Analysis of a new three-dimensional chaotic system

In this paper, a new three-dimensional autonomous chaotic system is presented. There are three control parameters and three different nonlinear terms in the governed equations. Basic dynamic properties of the new system are investigated via theoretical analysis and numerical simulation. The nonlinear characteristic of the new three-dimensional autonomous system versus the control parameters is illustrated by bifurcation diagram, Lyapunov-exponent spectrum, phase portraits, etc.     

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

Nonlinear Dynamics - Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its...     

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

Analysis and synchronization for a new fractional-order chaotic system with absolute value term

A new fractional-order chaotic system with absolute value term is introduced. Some dynamical behaviors are investigated and analyzed. Furthermore, synchronization of this system is achieved by utilizing the drive-response method and the feedback method. The suitable parameters for achieving synchronization are studied. Both the theoretical analysis and numerical simulations show the effectiveness of the two methods.     

Impulsive control of chaotic attractors in nonlinear chaotic systems

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented. By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, andwith it, the upper bond of the impulse interval for asymptotically stable control was given.Numerical results are presented, which are considered with important reference value for control of chaotic attractors.     

Further nonlinear dynamical analysis of simple jerk system with multiple attractors

Nonlinear Dynamics - This paper presents an analytical framework to investigate the dynamical behavior of a recent chaotic jerk model with multiple attractors. The methods of analytical analysis...     

A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems

In this paper, several smooth canonical 3-D continuous autonomous systems are proposed in terms of the coefficients of nonlinear terms. These systems are derived from the existing 3-D four-wing smooth continuous autonomous chaotic systems. These new systems are the simplest chaotic attractor systems which can exhibit four wings. They have the basic structure of the existing 3-D four-wing systems, which means they can be extended to the existing 3-D four-wing chaotic systems by adding some linear and/or quadratic terms. Two of these systems are analyzed. Although the two systems are similar to each other in structure, they are different in dynamics. One is sensitive to the initializations and sampling time, but another is not, which is shown by comparing Lyapunov exponents, bifurcation diagrams, and Poincaré maps.     

Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization

Nonlinear Dynamics - Memristor chaotic system has been attracted by many researchers because of the rich dynamical behaviors. However, some existed memristor chaotic systems have finite numbers of...     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

Dynamical properties and simulation of a new Lorenz-like chaotic system

This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rössler system, Chen system, and includes Lü systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.     

Analysis of a new simple one dimensional chaotic map

In this paper, a new one-dimensional map is introduced, which exhibits chaotic behavior in small interval of real numbers. It is discovered that a very simple fraction in a square root with one variable and two parameters can lead to a period-doubling bifurcations. Given the nonlinear dynamics of one-dimensional chaotic maps, it is usually seen that chaos arises when the parameter raises up to a value, however in our map, which seems reverse, it arises when the related parameter decreases and approaches to a constant value. Since proposing a new map entails solid foundations, the analysis is originated with linear stability analysis of the new map, finding fixed points. Additionally, the nonlinear dynamics analysis of the new map also includes cobweb plot, bifurcation diagram, and Lyapunov analysis to realize further dynamics. This research is mainly consisting of real numbers, therefore imaginary parts of the simulations are omitted. For the numerical analysis, parameters are assigned to given values, yet a generalized version of the map is also introduced.     

On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system

We consider the well-known Sprott A system, which depends on a single real parameter a and, for \(a=1\), was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for \(a=0\), the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For \(a\ne 0\), the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for \(a>0\) small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the \(\alpha \)- and \(\omega \)-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for \(a<1\). Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for \(a>0\), the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered.     

Analytical study of global bifurcations,stabilization and chaos synchronization of jerk system with multiple attractors

Nonlinear Dynamics - This paper investigates the fixed-time synchronization of complex dynamical networks with nonidentical nodes in the presence of bounded uncertainties and disturbances using...