Synchronization and circuit simulation of a new double-wing chaos

A new three-dimensional double-wing chaotic system with three quadratic terms was proposed. And the parameters which can induce the system are analyzed. The system with five equilibrium points has sophisticated dynamical behaviors and it is further investigated in details, including phase trajectory, Lyapunov exponent spectrum, Poincaré map, spectrogram map and dissipativity analysis. The circuit simulation results of the chaotic attractors are in agreement with numerical simulations. Furthermore, numerical simulations indicate that mismatch synchronization can be achieved and circuit simulations of the system synchronization are also presented.     

A simple inductor-free memristive circuit with three line equilibria

In this work, a novel inductor-free fourth-order two-memristor-based chaotic circuit is proposed. This new circuit is developed from a current feedback op amp-based sinusoidal oscillator through replacing a linear resistor with a memristor and adding another different parallel memristor to the cascaded memristor–capacitor net. The proposed circuit can perform chaotic, fixed point, and period behaviors. The most striking feature is that this system has three line equilibria and exhibits the extreme multistability phenomenon of the coexisting infinitely many attractors. Specially, amplitude death behavior and transient transition behavior can also be found in the proposed system. By using standard nonlinear analysis tools including system dissipation, equilibrium point stability, phase portrait, Lyapunov exponent spectrum, and bifurcation diagram, the fundamental dynamical characteristics of the circuit are investigated in detail. Moreover, a MULTISIM circuit is designed to verify the numerical simulations.     

Resonance and Bifurcation in a Nonlinear Duffing System with Cubic Coupled Terms

The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors.     

A more chaotic and easily hardware implementable new 3-D chaotic system in comparison with 50 reported systems

This paper attempts to construct a new 3-D chaotic system which is easily hardware realisable and fulfil the requirement of a real-life application. The proposed system is relatively more chaotic (based on the first Lyapunov exponent) and has larger bandwidth than 50 available chaotic systems. Lyapunov spectrum and bifurcation diagram of the system reveal that it has chaotic behaviour for a wider range of its parameters. Such characteristic is helpful for an easy hardware realisation of the system. It is to be noted that the reported systems with hidden attractors are not considered here for the comparison. The proposed system has more complexity and disorder due to several unique properties like asymmetry to principle coordinates, dissimilar and asymmetrical equilibria, and non-uniform contraction and expansion of volume in phase space. The proposed system also exhibits asymmetric pairs of coexisting attractors during its operation in two modes. The new system has different routes to chaos including crisis, an inverse crisis, period-doubling and reverse period-doubling routes to chaos with the variation of parameters. MATLAB simulation results confirm the claims, and the results of hardware circuit realisation validate the simulation results. An application of the new system is shown by masking and retrieving an information signal. It is also shown that the proposed system is better than a well-known Lorenz chaotic system for this application. A system with the above unique properties is rare in the literature.     

A new 5D hyperchaotic system based on modified generalized Lorenz system

This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincaré projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.     

A novel strange attractor with a stretched loop

This short paper introduces a new 3D strange attractor topologically different from any other known chaotic attractors. The intentionally constructed model of three autonomous first-order differential equations derives from the coupling-induced complexity of the well-established 2D Lotka?CVolterra oscillator. Its chaotification process via an anti-equilibrium feedback allows the exploration of a new domain of dynamical behavior including chaotic patterns. To focus a rapid presentation, a fixed set of parameters is selected linked to the widest range of dynamics. Indeed, the new system leads to a chaotic attractor exhibiting a double scroll bridged by a loop. It mutates to a single scroll with a very stretched loop by the variation of one parameter. Indexes of stability of the equilibrium points corresponding to the two typical strange attractors are also investigated. To encompass the global behavior of the new low-dimensional dissipative dynamical model, diagrams of bifurcation displaying chaotic bubbles and windows of periodic oscillations are computed. Besides, the dominant exponent of the Lyapunov spectrum is positive reporting the chaotic nature of the system. Eventually, the novel chaotic model is suitable for digital signal encryption in the field of communication with a rich set of keys.     

Dynamics of delay induced composite multi-scroll attractor and its application in encryption

Time delay feedback has been shown to produce chaos from non-chaotic systems. In this paper, besides the single and double scroll chaotic attractors, a new composite multi-scroll attractor is found in stable systems with time delay feedback. From the viewpoint of the local stability analysis, conservation analysis, Lyapunov exponent spectrum and power spectrum, the composite multi-scroll attractor is shown to be a hyper-chaotic attractor. The phase trajectory in the new composite hyper-chaotic multi-scroll attractor diverges in multiple eigen-directions, which improves the security of secure communication and chaotic encryption. A paradigm using the multi-scroll attractor for encryption is proposed, demonstrating its potential applicability.     

Dynamical analysis of the generalized Sprott C system with only two stable equilibria

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.     

Amplitude control approach for chaotic signals

A general approach based on the introduction of a control function for constructing amplitude-controllable chaotic systems with quadratic nonlinearities is discussed in this paper. We consider three control regimes where the control functions are applied to different coefficients of the quadratic terms in a dynamical system. The approach is illustrated using the Lorenz system as a typical example. It is proved that wherever control functions are introduced, the amplitude of the chaotic signals can be controlled without altering the Lyapunov exponent spectrum.     

The matric algorithm of Lyapunov exponent for the experimental data obtained in dynamic analysis

ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｏｓｉｓａｎｉｒｒｅｇｕｌａｒｐｈｅｎｏｍｅｎｏｎｇｅｎｅｒａｔｅｄｂｙｎｏｎｌｉｎｅａｒｍｏｄｅｌｓ．Ｉｔｅｘｔｅｎｓｉｖｅｌｙｅｘｉｓｔｓｉｎｎａｔｕｒｅ．Ｗｈｅｎａｒｅａｌｉｒｒｅｇｕｌａｒｔｉｍｅｓｅｒｉｅｓｉｓｇｉｖｅｎ,ｐｅｏｐｌｅｗｉｌｌｓｐｏｎｔａｎｅｏｕｓｌｙａｓｋｔｈｅｑｕｅｓｔｉｏｎ：ｗｈｅｔｈｅｒｔｈｅｔｉｍｅｓｅｒｉｅｓｉｓｐｒｏｃｅｓｓａｓｒａｎｄｏｍｏｒａｓｄｅｔｅｒｍｉｎｉｓｔｉｃｃｈａｏｓ．Ｉｆｔｈｅｔｉｍｅｓｅｒｉｅｓｉｓｔｈｅｒ…     

SF-SIMM high-dimensional hyperchaotic map and its performance analysis

Derived from Sine map and an iterative chaotic map with infinite collapse (ICMIC), a new high-dimensional hyperchaotic map, sinusoidal feedback Sine ICMIC modulation map (SF-SIMM), is proposed. Two-dimensional (2D) model of SF-SIMM is investigated as an example, and its chaotic performances are evaluated. Results show that it has complicated phase space trajectory, infinite equilibrium points, hyperchaotic behaviors, rather large maximum Lyapunov exponent, three typical bifurcations and multiple coexisting attractors with odd symmetry. Furthermore, it has advantages in complexity, distribution characteristics and zero correlation and can generate two independent pseudo-random sequences simultaneously. Therefore, it has good application prospects in secure communication.     

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.     

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

A new chaotic attractor from hybrid optical bistable system

In this work, a new three-dimensional autonomous chaotic system has been introduced by modifying a hybrid optical system. The single quadratic nonlinearity is replaced by a single cubic nonlinearity; the new system can display two 1-scroll chaotic attractors simultaneously or one 2-scroll chaotic attractor. The bifurcation diagram is obtained and Lyapunov spectrum is calculated for the proposed system. The results show that the new system exhibits rich complexity features such as stable, periodic, and chaotic dynamics.     

A time-varying hyperchaotic system and its realization in circuit

A hyperchaotic system is often used to generate secure keys or carrier wave for secure communication and the realistic hyperchaotic circuit often is made of capacitor, nonlinear resistor unit and induction coil. Parameters are often fixed in these hyperchaotic circuits and the hyperchaotic property of the system can be estimated by using a scheme of synchronization and time series analysis. In this paper, a time-varying hyperchaotic system is proposed by introducing changeable electric power source into the circuit; the changeable electric power source is combined with induction coil or capacitor in series to generate changeable output signals to excite the system. The diagrams of improved circuit are illustrated and critical parameters in experimental circuits are presented; the Lyapunov exponent spectrum vs. external applied electric power source is calculated. It is confirmed that the improved circuit always holds two positive Lyapunov exponents when the external electric power source works, and the chaotic attractors are much too different from the original one; thus, a more changeable hyperchaotic system is constructed in experiment.     

Designing a multi-scroll chaotic system by operating Logistic map with fractal process

Recently, chaotic systems have been widely investigated in several engineering applications. This paper presents a new chaotic system based on Julia’s fractal process, chaotic attractors and Logistic map in a complex set. Complex dynamic characteristics were analyzed, such as equilibrium points, bifurcation, Lyapunov exponents and chaotic behavior of the proposed chaotic system. As we know, one positive Lyapunov exponent proved the chaotic state. Numerical simulation shows a plethora of complex dynamic behaviors, which coexist with an antagonist form mixed of bifurcation and attractor. Then, we introduce an algorithm for image encryption based on chaotic system. The algorithm consists of two main stages: confusion and diffusion. Experimental results have proved that the proposed maps used are more complicated and they have a key space sufficiently large. The proposed image encryption algorithm is compared to other recent image encryption schemes by using different security analysis factors including differential attacks analysis, statistical tests, key space analysis, information entropy test and running time. The results demonstrated that the proposed image encryption scheme has better results in the level of security and speed.     

Non-lane-discipline-based car-following model incorporating the electronic throttle dynamics under connected environment

We investigate the nonlinear dynamics of a system of generalized Duffing-type MEMS resonator in the frame of simple analog electronic circuit. A mathematical model formed for the proposed generalized Duffing-type MEMS oscillator in which nonlinearities arising out of two different sources such as mid-plane stretching and electrostatic force can lead to variety of nonlinear phenomena such as period-doubling route, transient chaos and homo-/heteroclinic oscillations. These phenomena were confirmed through detailed numerical investigations such as phase portraits, bifurcation diagram, Poincaré map, Lyapunov exponent spectrum and finite-time Lyapunov exponent. The analog circuit realization for the Duffing-type MEMS resonator is constructed. The numerically simulated results are confirmed in the laboratory experimental observations which are closely matched with each other. The experimentally observed chaotic attractor confirmed through FFT spectrum, 0–1 test and Poincaré cross section. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio.     

A novel bounded 4D chaotic system

This paper presents a novel bounded four-dimensional (4D) chaotic system which can display hyperchaos, chaos, quasiperiodic and periodic behaviors, and may have a unique equilibrium, three equilibria and five equilibria for the different system parameters. Numerical simulation shows that the chaotic attractors of the new system exhibit very strange shapes which are distinctly different from those of the existing chaotic attractors. In addition, we investigate the ultimate bound and positively invariant set for the new system based on the Lyapunov function method, and obtain a hyperelliptic estimate of it for the system with certain parameters.