Generalized Lorenz System Research Articles

Synchronization of Generalized Lorenz Systems in a Loop

The conditions for the synchronization of three interconnected generalized Lorenz systems are given. The interconnection topology contains a loop. It is shown that, under certain conditions on the strength of the coupling of the systems, the full synchronization of all three systems is guaranteed. The results are illustrated by an example.

A generalized Lorenz system-based initialization method for deep neural networks

Deep neural networks (DNNs) are a powerful tool for solving complex problems. The effectiveness of DNNs largely depends on the initialization technique used. This research develops a new initialization method for DNNs that uses chaotic sequences from the generalized Lorenz system to improve their performance. The proposed method, termed Generalized Lorenz Initialization (GLI), has been compared with two established initialization methods (Kaiming and Xavier) across four different DNN architectures: Informer, Neural Basis Expansion Analysis for Interpretable Time Series, Long Short-Term Memory, and NeuRewriter. The performance of these methods has been evaluated on seven time series forecasting datasets and one combinatorial optimization dataset. Results show that the GLI method improved forecasting accuracy by up to 86.47% compared to the Kaiming method and 88.86% compared to the Xavier method across all time series datasets. For the combinatorial optimization task, the GLI method reduced computational time by up to 9.24% with the better solution quality. These indicate the superiority of the GLI method over the two representative initialization methods for different DNN architectures across different problem domains.

Jacobi Stability for T-System

In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

Complexity measurements for the thermal convection in a viscoelastic fluid saturated porous medium

Measuring complexity statistical indicators is a key method to analyze and characterize dynamical systems. In this work, we perform a comparative analysis among the López-Ruiz, Mancini & Calbet complexity indicator and the largest Lyapunov exponent for the convection problem of a viscoelastic fluid in a porous medium with feedback control based in an Oldroyd carrier liquid through a four-dimensional generalized Lorenz system. With both indicators can be distinguished from chaotic to periodic states. We perform intensive numerical simulations with 4×106 in the space parameters, finding good agreement between them, such that difference is close to 2%. We have also detected that the computing time is much faster in the case of complexity indicator than for Lyapunov exponents. Finally, we have also studied the effect of the initial conditions in the coexistence states, encountering multistability.

Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x^{2}y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: dot{x}=a(y - x), dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y, dot{z}= -cz + y^{2}, which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S_{x} = {(x, x, frac{x^{2}}{c})|xin mathbb {R}, cne 0} are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.

Synchronization of uncertain general fractional unified chaotic systems via finite-time adaptive sliding mode control.

This paper employs two adaptive sliding mode control (ASMC) strategies to accomplish finite-time synchronization of uncertain general fractional unified chaotic systems (UGFUCSs) when uncertainty and external disturbance exist. First, general fractional unified chaotic system (GFUCS) is developed. GFUCS may be transitioned from general Lorenz system to general Chen system, and the general kernel function could compress and extend the time domain. Furthermore, two ASMC methods are applied to finite-time synchronization of UGFUCSs, where system states arrive at sliding surfaces in finite-time. The first ASMC approach utilizes three sliding mode controllers to achieve synchronization between chaotic systems, while the second ASMC method needs just one sliding mode controller to produce synchronization between chaotic systems. Finally, the effectiveness of the proposed ASMC approaches is verified using numerical simulations.

Multi-directional annular multi-wing chaotic system based on Julia fractals

Fractals and chaos are inextricably linked, and relevant study has been done on both. However, the studies that combine fractals and chaotic systems to produce multi- scroll or multi-wing chaotic attractors are all extend multi-scroll or multi-wing in the even direction. To this end, by improving the traditional fractal mapping, a chaotic system that can extend multiple wings in any direction of odd and even is designed and analyzed. The main point of this method is as follows: Firstly, a new family of even-symmetric segmented composite functions is proposed and applied to a family of generalized Lorenz systems, which can generate multi-wing chaotic systems in two directions and the number of wings in each direction can also be increased with the parameter N. Secondly, the improved Julia fractal is applied to two-direction multi-wing chaotic systems to generate a class of single-direction multi-wing chaotic systems. Finally, the combination of this class of single-direction multi-wing chaotic systems and higher-order Julia fractals is used to realize chaotic systems with extended multi-wing in odd and even arbitrary directions. Computer simulations confirm the method's viability. In addition, the maximum Lyapunov exponent, bifurcation diagram, Poincare diagram, and PE complexity chaos diagram are also analyzed as examples of multi-directional multi-wing Lü chaotic systems, demonstrating the system's superiority in odd directions over even directions and the existence and effective scalability of multi-directional multi-wing chaotic systems.

Local Bifurcation and Center Problem for a More Generalized Lorenz System

Local Bifurcation and Center Problem for a More Generalized Lorenz System

Using the (3N)-Dimensional Generalized Lorenz Systems as a Testbed for Data Assimilation: The Ensemble Kalman Filter

AbstractThe feasibility of using a (3N)-dimensional generalization of the Lorenz system in testing a traditional implementation of the ensemble Kalman filter is explored through numerical experiments. The generalization extends the Lorenz system, known as the Lorenz ’63 model, into a (3N)-dimensional nonlinear system for any positive integerN. Because the extension involves inclusion of additional wavenumber modes, raising the dimension allows the system to resolve smaller-scale motions, a unique characteristic of the present generalization that can be relevant to real modeling scenarios. Model imperfections are simulated by assuming a high-dimensional generalized Lorenz system as the true system and a generalized system of dimension less than or equal to the dimension of the true system as the model system. Different scenarios relevant to data assimilation practices are simulated by varying the dimensional differences between the model and true systems, ensemble size, and observation frequency and accuracy. It is suggested that the present generalization of the Lorenz system is an interesting and flexible tool for evaluating the effectiveness of data assimilation methods and a meaningful addition to the portfolio of testbed systems that includes the Lorenz ’63 and ’96 models, especially considering its relationship with the Lorenz ’63 model. The results presented in this study can serve as useful benchmarks for testing other data assimilation methods besides the ensemble Kalman filter.

Generalized Lorenz Canonical Form Revisited

This paper completes the description of the generalized Lorenz system (GLS) and hyperbolic generalized Lorenz system (HGLS) along with their canonical forms (GLCF, HGLCF), mostly presented earlier, by deriving explicit state transformation formulas to prove the equivalence between GLS and GLCF, as well as between HGLS and HGLCF. Consequently, complete formulations of the generalized Lorenz canonical systems and forms, and their hyperbolic settings, are obtained and presented. Only potentially chaotic systems are classified, which significantly helps clarify the respective canonical forms. To do so, some tools for systems to exclude chaotic behavior are developed, which are interesting in their own right for general dynamical systems theory. The new insight may inspire future investigations of generalized and canonical formulations of some other types of chaotic systems.

Chaos synchronization in generalized Lorenz systems and an application to image encryption

Examples of synchronization, pervasive throughout the natural world, are often awe-inspiring because they tend to transcend our intuition. Synchronization in chaotic dynamical systems, of which the Lorenz system is a quintessential example, is even more surprising because the very defining features of chaos include sensitive dependence on initial conditions. It is worth pursuing, then, the question of whether high-dimensional extensions of such a system also exhibit synchronization. This study investigates synchronization in a set of high-dimensional generalizations of the Lorenz system obtained from the inclusion of additional Fourier modes. Numerical evidence supports that these systems exhibit self-synchronization. An example application of this phenomenon to image encryption is also provided. Numerical experiments also suggest that there is much more to synchronization in these generalized Lorenz systems than self-synchronization; while setting the dimension of the driver system higher than that of the receiver system does not result in perfect synchrony, the smaller the dimensional difference between the two, the more closely the receiver system tends to follow the driver, leading to self-synchronization when their dimensions are equal.

Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid

The paper deals with the study of effect of gravity modulation on double-diffusive convection in a dielectric liquid for the cases of rigid-rigid and free-free boundaries. Using a modified Venezian approach, expressions for the Rayleigh number and its correction are determined. Fourier–Galerkin expansion is employed for a weakly nonlinear stability analysis and this results in a fifth-order Lorenz system that retains the structure of the classical one in the limiting case. A local nonlinear stability analysis using the method of multiscales leads to the time-periodic Ginzburg–Landau equation from the time-periodic generalized Lorenz system and the numerical solution of this simpler equation helps in quantifying unsteady heat and mass transports. Influence of various non-dimensional parameters (Lewis number, solutal Rayleigh number, electrical Rayleigh number and Prandtl number), amplitude and frequency of gravity modulation on onset of convection and heat and mass transports is discussed. The study reveals that the influence of gravity modulation is to stabilize the system and enhance heat and mass transports. The results from free-free boundaries are qualitatively similar to that of rigid-rigid boundaries. Further, it is shown that in the case of free-free boundaries the heat and mass transports are less compared to those of rigid-rigid boundaries.

The effect of boundary conditions on the onset of chaos in Rayleigh–Bénard convection using energy-conserving Lorenz models

The influence of 16 boundary conditions on linear and nonlinear stability analyses of Rayleigh–Bénard system is reported. A Stuart–Landau amplitude equation for the Rayleigh–Bénard system between stress-free, isothermal boundary conditions is derived and the procedure used in this derivation serves as guidance for constructing an appropriate Fourier–Galerkin expansion for the other 15 boundary conditions to derive a generalized Lorenz model. The influence of the boundary conditions comes within the coefficients of the generalized Lorenz model. It is shown that the obtained generalized Lorenz model is energy conserving and for certain boundary conditions it retains features of the classical Lorenz model. Further, the principle of exchange of stabilities is shown to be valid for the present problem and hence it is the steady-state, linearized version of the generalized Lorenz model which yields an analytical expression for the Rayleigh number. On minimizing this expression with respect to wave number the critical Rayleigh number at which the onset of regular convective motion occurs in the form of rolls is determined for all 16 boundary conditions. It is found that these values are in good agreement with those of previous investigations leading to the conclusion that the chosen minimal Fourier–Galerkin expansion is a valid one. Exhibition of chaotic motion in the generalized Lorenz system at the Hopf Rayleigh number is studied. The phase-space plots which indicate a clear-cut visualization of the transition from regular convective motion to chaotic motion in the generalized Lorenz system are presented. Further, existence of a developing region for chaos (mildly chaotic motion) and windows of periodicity are captured using the bifurcation diagrams. It is concluded from the phase-space plots and the bifurcation diagrams that the generalized Lorenz model for certain sets of boundary conditions retains all the features of the classical Lorenz model. Such a conclusion cannot be made by a linear stability analysis and the need thus for a nonlinear analysis is highlighted in the paper.

A new image encryption algorithm based on optimized Lorenz chaotic system

SummaryIn this article, an improved Lorenz system was constructed, in the first stage, a nonlinear term in general Lorenz system was replaced by the sum of term of exponential function and term of the square of single variable, then the dynamic characteristics of the improved Lorenz system were analyzed through the analysis on the phase diagrams and its Lyapunov exponents, and it is proved that the new system has good chaotic characteristics. In the second stage, to improve the security and anti‐attack ability of image encryption algorithm, a new image encryption algorithm was proposed based on this improved Lorenz chaotic system, which was used to generate two groups of key sequences which were used in scrambling encryption and alternative encryption. In the scrambling encryption stage, the gray values of image pixels were sorted in ascending order, and combining with the sequence of scrambled encryption key, the pixel position scrambling was realized. In the phase of the pixel value replacement encryption stage, the gray values of the pixels were changed by the method of cipher text feedback. In the third stage, the quantitative analysis of the effect of encryption was carried out by using histogram, information entropy, correlation coefficient of adjacent pixels. The experimental results show that the improved Lorenz formula is a nonlinear chaotic system, the encrypted algorithm based on this chaotic system can effectively resist chosen plaintext attack, in the encrypted image, the grey value shave random‐like distribution behavior and the adjacent pixels have zero correlation. The algorithm has a 299 bit key space and it is very sensitive to the key.

