Connecting the Lorenz and Chen systems via nonlinear control

The Lü system is a new chaotic system, which connects the Lorenz system and the Chen system and represents the transition from one to the other. In this letter, based on the concept of nonresonant parametric perturbations, further detailed analysis about the forming mechanism and its compound structure for the chaotic Lü system are offered. The obtained results clearly reveal the intermediate chaotic system has another novel forming mechanism: the compression and pull forming mechanism, which provides an enlighten insight about the relationship of its vibration “mode” and the two-scroll “base” chaotic attractor. Then motivated by this novel forming mechanism, by adding a simple nonlinear term to the Lü system, its role as a joint function is revisited. With the gradual tuning the parameter of the nonlinear controller, the transition from the canonical Lorenz attractor to the Chen attractor through the Lü attractor is revived. The scheme herein goes beyond the traditional framework for studying the Lorenz-like systems, which can be very helpful in generating and analyzing of all similar and closely related chaotic systems.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Optimization in “self‐modeling” complex adaptive systems

When a dynamical system with multiple point attractors is released from an arbitrary initial condition, it will relax into a configuration that locally resolves the constraints or opposing forces between interdependent state variables. However, when there are many conflicting interdependencies between variables, finding a configuration that globally optimizes these constraints by this method is unlikely or may take many attempts. Here, we show that a simple distributed mechanism can incrementally alter a dynamical system such that it finds lower energy configurations, more reliably and more quickly. Specifically, when Hebbian learning is applied to the connections of a simple dynamical system undergoing repeated relaxation, the system will develop an associative memory that amplifies a subset of its own attractor states. This modifies the dynamics of the system such that its ability to find configurations that minimize total system energy, and globally resolve conflicts between interdependent variables, is enhanced. Moreover, we show that the system is not merely “recalling” low energy states that have been previously visited but “predicting” their location by generalizing over local attractor states that have already been visited. This “self‐modeling” framework, i.e., a system that augments its behavior with an associative memory of its own attractors, helps us better understand the conditions under which a simple locally mediated mechanism of self‐organization can promote significantly enhanced global resolution of conflicts between the components of a complex adaptive system. We illustrate this process in random and modular network constraint problems equivalent to graph coloring and distributed task allocation problems. © 2010 Wiley Periodicals, Inc. Complexity 16: 17–26, 2011     

Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes is proposed. The strongly coupled equations for the ion concentrations include drift terms involving the electric potential, which is coupled to the concentrations through the Poisson equation. The cross‐diffusion system possesses a formal gradient‐flow structure revealing nonstandard degeneracies, which lead to considerable mathematical difficulties. The finite‐volume scheme is based on two‐point flux approximations with “double” upwind mobilities. The existence of solutions to the fully discrete scheme is proved. When the particles are not distinguishable and the dynamics is driven by cross diffusion only, it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation. The degeneracy is overcome by proving a new discrete Aubin–Lions lemma of “degenerate” type. Numerical simulations of a calcium‐selective ion channel in two space dimensions show that the scheme is efficient even in the general case of ion transport.     

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.     

Bayesian optimal life‐testing plan under the balanced two sample type‐II progressive censoring scheme

Joint progressive censoring schemes are quite useful to conduct comparative life‐testing experiment of different competing products. Recently, Mondal and Kundu (“A New Two Sample Type‐II Progressive Censoring Scheme,” Commun Stat‐Theory Methods; 2018) introduced a joint progressive censoring scheme on two samples known as the balanced joint progressive censoring (BJPC) scheme. Optimal planning of such progressive censoring scheme is an important issue to the experimenter. This article considers optimal life‐testing plan under the BJPC scheme using the Bayesian precision and D‐optimality criteria, assuming that the lifetimes follow Weibull distribution. In order to obtain the optimal BJPC life‐testing plans, one needs to carry out an exhaustive search within the set of all admissible plans under the BJPC scheme. However, for large sample size, determination of the optimal life‐testing plan is difficult by exhaustive search technique. A metaheuristic algorithm based on the variable neighborhood search method is employed for computation of the optimal life‐testing plan. Optimal plans are provided under different scenarios. The optimal plans depend upon the values of the hyperparameters of the prior distribution. The effect of different prior information on optimal scheme is studied.     

Nonstandard methods for the convective‐dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian‐Lagrangian method is constructed for the one‐dimensional, transient convective‐dispersive transport equation with nonlinear reaction terms. An “exact” difference scheme is applied to the convection‐reaction part of the equation to produce a semi‐discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection‐dominated, reactive transport problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617–624, 1999     

Long—time behaviour of strong solutions of retarded nonlinear P.D.E.s

We prove the existence of strong solutions for a class of retarded partial differential equations of second order with respect to the time variable, and study the long-time behaviour of these solutions. We prove the existence of a global finite-dimensional attractor when the parameters of the system range over a “large” domain and investigate the dependence of the attractor on these parameters. MOS subject classification: 58F39, 58F12, 35B40, 73K70.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Bifurcation Behavior of a 1DOF Sliding Friction Oscillator

This article deals with different types of friction models and their influence on the behavior of a simple 1 degree-of-freedom (DOF) sliding friction oscillator which is in literature commonly referred to as “mass-on-a-belt”-oscillator. The examined friction characteristics are assumed to be proportional to the applied normal force and only dependend on the relative velocity between the mass and the belt. For an exponential and a generalized cubic friction characteristic, the linear stability of the steady-state and the bifurcation behavior in the sliding domain are examined. It is shown that the resulting phase plots of the observed system are strongly dependent on the chosen friction characteristic. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long time behavior of solutions of Davey-Stewartson equations

1. IntroductionIn the present paper we study the following Davey-Stewartson systemsupplemented with boundary conditionsand initial conditionwhere a = al la2, b = hi fo2, p = gi iap2, 7 = 71 iap and X = FI ixZ are complexconstallts, fi C RZ is a smooth bounded domain. The system was derived by Davey etalll] to model the evolution of a three-dimensional disturbance in the nonlinear regime ofplane Poiseuille flow (fully developed steady flow under a constallt pressure gradient betwe…     

On a Boussinesq Capillarity System: Hamiltonian Reductions and Associated Quartic Geometries

A nonlinear Boussinesq‐type capillarity model system is shown to admit exact reductions to coupled Hamiltonian subsystems associated with time‐modulated quartic density distributions. In particular, in 3+1 dimensions, time‐modulated “red blood cell” geometries for the density distribution are isolated.     

Irreducible representations of finite exceptional groups of lie type containing matrices with simple spectra

Absolutely irreducible representations of finite groups of exceptional Lie types in defining characteristic whose images contain matrices with simple spectra are determined. The term ”simple spectrum“ means that each eigenvalue has multiplicity 1. The similar question for the classical finite groups has been solved in the authors' previous paper [Comm. in Algebra 26 (1998), no 3, 863-888] where one can find general comments to the problem. For dimensions ≥ 100 all absolutely irreducible representations of finite groups of Lie type in defining characteristic containing matrices with simple spectra are tabulated.     

Psychometric Re‐evaluation of the Image of Science and Scientists Scale (ISSS)

An enduring concern among science education researchers is the “swing away from science” ( Osborne. 2003 ). One of their central dilemmas is to identify—or construct—a valid outcome measure that could assess curricular effectiveness, and predict students' choices of science courses, university majors, or careers in science. Many instruments have been created and variably evaluated. The primary purpose of this paper was to re‐evaluate the psychometric properties of the Image of Science and Scientists Scale (ISSS) ( Krajkovich 1978 ). In the current study, confirmatory factor analysis (CFA) was used to examine the dimensionality of the 29‐item ISSS, which was administered to 531 middle school students in three San Antonio. Texas school districts at the beginning of the 2004–2005 school year. The results failed to confirm the presumed 1‐factor structure of the ISSS. but instead showed a 3‐factor structure with only marginal fit with the data, even after removal of 12 inadequate items. The three dimensions were “Positive Images of Scientists” (5 items). “Negative Images of Scientists” (9 items), and “Science Avocation” (3 items). The results do not support use of the original form of the ISSS for measuring “attitudes toward science,”“images of scientists. “or “scientific attitudes. “Shortening the scale from 29 to 17 items makes it more feasible to use in a classroom setting. Determining whether the three dimensions identified in our analysis. “Positive Images of Scientists. ““Negative Images of Scientists. “and “Science Avocation “contain useful assessments of middle school student impressions and attitudes will require independent investigation in other samples.     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

A high‐order compact boundary value method for solving one‐dimensional heat equations

We combine fourth‐order boundary value methods (BVMs) for discretizing the temporal variable with fourth‐order compact difference scheme for discretizing the spatial variable to solve one‐dimensional heat equations. This class of new compact difference schemes achieve fourth‐order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second‐order Crank‐Nicolson scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 846–857, 2003.     

A preliminary investigation on an ELLAM scheme for linear transport equations

We present an Eulerian‐Lagrangian localized adjoint method (ELLAM) for linear advection‐reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to investigate the performance of the ELLAM scheme with a range of well‐perceived and widely used methods in fluid dynamics including the monotonic upstream‐centered scheme for conservation laws (MUSCL), the minmod method, the flux‐corrected transport method (FCT), and the essentially non‐oscillatory (ENO) schemes and weighted essentially non‐oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is very competitive with these methods in the context of linear transport PDEs, and suggest/justify the development of ELLAM‐based simulators for subsurface porous medium flows and other applications. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 22–43, 2003     

Characteristic exponents of semialgebraic singularities

Let X be a semialgebraic (or algebraic) set and let x₀ ∈ X be a singular point. There are some topological cycles of different dimensions contained in a small neighbourhood of x₀ in X. All these cycles vanish in x₀. The paper is devoted to “vanishing rates” of these cycles, which we call “characteristic exponents”. We prove that the characteristic exponents are invariant under bi‐Lipschitz transformations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The computational complexity of two‐state spin systems

The subject of this article is spin‐systems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hard‐core gas model. There are three degrees of freedom, corresponding to our parameters β, γ, and μ. Informally, β represents the weights of edges joining pairs of “spin blue” sites, γ represents the weight of edges joining pairs of “spin green” sites, and μ represents the weight of “spin green” sites. We study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case μ = 1. Exact computation of the partition function Z is NP‐hard except in the trivial case βγ = 1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the “ferromagnetic” region βγ ≥ 1, but (unless RP = NP) there is no FPRAS in the “antiferromagnetic” region corresponding to the square defined by 0 < β < 1 and 0 < γ < 1. Neither of these “natural” regions—neither the hyperbola nor the square—marks the boundary between tractable and intractable. In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola. In the other direction, we exhibit two tiny, symmetric, intractable regions extending beyond the antiferromagnetic region. We also extend our results to the asymmetric case μ ≠ 1. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 133–154, 2003