Evolving chaos: Identifying new attractors of the generalised Lorenz family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor's which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are derived from the original Lorenz system of equations. Compared to the previous exploration with sinusoidal type transcendental nonlinearity, here we focus on only cross-product and higher-power type nonlinearities in the three state equations. We here report over 150 different structures of chaotic attractors along with their one set of parameter values, phase space dynamics and the Largest Lyapunov Exponents (LLE). The expressions of these new Lorenz-like nonlinear dynamical systems have been automatically evolved through multi-gene genetic programming (MGGP). In the past two decades, there have been many claims of designing new chaotic attractors as an incremental extension of the Lorenz family. We provide here a large family of chaotic systems whose structure closely resemble the original Lorenz system but with drastically different phase space dynamics. This advances the state of the art knowledge of discovering new chaotic systems which can find application in many real-world problems. This work may also find its archival value in future in the domain of new chaotic system discovery.     

Hyperchaotic set in continuous chaos–hyperchaos transition

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.     

A new butterfly-shaped attractor of Lorenz-like system

In this letter a new butterfly-shaped chaotic attractor is reported. Some basic dynamical properties, such as Poincare mapping, Lyapunov exponents, fractal dimension, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, we clarify that the chaotic attractors of the system is a compound structure obtained by merging together two simple attractors through a mirror operation.     

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.     

Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor

Exact Lyapunov dimension of attractors of many classical chaotic systems (such as Lorenz, Henon, and Chirikov systems) is obtained. While exact Lyapunov dimension for Rössler system is not known, Leonov formulated the following conjecture: Lyapunov dimension of Rössler attractor is equal to local Lyapunov dimension in one of its stationary points. In the present work Leonov’s conjecture on Lyapunov dimension of various Rössler systems with standard parameters is checked numerically.     

A three-dimensional nonlinear system with a single heteroclinic trajectory

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.     

Generating a new chaotic attractor by feedback controlling method

A new chaotic system is found by feedback controlling method in this paper. According to the definition of the generalized Lorenz system, the new chaotic system does not belong to generalized Lorenz systems. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map. The particular interest is that this novel system can generate two one‐scroll and one two‐scroll chaotic attractors with the variation of a single parameter. The obtained results show clearly that the system is a new chaotic system and deserves a further detailed investigation. Copyright © 2011 John Wiley & Sons, Ltd.     

A novel chaotic attractor

In this letter, a novel chaotic attractor is reported. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincare mapping, the continuous spectrum and chaotic behavior of this new transverse butterfly attractor are studied. Meanwhile, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis. Furthermore, the complex chaotic dynamical behavior of the system has been also proofed by experimental simulation of a designed electronic oscillator based on EWB.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

On the influence of noise on the coexistence of chaotic attractors

In this paper, we study the influence of noise on a mathematical model which contains the coexistence of chaotic attractors. In particular, application to the case of a coexistence of a Lorenz and a Rössler dynamic system is provided. Furthermore, numerical investigations concerning the stability of the Chua's double chaotic attractor under stochastic perturbations of the initial conditions are proposed.     

Generation and circuit implementation of fractional-order multi-scroll attractors

This paper considers the generating of multi-scroll chaotic attractors for a new fractional-order linear system by using the piecewise-linear function. Multi-scroll chaotic attractors are generated by extending the number of saddle equilibrium points with index 2. Poincaré map and maximum Lyapunov exponents are applied to verifying the chaotic behaviors of the generated multi-scroll chaotic attractors. A circuit for the multi-scroll attractor is designed and simulated. Moreover, physical experiment of 3-scroll attractors and 5-scroll attractors are implemented. The numerical simulation, the circuit simulation and hardware experimental results are in accordance with each other, which verifies the effectiveness and physical realization of the approach.     

Optimizing Markovian modeling of chaotic systems with recurrent neural networks

In this paper, we propose a methodology for optimizing the modeling of an one-dimensional chaotic time series with a Markov Chain. The model is extracted from a recurrent neural network trained for the attractor reconstructed from the data set. Each state of the obtained Markov Chain is a region of the reconstructed state space where the dynamics is approximated by a specific piecewise linear map, obtained from the network. The Markov Chain represents the dynamics of the time series in its statistical essence. An application to a time series resulted from Lorenz system is included.     

Qualitative behavior of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres

In this article, the bounds of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres have been studied. Based on Lagrange multiplier method, the function extremum theory and the generalized positive definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and the globally exponentially attractive set for this system. The results that obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension and the Lyapunov dimension of chaotic attractors. © 2016 Wiley Periodicals, Inc. Complexity 21: 67–72, 2016     

Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria

This paper presents a new 3-D autonomous chaotic system, which is topologically non-equivalent to the original Lorenz and all Lorenz-like systems. Of particular interest is that the chaotic system can generate double-scroll chaotic attractors in a very wide parameter domain with only two stable equilibria. 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. Finally, the complicated dynamics are studied by virtue of theoretical analysis, numerical simulation and Lyapunov exponents spectrum. The obtained results clearly show that the chaotic system deserves further detailed investigation.     

Controlling the Lorenz system: combining global and local schemes

The Lorenz equations are one of the best-known and analyzed systems exhibiting chaotic behavior. In this paper, a new control scheme for the Lorenz system combining local and global techniques is introduced. This scheme is based on a feedback law which is only applied in a bounded state space region of control (SSRC). The SSRC is determined by the enclosure of the Lorenz attractor.     

动力系统实测数据的Lyapunov指数的矩阵算法

Lyapunov指数l是定量描述混沌吸引子的重要指标,自从1985年Wolf提出Lyapunov指数l的轨线算法以来,如何准确、快速地计算正的、最大的Lyapunov指数l_max便成为人们关注的问题,虽有不少成功计算的报导,但一般并不公开交流.在Zuo Bingwu理论算法的基础上,给出了Lyapunov指数l的具体的矩阵算法,并与Wolf的算法进行了比较,计算结果表明:算法能快速、准确地计算(主要是正的、最大的)Lyapunov指数l_max.并对Lyapunov指数l的大小所反应的吸引子的特性进行了分析,并得出了相应的结论.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

The chaotic dynamics of vibrational mechanisms with energy sources of limited power

The dynamics of a vibrational mechanism with an energy source of limited power is considered. A system of two degrees of freedom is reduced to a system of the Lorenz type by the method of averaging. The existence of one of the types of chaotic attractors in a dynamical system which is a vibrational mechanism, that is, a Lorenz attractor, is established by this. The existence of a Feigenbaum attractor and intermittence is also established. Chaotic limit sets determine the chaotic behaviour of the instantaneous frequency of rotation of an asynchronous motor. The qualitative patterns of the rotational characteristic are constructed for different values of the parameters of the system and a physical interpretation of the results is given.