A preprocessing method for parameter estimation in ordinary differential equations

Parameter estimation for nonlinear differential equations is notoriously difficult because of poor or even no convergence of the nonlinear fit algorithm due to the lack of appropriate initial parameter values. This paper presents a method to gather such initial values by a simple estimation procedure. The method first determines the tangent slope and coordinates for a given solution of the ordinary differential equation (ODE) at randomly selected points in time. With these values the ODE is transformed into a system of equations, which is linear for linear appearance of the parameters in the ODE. For numerically generated data of the Lorenz attractor good estimates are obtained even at large noise levels. The method can be generalized to nonlinear parameter dependency. This case is illustrated using numerical data for a biological example. The typical problems of the method as well as their possible mitigation are discussed. Since a rigorous failure criterion of the method is missing, its results must be checked with a nonlinear fit algorithm. Therefore the method may serve as a preprocessing algorithm for nonlinear parameter fit algorithms. It can improve the convergence of the fit by providing initial parameter estimates close to optimal ones.     

Transition to chaos in a lorenz system via a complete double homoclinic bifurcation cascade

For the Lorenz system of equations we prove the existence of a complete double homoclinic attractor and determine the region in the parameter space where this attractor is observed. A scenario is proposed illustrating the transition to chaos in a Lorenz system via a complete double homoclinic bifurcation cascade, which produces a complete double homoclinic attractor in general different from the Lorenz attractor.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 179–194, 2002.     

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

四模Lorenz系统的动力学行为及其数值模拟

<正>1引言美国气象学家E.N.Lorenz在1963年提出的用来刻画热对流不稳定性的模型,即Lorenz混沌模型,后来人们对这个系统进行了大量而深入的研究.到目前为止,Lorenz系统的动力学行为已经比较清楚.1998年,Mischaikow和Mrozek用Conley指标理论和严格的计算机辅助计算方法证明了Lorenz系统存在混沌行为.1999年,Tucker则用规范型理     

Singular hyperbolic systems

We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.

     

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.     

Attractors synthesis for a Lotka-Volterra like system

The aim of this paper is to prove numerically, via computer graphic simulations, that the synthesis algorithmprovided by Danca et al. [1] can be utilized to synthesize any attractor of a dynamical system modeling a two-predator, one prey Lotka-Volterra like system. The algorithm switches in a periodic deterministic or a random way the control parameter inside a set of a chosen values. The obtained attractor is the same with the attractor obtained for parameter value taken as averaged value of the switched control values. This simple and effective algorithm relies on a convex property induced in the set of the attractors corresponding to the chosen switching parameters. The algorithm was tested successfully on systems depending linearly on the control parameter like Lorenz, Chen, Rossler, networks and other systems.     

Bifurcation of infinite Prandtl number rotating convection

We consider infinite Prandtl number convection with rotation which is the basic model in geophysical fluid dynamics. For the rotation free case, the rigorous analysis has been provided by Park (2005, 2007, revised for publication) [5], [6] and [25] under various boundary conditions. By thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u=0 to an attractor Σ_R which consists of only one cycle of steady state solutions and is homeomorphic to S¹. We also see how intensively the rotation inhibits the onset of convective motion. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation which was developed by Ma and Wang (2005); see [15].     

Attractivity,multistability, and bifurcation in delayed Hopfieldʼs model with non-monotonic feedback

For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients.