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.     

A novel study of parity and attractor in the time reversed Lorentz system

In this Letter, a new effective approach to achieve adaptive synchronization is proposed. Via using Ge-Yao-Chen (GYC) partial region stability theory and pragmatical asymptotically stability theorem, the numerical simulation results show that the states errors and parameter errors approach to zero much more exactly and efficiently than traditional method. The time reversed Lorenz system (called historical Lorenz system in this Letter) is introduced and used for example in this Letter. The simulation results are given in figures and tables for comparison between the new approach and traditional one to show the effectiveness and feasibility of our new strategy.     

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.     

Critical curves and coexisting attractors in a quasiperiodically forced delayed system

We study three critical curves in a quasiperiodically driven system with time delays, where occurrence of symmetry-breaking and symmetry-recovering phenomena can be observed. Typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished. A striking phenomenon that can be discovered is multistability and coexisting attractors in some tongues surrounding by critical curves. The blowout bifurcation accompanying with on-off intermittency can also be observed. We show that collision of attractors at a symmetric invariant subspace can lead to the appearance of symmetry-breaking.     

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.     

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.     

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

The stability conditions in fractional order hyperchaotic systems are derived. These conditions are applied to a novel fractional order hyperchaotic system. The proposed system is also shown to exhibit hyperchaos for orders less than 4. Based on the Routh-Hurwitz conditions, the conditions for controlling hyperchaos via feedback control are also obtained. A specific condition for controlling only fractional order hyperchaotic systems is achieved. Numerical simulations are used to verify the theoretical analysis.     

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.     

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.     

Initial State Dependent Dynamical Behaviors in a Memristor Based Chaotic Circuit

Unlike conventional chaotic systems, a memristor based chaotic circuit has an equilibrium set, whose stability is dependent on the initial state of the memristor. The initial state dependent dynamical behaviors of the memristor based chaotic circuit are investigated both theoretically and numerically.     

The role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables

In this Letter we analyse the behaviour of the probability density function of the sum of N deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. However, when the map is weakly chaotic, and fuzzily not mixing, the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables N increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of N going to infinity the distribution has the same asymptotic decay as a Lorentzian (or a q=2-Gaussian).     

Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles

We study the Hamiltonian dynamics of a one-dimensional chain of linearly coupled particles in a spatially periodic potential which is subjected to a time-periodic mono-frequency external field. The average over time and space of the related force vanishes and hence, the system is effectively without bias which excludes any ratchet effect. We pay special attention to the escape of the entire chain when initially all of its units are distributed in a potential well. Moreover for an escaping chain we explore the possibility of the successive generation of a directed flow based on large accelerations. We find that for adiabatic slope-modulations due to the ac-field transient long-range transport dynamics arises whose direction is governed by the initial phase of the modulation. Most strikingly, that for the driven many particle Hamiltonian system directed collective motion is observed provides evidence for the existence of families of transporting invariant tori confining orbits in ballistic channels in the high-dimensional phase spaces.     

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.     

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.     

Frontward-response control of sudden occurrence of chaos

In this Letter, a novel approach to controlling chaos in one-dimensional discrete-time nonlinear autonomous systems is proposed. The method is validated for sudden occurrence of chaos (SOC); its efficacy is demonstrated via numerical simulations of the mappings. The method is simple in implementation. The approach looks highly promising and may have diverse applications.     

Experimental Confirmation of a Modified Lorenz System

We experimentally demonstrate the butterfly-shaped chaotic attractor we have proposed before lint. J. Nonlin. Sci. Numerical Simulation 7 （2006） 187]. Some basic dynamical properties and chaotic behaviour of this new butterfly attractor are studied and they are in agreement with the results of our theoretical analysis. Moreover, the proposed system is experimental demonstrated.     

Secure communication based on a four-wing chaotic system subject to disturbance inputs

In this paper, a two-input two-output secure communication scheme based on a four-wing four-dimensional chaotic system with disturbance inputs is discussed. Based on parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver are achieved and two message signals are recovered via a convenient robust high-order sliding mode adaptative controller. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the disturbance inputs can be suppressed simultaneously by the proposed adaptative controller. Synchronization under the effect of noise is also considered. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

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.