A new five-term simple chaotic attractor

A new chaotic attractor is presented with only five terms in three simple differential equations having fewer terms and simpler than those of existing seven-term or six-term chaotic attractors. Basic dynamical properties of the new attractor are demonstrated in terms of equilibria, Jacobian matrices, non-generalized Lorenz systems, Lyapunov exponents, a dissipative system, a chaotic waveform in time domain, a continuous frequency spectrum, Poincaré maps, bifurcations and forming mechanisms of its compound structures.     

A three-scroll chaotic attractor

This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.     

A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors

In this Letter a novel three-dimensional autonomous chaotic system is proposed. Of particular interest is that this novel system can generate two, three and four-scroll chaotic attractors with variation of a single parameter. By applying either analytical or numerical methods, basic properties of the system, such as dynamical behaviors (time history and phase diagrams), Poincaré mapping, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

In this paper we study different aspects of the paradigmatic Rössler model. We perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles. This provides us a global idea of the three-parametric evolution of the system. We also study the regions of parameters where we may expect a chaotic behavior by the use of different Chaos Indicators. The combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system.     

Control-oriented approaches to anticipating synchronization of chaotic deterministic ratchets

In virtue of techniques derived from nonlinear control system theory, we establish conditions under which one could obtain anticipating synchronization between two periodically driven deterministic ratchets that are able to exhibit directed transport with a finite velocity. Criteria are established in order to guarantee the anticipating synchronization property of such systems as well as characterize phase space dynamics of the ratchet transporting behaviors. These results allow one to predict the chaotic direct transport features of particles on a ratchet potential using a copy of the same system that performs as a slave, which are verified through numerical simulation.     

Dynamics analysis and synchronization of a new chaotic attractor

In this paper, a new simple chaotic system is discussed. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria and stability, Lyapunov exponents, a dissipative system, Poincaré mapping, bifurcation diagram, especially Hopf bifurcation. Next, based on well-known Lyapunov stability theorem, backstepping design is proposed for synchronization of the new chaotic system. At last, numerical studies are provided to illustrate the effectiveness of the presented scheme.     

The width of a chaotic layer

A model of nonlinear resonance as a periodically perturbed pendulum is considered, and a new method of analytical estimating the width of a chaotic layer near the separatrices of the resonance is derived for the case of slow perturbation (the case of adiabatic chaos). The method turns out to be successful not only in the case of adiabatic chaos, but in the case of intermediate perturbation frequencies as well.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Synchronization, anti-synchronization and current transports in non-identical chaotic ratchets

The synchronization (CS) and anti-synchronization (AS) of two non-identical inertial ratchets moving in different asymmetric potentials and transporting particles chaotically is demonstrated based on a technique derived from nonlinear control theory. It is shown that in the CS state, particle current is enhanced; while in the AS state, the particles transport current in different directions, suggesting that AS phenomenon could be exploited as a mechanism for particle separation.     

一个四翼混沌吸引子

在新的四维混沌系统中数值观察到四翼混沌吸引子，然而，通过进一步分析发现，该四翼吸引子并非真实的，实际上它是上、下两个共存的双翼混沌吸引子，他们各自有独立的混沌吸引域，由于其位置靠得太近和数值误差产生的一种假象.通过引入一个线性状态反馈控制项，系统的一些相似性被破坏，受控系统能产生穿越上下吸引域界限的对角双翼混沌吸引子，进一步，随着动力学模态的演化，上下混沌吸引子与对角混沌吸引子融合成一个真正的四翼混沌吸引子.最后，通过比较该四翼混沌吸引子的系统、Lorenz系统、Chua氏电路等混沌信号的频谱发现，四翼混沌吸引子的系统信号具有极宽的频谱带宽，该特性在通讯加密等工程应用中具有重要价值. 关键词： 四维混沌系统 双翼吸引子 四翼吸引子 频谱分析     

The evolution of chaotic dynamics for fractional unified system

Based on reliable numerical approach, this Letter studies the chaotic behavior of the fractional unified system. The lowest orders for this system to have a complete chaotic attractor (the attractor covers the three equilibrium points of the classical unified system) at different parameter values are obtained. A striking finding is that with the increase of the parameter α of the fractional unified system from 0 to 1, the lowest order for this system to have a complete chaotic attractor monotonically decreases from 2.97 to 2.07. Because of the inherent attribute (memory effects) of fractional derivatives, this finding reveals that the chaotic behavior of fractional (classical) unified system becomes stronger and stronger when α increases from 0 to 1. Furthermore, this Letter introduces a novel measure to characterize the chaos intensity of fractional (classical) differential system.     

Adaptive function projective synchronization of two-cell Quantum-CNN chaotic oscillators with uncertain parameters

This work investigates function projective synchronization of two-cell Quantum-CNN chaotic oscillators using adaptive method. Quantum-CNN oscillators produce nano scale chaotic oscillations under certain conditions. By Lyapunove stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Cryptanalysis of a discrete-time synchronous chaotic encryption system

Recently a chaotic cryptosystem based on discrete-time synchronization has been proposed. Some weaknesses of that new encryption system are addressed and exploited in order to successfully cryptanalyze the system.     

Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales

By employing a special feedback controlling scheme, a hyperchaotic Lorenz system with the structure of two time scales is constructed. Two kinds of bursting phenomena, symmetric fold/fold bursting and symmetric sub-Hopf/sub-Hopf bursting, can be observed in this system. Their respective dynamical behaviors are investigated by means of slow-fast analysis. In particular, symmetric fold/fold bursting is of focus-focus type, namely, both the up-state and the down-state are stable focus, which is different from the usual fold/fold bursting; Symmetric sub-Hopf/sub-Hopf bursting is also of focus-focus type, which has not been reported in previous work. Furthermore, phase plane analysis has been introduced to explore the evolution details of the fast subsystem for symmetric sub-Hopf/sub-Hopf bursting. With the variation of the parameter, symmetric sub-Hopf/sub-Hopf bursting can evolve to symmetric chaotic bursting or even hyperchaos.     

Spiking Behavior in Chua's Circuit

Chua ＇s circuit with a slow-fast effect is established under certain parameter conditions. The dynamics of this slow- fast system is investigated. A spiking phenomenon can be observed in the numerical simulation. By introducing slow-fast analysis and a generalized Jacobian matrix at the non-smooth boundaries, the bifurcation mechanism for the periodic spiking solution, different from the smooth case, is discussed.     

Fermi acceleration with memory-dependent excitation

Some scaling properties for a classical particle confined to bounce between two walls, where one wall is fixed and the other one moves in time according to a random signal with a memory length are studied. We have considered two different kinds of collisions of the particle with the moving wall namely: (i) elastic and (ii) inelastic. The dynamics of the model is described in terms of a two-dimensional nonlinear mapping. For the case of elastic collisions, we show that the memory of the stochastic signal affects directly the behaviour of the average velocity of the particle. It then exhibits different slopes for the average velocity at different stages of the series with β≅3/4 for a short time, β≅1 for the average stage and β≅1/2 for a long time, as predicted by the Central Limit Theorem, therefore leading to the Fermi acceleration. The situation where inelastic collisions are taken into account yields a more drastic change, particularly suppressing the Fermi acceleration.     

Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system

A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries.