Lorenz Stenflo Research Articles

Chaotic model and control of an atmospheric convective system coupled with large-scale circulation

Large-scale weather systems affect or determine the generation, development and movement of small- and medium-scale convective weather systems, and the understanding of their influence mechanism is of great significance for the prediction and forecasting of small- and medium-scale convective systems. From the perspective of dynamics, based on the Lorenz–Stenflo model describing convective systems and the Lorenz84 model describing low-order atmospheric circulation, we constructed a chaotic model of an atmospheric convective system considering large-scale circulation, analyzed the influence of the system parameters on the convective system, and designed a state feedback controller to adjust the convective motion. For the constructed convective weather system model, the existence and boundedness of its solution were analyzed, the Hamiltonian quantity was solved, the dynamic characteristics of the Lyapunov exponent, bifurcation diagram, attractor phase diagram and other methods were used to analyze its dynamic characteristics, the physical significance of its dynamic behavior was discussed, and the influence mechanism of the system parameters on convective weather was given. Additionally, the temperature difference between the upper and lower air currents is changed by the state feedback method, which provides a feasible scheme for adjusting the intensity of convective motion. The simulation experiments show that the chaotic model can explain and reveal the complex dynamic behavior of large-scale weather circulation in small- and medium-scale convective systems, which is of great significance for improving the prediction accuracy of convective systems in local areas, and feedback control can obtain the desired convective motion state.

Decision support system for managing multi-echelon supply chain networks against disruptions using adaptive fractional order control algorithm

Managing highly evolving supply chains can be challenging, especially when vulnerable to disruptions and risks. This paper deals with a supply chain system’s dynamical analysis and efficient management strategy using a four-stage hyperchaotic Lorenz–Stenflo equation under disruptive events. Nonlinear behaviors are intensely investigated by eigenvalue and bifurcation analysis to identify supply chain risks. Then phase portraits are presented to illustrate the bullwhip effect negatively influencing the performance of various stages of multi-echelon supply chains. Resilient supply chains have been developed along with dynamic identification by realizing an adaptive fractional-order controller. An efficient control algorithm can optimize the management system while reducing potential risks by employing control theory in a decision support system. Performance criteria have been exploited to validate the control methodology. Using digital management algorithms, decision-makers might effectively cope with chaos suppression and synchronization problems, ensuring productivity and sustainability. Finally, the novel decision-making strategy can offer new insights into effectively managing digital supply chain networks against market volatility.

Climate System: A Global Sensitivity Approach

This article is a first attempt to develop a numerical approach to solving differential equations based on Galerkin projections and extensions of polynomial chaos to analyze the sensitivity of input parameters in the Lorenz–Stenflo climate model (Leonov in Commun Pure Appl Anal 16(6):2253–2267, 2017). The sensitivity analysis was undertaken to measure the influence of key parameters (chemical properties of the atmosphere, rotation, temperature gradient, convection motion). In addition, we do simulations of the climate model in the non-chaotic case and in the chaotic case and we calculate the Sobol indices when the parameters follow the uniform law.

Fractional dynamic sliding mode control for non-identical uncertain fractional chaotic systems

Using the fractional calculus a novel dynamic sliding mode control is proposed for control and synchronisation between different fractional chaotic systems with matched disturbances. Lyapunov stability theory has guaranteed the stability of the closed-loop system. The synchronisation and control of two chaotic Lorenz-Stenflo (LS) and Qi systems in master-slave configuration are realised by the presented controller. Furthermore, the obtained chaotic fractional LS and Qi motions are sorted out for qualitative and quantitative study using Lyapunov exponents and bifurcation diagrams with respect to fractional-order of the systems. In the fractional-order LS and Qi systems chaos can exist with order as low as 3.76 and 3.48, respectively. The control method is presented for eliminating chattering disadvantage of sliding mode control in a finite time.

On the dynamics of a modified Lorenz–Stenflo system

The Lorenz–Stenflo system is a four-parameter four-dimensional autonomous nonlinear continuous-time dynamical system, derived to model the time evolution of finite amplitude acoustic gravity waves in a rotating atmosphere. In this paper, we propose a modified Lorenz–Stenflo system, where the variable [Formula: see text] in the fourth equation of the original Lorenz–Stenflo system was replaced by [Formula: see text]. We investigate cross-sections of the parameter-space of this new system, characterizing regions of different dynamical behaviors. We show that the aforementioned replacement may promote the emergence of organized periodic structures in places of these cross-sections, where they did not exist before modification.

Combination Control of Chaotic Systems

In this paper, the tracking control of a combination of two, three, four and five chaotic systems is achieved for the 2-D Duffing oscillator, 3-D Lorenz chaotic system and 4-D Lorenz–Stenflo chaotic system via recursive backstepping technique. The combination tracking control could find applications in situations where the total output of several chaotic systems is required in a desired form. Numerical simulations are carried out to validate the schemes.