Chaotic convection in an Oldroyd viscoelastic fluid in saturated porous medium with feedback control.

The control effects on the convection dynamics in a viscoelastic fluid-saturated porous medium heated from below and cooled from above are studied. A truncated Galerkin expansion was applied to balance equations to obtain a four-dimensional generalized Lorenz system. The dynamical behavior is mainly characterized by the Lyapunov exponents, bifurcation, and isospike diagrams. The results show that within a range of moderate and high Rayleigh numbers, proportional controller gain is found to enhance the stabilization and destabilization effects on the thermal convection. Furthermore, due to the effect of viscoelasticity, the system exhibits remarkable topological structures of regular regions embedded in chaotic domains.

Hopf Bifurcations, Periodic Windows and Intermittency in the Generalized Lorenz Model

In the present study, we analyze the dynamics of a four-dimensional generalized Lorenz system with one variable describing the profile of the magnetic field induced in a convected magnetized fluid. In particular, we identify the subcritical Hopf bifurcation, at which the dimension of the unstable manifold is increased or reduced by two. Moreover, the new four-dimensional system behavior depending on the control parameters is considered and bidirectional bifurcation structures are revealed. The results show the existence of several windows of nonchaotic variation (windows of order), in particular period-3 windows at the edge of which type I intermittency is observed.

Homoclinic orbits and chaos in the generalized Lorenz system

This paper investigates the homoclinic orbits and chaos in the generalized Lorenz system. Using center manifold theory and Lyapunov functions, we get non-existence conditions of homoclinic orbits associated with the origin. The existence conditions of the homoclinic orbits are obtained by Fishing Principle. Therefore, sufficient and necessary conditions of existence of homoclinic orbits associated with the origin are given. Furthermore, with the broken of the homoclinic orbits, we show that the chaos is in the sense generalized Shil'nikov homoclinic criterion.

Analysis and Control of Fractional Order Generalized Lorenz Chaotic System by Using Finite Time Synchronization

In this paper, we present a corresponding fractional order three-dimensional autonomous chaotic system based on a new class of integer order chaotic systems. We found that the fractional order chaotic system belongs to the generalized Lorenz system family by analyzing its linear term and topological structure. We also found that the equilibrium point generated by the fractional order system belongs to the unstable saddle point through the prediction correction method and the fractional order stability theory. The complexity of fractional order chaotic system is given by spectral entropy algorithm andC0algorithm. We concluded that the fractional order chaotic system has a higher complexity. The fractional order system can generate rich dynamic behavior phenomenon with the values of the parameters and the order changed. We applied the finite time stability theory to design the finite time synchronous controller between drive system and corresponding system. The numerical simulations demonstrate that the controller provides fast and efficient method in the synchronization process.

具有隐藏吸引子的广义Lorenz系统动力学分析

具有隐藏吸引子的广义Lorenz系统动力学分析

Study on noise control effect based on generalized Lorenz system

The suppression of ship radiation noise is studied through a generalized Lorenz system. The system integrates a classical Lorentz system with a time-delay feedback control and an external acoustic excitation. The output of the system can be categorized into three different states: the quasi-periodic state, the periodic state and the chaotic state. The output state is characterized by the trajectory, the bifurcation diagram and the Lyapunov exponent of the system. The dependence of the output state on different excitation parameters like amplitude, frequency, time delay and the number of external excitations are simulated. It is found that the chaotic state and noise suppression is largely dependent on the amplitude of the external excitation. The noise suppression in the generalized Lorenz system is characterized by the changes in power spectrum of the acoustic energy, which is investigated through Fourier transformation. The suppression is optimized by specified excitation parameters: an acoustic energy reduction up to 26.67 dB is realized with excitation source of small amplitude, high frequency, and a phase synchronized with the noise phase. The noise suppression through multiple excitations in the system is also investigated.