlinear term as compared with the existing no-equilibrium continuous chaotic ones with two quadratic or higher nonlinearity. Despite the simplicity of its model, this new system exhibits complicated chaotic behaviors. Many interesting phenomena including the coexistence of different hidden attractors, offset boosting and antimonotonicity have been discovered by various numerical analysis such as bifurcation diagrams, Lyapunov exponent spectra, and phase portraits. The further circuit simulation of this new system verifies its feasibility and practicality. Moreover, the new system is applied to construct Pseudo-Random Number Generators (PRNG), following a thorough assessment of its complexity and randomness by sample entropy. The generated numbers are subsequently assessed using 15 statistical tests. The results confirm that the new system can produce unpredictable numbers with high randomness, making it ideal for information encryption.

Analysis and implementation of no-equilibrium chaotic system with application in image encryption

Analysis and implementation of no-equilibrium chaotic system with application in image encryption

Modelling and circuit realisation of a new no‐equilibrium chaotic system with hidden attractor and coexisting attractors

This Letter reports a new no-equilibrium chaotic system with hidden attractors and coexisting attractors. Bifurcation diagram shows that the proposed system generates chaos through period-doubling bifurcation with the variation of system parameters, and the hidden chaotic and periodic attractors are visually given by phase portraits. The coexisting chaotic and periodic attractors from different initial conditions are observed in the system. The analogue circuit realisation and microcontroller-based experimental implementation of the system are given to verify its physical existence.

A simple no-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation.

This paper proposes a simple no-equilibrium chaotic system with only one signum function as compared with the existing no-equilibrium chaotic ones with at least one quadratic or higher nonlinearity. The system has the offset boosting of three variables through adjusting the corresponding controlled constants. The resulting hidden attractors can be distributed in a 1D line, a 2D lattice, a 3D grid, and even in an arbitrary location of the phase space. Particularly, a hidden chaotic bursting oscillation is also observed in this system, which is an uncommon phenomenon. In addition, complex hidden dynamics is investigated via phase portraits, time series, Kaplan-Yorke dimensions, bifurcation diagrams, Lyapunov exponents, and two-parameter bifurcation diagrams. Then, a very simple hardware circuit without any multiplier is fabricated, and the experimental results are presented to demonstrate theoretical analyses and numerical simulations. Furthermore, the randomness test of the chaotic pseudo-random sequence generated by the system is tested by the National Institute of Standards and Technology test suite. The tested results show that the proposed system has good randomness, thus being suitable for chaos-based applications such as secure communication and image encryption.

Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

This paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. The circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method.

A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees

Abstract The topic associated with hidden attractor and multistability has been received considerable attention recently. In this paper, a novel no-equilibrium Jerk-like chaotic system is constructed and explored. Particularly, owing to the absence of the equilibria, such a new system can be categorized as a system with hidden attractors. More interestingly, this system holds three conspicuous characteristics. The first one is that various asymmetric coexisting hidden attractors and complicated transient chaos behaviors are obtained. The second one is the new finding of the periodic bursting oscillation and unusual phenomenon of transient periodic bursting oscillation in the system. The third one is the observation of the amazing and rare phenomenon of one to two full Feigenbaum remerging trees, namely, antimonotonicity. To the best knowledge of us, the last two special features are first discovered and have never been reported, especially in such no-equilibrium chaotic system that exhibits hidden attractors. With the help of phase portraits, time series, bifurcation diagram, Lyapunov exponents, chaotic dynamical diagram, basin of attraction and so forth, the rich hidden dynamical properties of this system are systematically analyzed and investigated. Additionally, a hardware electronic circuit on a breadboard is carried out. A very good similarity between the hardware experimental results and the theoretical analysis testifies the feasibility and practicality of this original system.

A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation

In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations.

A novel simple no-equilibrium chaotic system with complex hidden dynamics

By utilizing a linear state feedback controller in the famous Sprott-S system, a novel no-equilibrium autonomous chaotic system is introduced in this paper. Although the newly proposed system is simple and elegant with eight terms including only one nonlinear term, it exhibits extremely rich and complex hidden dynamical behaviors such as hidden multistability and invariable Lyapunov exponents. Interestingly, the freedom of offset boosting of a variable is obtained by using a controlled constant. The complicated hidden dynamic characteristics of this no-equilibrium system are investigated by means of phase portraits, bifurcation diagrams, Lyapunov exponents, chaos diagram and so on. Furthermore, the corresponding implementation circuit is designed. The Multisim simulations and the hardware experimental results are in agreement with the numerical simulations of the same system on the Matlab platform, which verifies the feasibility of this new no-equilibrium system.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

Dynamics and circuit realization of a no-equilibrium chaotic system with a boostable variable

Recent evidence suggests that there exists chaos in a few no-equilibrium systems. A chaotic system without equilibrium is proposed and studied in this work. It is worth noting that due to the absence of equilibrium, such a system belongs to a class of systems with hidden attractor. Dynamics properties and the feasibility of the system are investigated by using numerical simulations and circuit implementation. Interestingly, this no-equilibrium system has one variable with the freedom of offset boosting.

Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors

Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its application are introduced. The interesting aspects of this chaotic system are the absence of equilibrium points and the coexisting of limit cycle and torus. Basic dynamics of the no-equilibrium system have been executed by means of phase portraits, bifurcation diagram, continuation, and Lyapunov exponents. Experimental results of the electronic circuit realizing the no-equilibrium system have been reported to show system’s feasibility. By using the chaoticity of the new system without equilibrium, we have developed a random bit generator for practical signal encryption application. Numerical results illustrate the usefulness of the random bit generator.

FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization

This paper introduces two novel fractional-order chaotic systems with cubic nonlinear resistor and investigates its adaptive sliding mode synchronization. Firstly the novel two equilibrium chaotic system with cubic nonlinear resistor (NCCNR) is derived and its dynamic properties are investigated. The fractional-order cubic nonlinear resistor system (FONCCNR) is then derived from the integer-order model and its stability and fractional-order bifurcation are discussed. Next a novel no-equilibrium chaotic cubic nonlinear resistor system (NECNR) is derived from NCCNR system. Dynamic properties of NECNR system are investigated. The fractional-order no equilibrium cubic nonlinear resistor system (FONECNR) is derived from NECNR. Stability and fractional-order bifurcation are investigated for the FONECNR system. The non-identical adaptive sliding mode synchronization of FONCCNR and FONECNR systems are achieved. Finally the proposed systems, adaptive control laws, sliding surfaces and adaptive controllers are implemented in FPGA.

A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system

AbstractFirst, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1= 0.0395,L2= 0 and L3= −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY=3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.

A Novel No-Equilibrium Chaotic System with Multiwing Butterfly Attractors

Discovering unknown features of no-equilibrium systems with hidden strange attractors is an attractive research topic. This paper presents a novel no-equilibrium chaotic system that is constructed by using a state feedback controller. Interestingly, the new system can exhibit multiwing butterfly attractors. Moreover, a new chaotic system with an infinite number of equilibrium points, which can generate multiscroll attractors, is also proposed by applying the introduced methodology.

Constructing a Novel No-Equilibrium Chaotic System

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.