Chaos Suppression in Uncertain Generalized Lorenz–Stenflo Systems via a Single Rippling Controller with Input Nonlinearity

In this paper, a robust control design of chaos suppression is considered for generalized four-dimensional (4D) Lorenz–Stenflo systems subjected to matched/mismatched uncertainties and input nonlinearity. It is implemented by using rippling sliding mode control (SMC). A proportional-integral (PI) type scalar switching surface is designed such that the controlled dynamics in the sliding manifold becomes easy to analyze. Furthermore, only by using single rippling SMC even with input nonlinearity can we ensure the existence of the sliding mode for the controlled dynamics and suppress the chaotic behavior in a manner of rippling. Under the proposed control scheme, the chaos behavior in uncertain generalized 4D Lorenz–Stenflo systems subjected to mismatched uncertainties can be robustly suppressed to predictable bounds, which is not addressed in the literature. The numerical simulation results including matched/mismatched uncertainties and nonlinear inputs are presented to verify the robustness and validity of the rippling sliding mode controller.

A Novel Self-Adaptive Wind Speed Prediction Model Considering Atmospheric Motion and Fractal Feature

Many of the previous investigations predicted wind speed by directly using wind speed data, which rarely considered physical characteristics of wind speed and was difficult to improve prediction accuracy further. Therefore, a novel self-adaptive wind speed prediction model considering atmospheric motion and fractal feature is developed in this paper. Lorenz-Stenflo (LS) equation is employed to describe the disturbances and chaos effect caused by atmospheric motion on wind speed. One-dimension LS motion series obtained by LS equation is adopted to improve the decomposition effect of wind speed by ensemble empirical mode decomposition (EEMD). The fractal feature of wind speed series is primitively adopted to determine the key parameter in LS equation. Then back propagation (BP) neural network model optimized by genetic algorithm (GA), as a fundamental prediction model, is used for prediction. Eight groups of wind speed series on different time scales from two wind farms are tested and evaluated. The proposed model effectively overcomes the disturbances of atmospheric motion and achieves promising prediction accuracy. Meanwhile, the criterion based on fractal feature ensures accurate selection of the key parameter in atmospheric motion equation according to different features of sampled wind data.

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.

Multi-Switching Combination Anti-synchronization of Unknown Hyperchaotic Systems

The security of the information signal in digital communication systems is an active research problem. This paper proposes a new robust adaptive anti-synchronization control (RAASC) technique and studies the multi-switching combination anti-synchronization (MSCAS) of different unknown hyperchaotic (UH) systems. This controller reduces the overshoots in the transient phase and establishes the anti-synchronization convergence quickly. The proposed MSCAS scheme anti-synchronizes two master UH systems with a slave UH system using the multi-switching mechanism that increases the complexity and enhances the security of the information signal in digital communications. The Lyapunov direct method assures the robust stability of the closed-loop dynamical system. This work provides adaptive laws that estimate the true values of unknown parameters. A numerical example of three different hyperchaotic Pang, Lu and Lorenz–Stenflo systems is simulated. The simulation results verify the theoretical findings. The proposed MSCAS approach is used for the secret communication of digital messages using the chaotic masking technique. The receiver retrieves the information signal with high security and good precession qualitatively, under the effect of different types of disturbances. This paper also discusses the implications of controller constituents of the proposed RAASC algorithm that establishes fast convergence. The article suggests some future research problems related to this work.

Multistability in coupled different-dimensional dynamical systems

Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to design multistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents.

Controller design scheme for stabilization and synchronization of a class of chaotic and hyperchaotic systems in uncertain environment using SMC approach

This paper presents sliding mode control (SMC) technique for stabilization and synchronization of a class of chaotic/hyperchaotic systems in master–slave configuration. Here, a detailed procedure for deriving a single controller using SMC technique based proportional-integral sliding surface is proposed. Adaptation laws have been derived for the system parameters when the systems are subject to parametric uncertainties. Further, explicit criteria to decide the minimum number of control inputs required to meet out the desired operation of stabilization and synchronization is also proposed. Lyapunov stability theory is utilized to accomplish the desired objective. The proposed controller ensures the occurrence of sliding motion and achieves synchronization for the addressed class of chaotic/hyperchaotic systems in master–slave configuration. To validate the analytical results, example of four dimensional Lorenz–Stenflo hyperchaotic system is considered. Finally, detailed simulation results are provided to illustrate the effectiveness of the proposed controller.

NAC-based Synchronisation and Anti-synchronisation Between Hyperchaotic and Chaotic Systems, Its Analogue Circuit Design and Application

ABSTRACTThe objectives of this paper are synchronisation and anti-synchronisation between a hyperchaotic and a chaotic systems using nonlinear active control (NAC) scheme, its analogue circuit design and application to secure communication. Hyperchaotic Lorenz–Stenflo system is used as a master and the chaotic (a hyperchaotic system in its chaotic state) Chen system is used as a slave in this paper. Chaotic slave system is controlled to make it synchronised and anti-synchronised with the hyperchaotic master system by designing a modified NAC law via suitable forcing. Proposed scheme is applied in communication to ensure more security of the message signal through hyperchaotic masking. The MATLAB environment is used to simulate synchronisation, anti-synchronisation of the proposed hyperchaotic-chaotic pair using the modified NAC law and its application to secure communication application. The simulation results reflect the successful achievement of the objectives. Further, analogue circuit designs of the hyperchaotic Lorenz–Stenflo, the chaotic Chen systems as well as their synchronisation and anti-synchronisation of proposed pair using modified NAC are presented in this paper and realised in NI Multisim. NI Multisim circuit realisation results validate the simulation results and the fulfilment of the desired objectives of the paper.

Dynamical Behaviors of a Modified Lorenz–Stenflo Chaotic System

In this paper, the ultimate bound and globally exponentially attractive sets of a modified Lorenz–Stenflo system are studied based on the Lyapunov stability theory and optimization theory. Comparing with the best results in the current literature, our new results include the existing results as special cases. Furthermore, the new results offer a theoretical support to studying the Hausdorff dimension of attractor of this modified Lorenz–Stenflo system. These theoretical results are also important and useful for chaos control and chaos synchronization.

Analysis of a Generalized Lorenz–Stenflo Equation

Although the globally attractive sets of a hyperchaotic system have important applications in the fields of engineering, science, and technology, it is often a difficult task for the researchers to obtain the globally attractive set of the hyperchaotic systems due to the complexity of the hyperchaotic systems. Therefore, we will study the globally attractive set of a generalized hyperchaotic Lorenz–Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere in this paper. Based on Lyapunov-like functional approach combining some simple inequalities, we derive the globally attractive set of this system with its parameters. The effectiveness of the proposed methods is illustrated via numerical examples.

On the dynamics of a high-order Lorenz–Stenflo system

Results presented in a recent paper in this journal concerning a continuous-time dynamical system, namely that involving high-order Lorenz–Stenflo equations, are extended in this paper. More specifically, the present paper reports on nonlinear dynamics of a six-variable, four-parameter high-order Lorenz–Stenflo system. Six cross-sections of a four-dimensional parameter-space are considered. By using Lyapunov exponents spectra to characterize the dynamical behavior at each point of each of these plots, it is shown that different regions are allowed, from equilibrium point to chaos regions. It is also shown that hyperchaos is not an allowed behavior in a high-order Lorenz–Stenflo system. In addition, new results reported here are compared with those obtained for the original Lorenz–Stenflo system.

Zero‐zero‐Hopf bifurcation and ultimate bound estimation of a generalized Lorenz–Stenflo hyperchaotic system

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero‐zero‐Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Mixed Synchronization Scheme for Coupled Different Dimensional Dynamical Systems

In this paper, a scheme for mixed synchronization between two coupled different dimensional dynamical system is proposed. The sufficient conditions for mixed synchronization are derived analytically using Lyapunov stability theory. The proposed scheme is illustrated considering coupled Lorenz (three dimensional) and Lorenz–Stenflo (four dimensional) systems. Numerical simulation results are given to show the efficiency of the proposed scheme.

Spiral organization of periodic structures in the Lorenz–Stenflo system

This paper reports the existence of organized periodic structures embedded in chaotic regions of a parameter plane of the Lorenz–Stenflo system. More specifically, this work reports on spiral organization of periodic structures observed in the (σ, s) parameter plane of the Lorenz–Stenflo system.

Stability and periodicity of high-order Lorenz–Stenflo equations

In this paper, we derive high-order Lorenz–Stenflo equations with 6 variables and investigate periodic behaviors as well as stability of the equations. The stability of the high-order Lorenz–Stenflo equations is investigated by the linear stability analysis for various parameters. A periodicity diagram is also computed and it shows that the high-order Lorenz–Stenflo equations exhibit very different behaviors from the original Lorenz–Stenflo equations for both periodic and chaotic solutions. For example, period 3 regime for large parameters and scattered periodic regime are newly observed, and chaotic regimes exist for smaller values of r but for larger values of s than the original equations. In contrast, similarities such as the enclosure of the chaotic regime by the periodic regime or complex periodic regimes inside the chaotic regime are also observed for both the original and high-order Lorenz–Stenflo equations.