Numerical stabilization of the Lorenz system by a small external perturbation

The Lorenz system perturbed by noise and its invariant measure whose density obeys the stationary Fokker-Planck equation are analyzed numerically. A linear functional of the invariant measure is considered, and its variation caused by a variation in the right-hand side of the Lorenz system is calculated. A small (in modulus) external perturbation is calculated under which the strange attractor of the Lorenz system degenerates into a stable fixed point.     

Numerical modeling of the transmission dynamics of drug-sensitive and drug-resistant HSV-2

IntroductionThe prevalence of genital herpes infection (caused by HSV-2) in the United States is approximately 22% II]. In developing nations, the infection rate is between 40%A10% I2' 3]. Althoughthere is currently no cure for genital herpes, drug treatment with agents such as acyclovir isknown to significantly relieve symptoms and reduce viral shedding['1. Most cases of HSV-2infections remain untreated in developing nations, and even in the USA only 5%--10% aretreated[4]. Whilst incre…     

The dynamics of perturbations of the contracting Lorenz attractor

We show here that by modifying the eigenvalues ₂ < ₃ < 0 < ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition ₃+₁ > 0 by acontracting condition ₃+₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.     

Dynamical analysis of the hyperchaos Lorenz system

This article deals with the ultimate bound on the trajectories of the hyperchaos Lorenz system based on Lyapunov stability theory. The innovation of this article lies in that the method of constructing Lyapunov functions applied to the former chaotic systems is not applicable to this hyperchaos system, and moreover, one Lyapunov function can not estimate the bounds of this hyperchaos Lorenz system. We successfully estimate the bounds of this hyperchaos system by constructing three generalized Lyapunov functions step by step. Some computer simulations are also given to show the effectiveness of the proposed scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 440–445, 2016     

A universal unfolding of the Lorenz system

A universal unfolding of the Lorenz system is derived and studied in this paper. Both rigorous theoretical analysis and numerical simulations show that the Lorenz system, the Chen system, and the Lü system belong to the same universal unfolding. Therefore, they all have similar dynamical behaviors in the sense that if the Lorenz system has limit cycles produced from a Hopf bifurcation for a certain set of parameter values, then the other two systems also have limit cycles from the same set of parameter values; and if the Lorenz, Chen, and Lü systems are chaotic for some parameter values (for example, some typical parameter values), respectively, then the homotopic system for the Lorenz system and the Chen system, and the homotopic system for these three systems, are all chaotic within the entire domain of these homotopic parameters.     

Diffusive Lorenz dynamics: Coherent structures and spatiotemporal chaos

In this paper, we are interested in collective behaviors of many interacting Lorenz strange attractors. With an intermediate diffusion coupling between the attractors, a new remarkable synchronization of well organized structures merges as a result of two competing mechanisms: temporal chaos and spatial diffusive stabilization. A window of the coupling parameter for coherent structures is found numerically. Different from all existing scenarios of routes to chaos (period doubling, intermittency and strange attractors), an algorithmetic increase of wavenumbers before an abrupt change to chaos (compared to the periodic doubling geometrical) is unexpectedly discovered. Meta-stable states axe also observed in simulations.     

Computational fluid dynamics based dynamic modeling of parafoil system

The calculation of aerodynamic coefficients has been one of the key issues when modeling parafoil systems, that directly affects model precision. This study relates to investigate limitations of traditional calculation methods. As a result, we achieve aerodynamic parameters of a parafoil using computational fluid dynamics simulations. Also we employ the least square method as a tool for the rapid identification of deflection factors of aerodynamic coefficients. The estimated aerodynamic coefficients of the system were incorporated into the dynamic equations of the parafoil to implement a six degree of freedom model of a parafoil system according to the Kirchhoff equations. Numerical results generated by simulation and airdrop testing demonstrate that the established model can accurately describe the flight characteristics of the parafoil system.     

Invariant algebraic surfaces of the generalized Lorenz system

In this paper, according to the idea of the weight of a polynomial introduced by Swinnerton-Dyer(Math Proc Camb Philos Soc 132:385–393, 2002), we successfully find all the invariant algebraic surfaces of the generalized Lorenz system x′ = a(y ? x), y′ = bx + cy ? xz, z′ = xy + dz.     

Hydrodynamical limit for a drift-diffusion system modeling large-population dynamics

In this paper we study the stability of the following nonlinear drift-diffusion system modeling large population dynamics ∂_tρ+div(ρU−ε∇ρ)=0, divU=±ρ, with respect to the viscosity parameter ε. The sign in the second equation depends on the attractive or repulsive character of the field U. A proof of the compactness and convergence properties in the vanishing viscosity regime is given. The lack of compactness in the attractive case is caused by the blow-up of the solution which depends on the mass and on the space dimension. Our stability result is connected, depending of the character of the potentials, with models in semiconductor theory or in biological population.     

Three control strategies for the Lorenz chaotic system

This paper suggests three strategies of the dislocated feedback control, so that enhancing feedback control and speed feedback control of the Lorenz chaotic system to its unstable equilibrium points can be enhanced. When the coefficients of enhancing feedback control and speed feedback control are smaller than those of ordinary feedback control, the complexity and cost of the system control are reduced. Theoretical analysis and numerical simulation are given, revealing the effectiveness of these strategies.     

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.     

Chaos in a generalized Lorenz system

A three-component dynamic system describing a quantum cavity electrodynamic device with a pumping and nonlinear dissipation is studied. Various dynamical regimes are investigated in terms of divergent trajectories approaches and fractal statistics. It has been shown that stable and unstable dissipative structures type of limit cycles can be formed in such system, with variation of pumping and nonlinear dissipation rates. Transitions to chaotic regime and the corresponding chaotic attractor are studied in detail.     

New estimate the bounds for the generalized Lorenz system

The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Coexistence of anti-phase and complete synchronization in the generalized Lorenz system

This paper investigates a class of new synchronization phenomenon. Some control strategy is established to guarantee the coexistence of anti-phase and complete synchronization in the generalized Lorenz system. The efficiency of the control scheme is revealed by some illustrative simulations.     

Dynamics and bifurcation study on an extended Lorenz system

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.     

Accuracy of the Adomian decomposition method applied to the Lorenz system

In this paper, the Adomian decomposition method (ADM) is applied to the famous Lorenz system. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the fourth-order Runge–Kutta (RK4) numerical solutions are made for various time steps. In particular we look at the accuracy of the ADM as the Lorenz system changes from a non-chaotic system to a chaotic one.     

Navier-Stokes方程五模类Lorenz方程组的动力学行为及数值仿真

本文对平面正方形区域上不可压缩的Navier-Stokes方程,进行傅立叶展开后,截断得到五模类Lorenz方程组.给出了该方程组定常解及其稳定性的讨论,证明了该方程组吸引子的存在性,并对其全局稳定性进行了分析和讨论,数值模拟了雷诺数在一定范围内变化时,类Lorenz方程组的动力学行为.     

Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem.     

Numerical analysis of a parabolic variational inequality system modeling biofilm growth at the porescale

In this article, we consider a system of two coupled nonlinear diffusion–reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions.