Lorenz Family Research Articles

Some Notes on the Gini Index and New Inequality Measures: The nth Gini Index

A new family of inequality indices based on the deviation between the expected maximum and the expected minimum of random samples, called the nth Gini index is presented. These indices generalize the Gini index. At the same time, this family of indices and the S-Gini index are generalized by proposing the uv-Gini index, which turns out to be a convex combination of the S-Gini index and the Lorenz family of inequality measures. This family of Gini indices provides a methodology for achieving perfect equality in a given distribution of incomes. This is achieved through a series of successive and equal increases in the incomes of each individual.

Synchronization of fractional-order chaotic networks in Presnov form via homogeneous controllers

The Presnov decomposition for continuously differential vector fields consists of a gradient of a scalar field (dissipative term) and a vector field orthogonal to a radial vector of the origin (circulative or non-dissipative term). This paper uses the Presnov decomposition associated with a fractional-order Lorenz family to design algorithms that solve the multi-synchronization problem. Assuming that the whole state is known, two active control solutions are proposed to achieve the desired synchronization. Both control algorithms take advantage of homogeneous functions to have a powerful and simple Lyapunov analysis to show Mittag-Leffler stabilization in the first algorithm. In contrast, the second one achieves synchronization before a predefined time defined by the user. The combination of homogeneous functions and Riemann–Liouville integral in the controllers improve the behavior of the dynamic systems due to the impact on the robustness properties in the control scheme. Numerical simulations are presented to show the reliability of the proposed methods.

Multiple dynamics analysis of Lorenz-family systems and the application in signal detection

Inspired by the non-autonomous chaos system and the transient chaotic oscillation, based on the chaos systems of the Lorenz family, the system structure, parameters, and orders are adjusted, and the damping oscillation, sustained chaotic oscillation, transient chaotic oscillation, and chaotic oscillation with multiple chaotic trajectories are found. Firstly, the constrain conditions of the system parameters and fractional-orders are derived. Secondly, one external driving force signal and two control parameters are added to the system, different dynamic behaviors of the system under the excitement of different types, features of the external driving force signals are analyzed, simulated, and discussed. The phase diagram, bifurcation diagram, and Lyapunov exponents, etc are adopted for analysis. Finally, based on the adjusted fractional-order nonlinear systems, a kind of signal detection scheme that can be used under a strong noise environment is designed, some key information of the signal to be detected can be acquired. The simulation results show the effectiveness of the designed signal detection scheme.

Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination

AbstractThe Lorenz family is a class of dynamic systems which is parameterized in one variable and includes both the classic Lorenz system and the Chen system. Both systems (and therefore the Lorenz family) can show chaotic behaviour. For different questions of system analysis, e.g. for the estimation of various types of dimensions, it is helpful to be able to determine bounds on the attractor. One typical approach ist the usage of Lyapunov functions to describe positive invariant sets. However, this approach usually requires a good knowledge of the system to perform the necessary calculations. For polynomial systems, these calculations can be carried out using a computation technique known as quantifier elimination. In this contribution we describe this approach and employ it to calculate bounds on the Lorenz family.

Design and Synchronization of Chaotic System using Threshold Controller Coupling

A new autonomous chaotic system is designed from the Lorenz family of systems and studied its dynamics both theoretically and numerically. Then two such systems are synchronized using a threshold controller as the coupling element by which complete and asynchronization is achieved. The new system is investigated using MATLAB Simulink and Multisim. The results show that based on the control parameter value the system switches from complete to asynchronization.

Center cyclicity of Lorenz, Chen and Lü systems

This work provides upper bounds on the cyclicity of the centers on center manifolds in the well-known Lorenz family, and also in the Chen and Lü families. We prove that at most one limit cycle can be made to bifurcate from any center of any element of these families, perturbing within the respective family, with the exception of one specific Lorenz system where the cyclicity increases. We also show that this bound is sharp.

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.

Dynamical behaviors of a generalized Lorenz family

In this paper, the ultimate bound set and globally exponentially attractive set of a generalized Lorenz system are studied according to Lyapunov stability theory and optimization theory. The method of constructing Lyapunov-like functions applied to the former Lorenz-type systems (see, e.g. Lorenz system, Rossler system, Chua system) isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term to the Lyapunov-like functions that used for the Lorenz system to study this generalized Lorenz system. The authors in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853] obtained the ultimate bound set of this generalized Lorenz system but only for some cases with $0 ≤ α

Dynamics of a Class of Nonautonomous Lorenz-Type Systems

The dynamical properties of a kind of Lorenz-type systems with time-varying parameters are studied. The time-varying ultimate bounds are estimated, and a simple method is provided to generate strange attractors, including both the strange nonchaotic attractors (SNAs) and chaotic attractors. The approach is: (i) take an autonomous system with an ultimate bound from the Lorenz family; (ii) add at least two time-varying parameters with incommensurate frequencies satisfying certain conditions; (iii) choose the proper initial position and time by numerical simulations. Three interesting examples are given to illustrate this method with computer simulations. The first one is derived from the classical Lorenz model, and generates an SNA. The second one generates an SNA or a Lorenz-like attractor. The third one exhibits the coexistence of two SNAs and a Lorenz-like attractor. The nonautonomous Lorenz-type systems present more realistic models, which provide further understanding and applications of the numerical analysis in weather and climate predication, synchronization, and other fields.

Circuit Design Contains a Class of Squared Term Lorenz Family Chaotic System

A new class of three-dimensional chaotic system is constructed by algebraic methods, which has a similar structure with the classic Lorenz system but contains the square term. The equilibrium point of the system stability is analyzed, and the numerical simulation is carried on the bifurcation diagram and Lyapunov exponent. The chaotic circuit of these systems is designed by using the software of EWB. The results of the experimental simulation verify the existence of the chaotic attractor, which provides theoretical reference to the application of such system.

Simulation studies on the design of optimum PID controllers to suppress chaotic oscillations in a family of Lorenz-like multi-wing attractors

Multi-wing chaotic attractors are highly complex nonlinear dynamical systems with higher number of index-2 equilibrium points. Due to the presence of several equilibrium points, randomness and hence the complexity of the state time series for these multi-wing chaotic systems is much higher than that of the conventional double-wing chaotic attractors. A real-coded genetic algorithm (GA) based global optimization framework has been adopted in this paper as a common template for designing optimum Proportional-Integral-Derivative (PID) controllers in order to control the state trajectories of four different multi-wing chaotic systems among the Lorenz family viz. Lu system, Chen system, Rucklidge (or Shimizu Morioka) system and Sprott-1 system. Robustness of the control scheme for different initial conditions of the multi-wing chaotic systems has also been shown.

Dynamics of stochastic Lorenz family of chaotic systems with jump

This paper discusses the Lorenz family of chaotic systems under the influence of poisson noise. We obtain the sufficient and necessary conditions for stochastic stability under some assumptions for stochastic Lorenz family of chaotic systems with jump. We then investigate the estimation of the global attractive set and stochastic bifurcation behavior of the family of stochastic Lorenz system. Numerical experiments illustrate the results.

Comments on “Dynamics of the general Lorenz family” by Y. Liu and W. Pang

A paper, “Dynamics of the general Lorenz family,” was published in the journal Nonlinear Dynamics. The authors investigate the general Lorenz family \(\dot{x} = a(y-x)\), \(\dot{y} = dx + cy - xz\), \(\dot{z} = -bz + xy \), considering that it contains four independent parameters. However, we show that, generically (for c≠0), this family is equivalent to the Lorenz system and, thus, the results they provide for the general Lorenz family are easily obtained from the corresponding results on the Lorenz system. Moreover, in the case c=0, it is sufficient to consider a two-parameter system.

Comment on “A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family”, P. Yu, X.X. Liao, S.L. Xie, Y.L. Fu [Commun Nonlinear Sci Numer Simulat 14 (2009) 2886–2896

In the commented paper the authors study some aspects of boundedness in the general Lorenz family, ẋ=σ(y-x),ẏ=ρx-γy-xz,ż=-βz+xy, considering that it contains four independent parameters. However, as we show here by means of a linear scaling in time and coordinates, they are dealing with a system homothetically equivalent to the Lorenz system. Consequently, the novel and interesting results they provide for the general Lorenz family can be obtained working directly with the Lorenz equations, that is, dealing only with three independent parameters.

Classification of dynamical systems based on a decomposition of their vector fields

We present a method for the global classification of dynamical systems based on a specific decomposition of their vector fields. Every differentiable vector field on Rn can be decomposed uniquely in the sum of 2 systems: one gradient and one that leaves invariant the spheres Sn−1. We show that, under some conditions, the topological class of a vector field is determined by the topological classes of its summands. We illustrate this method by applying it to a number of vector fields, among them being some members of the so-called Lorenz family. The advantage of such a classification is that equivalent flows exhibit qualitatively the same dynamical phenomena as their parameters are varied.

Ein Heldenleben: A Life in Neurosurgery

New Haven is a smallish Connecticut town rooted deep in American colonialism and uniquely blessed by the presence of Yale University within its limits. It is a physically beautiful setting on Long Island sound, multicultural but predominantly Italian in its inhabitants’ cultural origins. I was born just before the Second World War. My father, Dominic, a machinist and fine craftsman, was a devoted husband and exemplary father. Intelligent and diligent, he at times worked as many as three jobs to maintain the upward mobility of the family. Our paternal ancestors were from Amalfi, Italy. Sailors for generations, with the advent of steam, they left Italy for Argentina in the early twentieth century, but an Atlantic storm and damaged ship brought them to Ellis Island in New York. My mother, Ann Lawrence, was a nurse. Multifaceted in talents, she was a mother who challenged her children to excel and left no stone unturned to provide an intellectually fertile environment directed to music, art painting, and sports. The Lorenz family, Austrian in origin, had originally migrated to Mahoney City, Pennsylvania, where my grandfather worked in coal mines; later, with my grandmother, they migrated to New Haven.

CLOSED ORBITS IN THE GENERAL LORENZ FAMILY

This paper proves that all the closed orbits (including limit cycles) of the general Lorenz family must be planar, but only to be curves in space based on the technique of classification and identification for the quadratic surface in three-dimensional space. This result indicates that the qualitative property of the closed orbit for the systems is very complicated. We hope that this study would be beneficial for further studies of the dynamically rich chaotic system.

Dynamics of the general Lorenz family

The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lu system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

Chaos Synchronization between Two Different Fractional Systems of Lorenz Family

This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz‐like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.

A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family

In this paper, we give a constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, which contains four independent parameters and is more general than any Lorenz systems studied so far in the literature. The system considered in this paper not only contains the classical Lorenz system and the generalized Lorenz family as special cases, but also provides three new Lorenz systems, which do not belong to the generalized Lorenz system, but the general Lorenz system. The results presented in this paper contain all the existing relative results as special cases.