The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II

We provide a qualitative analysis of the -dimensional dynamical system:

where is an arbitrary positive integer. Under mild algebraic conditions on the constant matrix , we show that every solution , , extends to a solution on , such that , for . Moreover, the difference between any two solutions approaches as . We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.

     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

Identification of Linear Error-Models with Projected Dynamical Systems

Linear error models are an integral part of several parameter identification methods for feedforward and feedback control systems and lead in connection with the L ₂-norm to a convex distance measure which has to be minimised for identification purposes. The parameters are hereby often subject to specific restrictions whose intersections span a convex solution set with non-differentiability points on its boundary. For solving these well conditioned problems on-line the paper formulates the solution of the bounded convex minimisation problem as a stable equilibrium set of a proper system of differential equations. The vector field of the corresponding system of differential equations is based on a projection of the negative gradient of the distance measure. A general drawback of this approach is the discontinuous right-hand side of the differential equation caused by the projection transformation. The consequence are difficulties for the verification of the existence, uniqueness and stability of a solution trajectory. Therefore the first subject of this paper is the derivation of an alternative formulation of the projected dynamical system, which exhibits, in contrast to the original formulation, a continuous right-hand side and is thus accessible to conventional analysis methods. For this purpose the multi-dimensional stop operator is used and the existence, uniqueness and stability properties of the solution trajectories are established. The second part of this paper deals with the numerical integration of the projected dynamical system which is used for an implementation of the identification method on a digital signal processor for example. To demonstrate the performance the application of this on-line identification method to the hysteretic filter synthesis with the modified Prandtl-Ishlinskii approach is presented in the last part of this paper.     

Persistent oscillations in a threshold adjustment model

In this paper we present a simple time-continuous behavioural model of habit formation. Addictive behaviour is damped by a threshold which adapts itself to the habit. This adaptive behaviour of the threshold may lead to periodic fluctuations of the consumption rate, the habit and the threshold. It turns out that both a low adjustment rate of the threshold as well as a steep consumption function favour oscillatory patterns.     

Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious diseases with periodic contact rates and disease-dependent demographic factors in the population

Within the framework of SEIR-like epidemic models, we studied the conditions for the stable eradication of some families of vertically and horizontally transmitted infectious diseases in the case of periodically varying contact rate. By means of Floquet’s theory, we found a condition for the eradication solution to be locally asymptotically stable. We then demonstrated that the same condition guarantees also that this vaccine-induced disease-free solution is globally asymptotically stable. A model with interacting populations is also considered. In the final part of this work, we extended the model by taking into account the variation of population size, the impact of disease-related deaths and reduction of fertility.     

A mathematical model for Babesiosis disease in bovine and tick populations

In this paper, we analyze the Babesiosis transmission dynamics on bovine and tick populations. Ticks play a role of infectious agents and vector of the protozoan Babesia hemo‐parasite. In this sense, we set out a mathematical model with constant size population for the evolution of the infected bovines with Babesiosis and analyze its qualitative dynamics. Statistical data are used to estimate some of the parameters of the model. Numerical simulations of the model varying the parameters show different scenarios about the spread of the disease. Copyright © 2011 John Wiley & Sons, Ltd.     

Controlled random fields,von Neumann–Gale dynamics and multimarket hedging with risk

We develop a model of asset pricing and hedging for interconnected financial markets with frictions – transaction costs and portfolio constraints. The model is based on a control theory for random fields on a directed graph. Market dynamics are described by using von Neumann–Gale dynamical systems first considered in connection with the modelling of economic growth [13,24]. The main results are hedging criteria stated in terms of risk-acceptable portfolios and consistent price systems, extending the classical superreplication criteria formulated in terms of equivalent martingale measures.     

Real Lax spectrum implies spectral stability

We consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset of the real line that gives rise to stable eigenvalues is contained in the Lax spectrum. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and it maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples, both in and not in the AKNS hierarchy.     

Parties and networks: Stochastic models for relationship networks

This paper argues that parties and other gatherings are important for the development of friendship networks. It proposes a stochastic model for the evolution of networks over time, the distinctive feature of which is the party event. A party event occurs when a person in the network has a gathering and invites all of his/her friends who then also become friends. The Party Models discussed are all based upon this simple assumption. After formulating basic assumptions, various differential equations describing Party Models are derived. Subsequently several concepts useful for model analysis are defined and briefly explored. These include the concepts of potential, equivalence class form, and degenerate models. The penultimate section considers models for three person networks in some detail and with numerical illustrations. All extended mathematical arguments are placed in the Mathematical Appendix.     

Modelling and estimation of social interaction effects in new product diffusion

We present a new product choice model in which individual agents are assumed to interact with each other across spatial hierarchies under random Gaussian laws. A restricted form that allows estimation of the model and inference on the random interaction structure on available data is then derived. This is essentially a random coefficients model which we estimate on repeated cross-sectional data for the adoption of video cassette recorders in Britain. It is argued that the random coefficient approach, based on economic and social-theoretic arguments allows for more accurate inference on the structure of social interactions, and thus may provide a useful way to help alleviate the inferential problem caused by the identification issues stated in the literature. This very same problem has for long been at the core of the debate between alternative economic and marketing theories of new product diffusion.     

Asymptotic and Lyapunov stability of constrained and Poisson equilibria

This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in constrained dynamical systems. First, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account the asymptotically stable behavior that may occur in certain constrained systems, such as Poisson and Leibniz dynamical systems. Second, this method is specifically adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained mechanical systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. (Arch. Rational Mech. Anal., 2004, preprint arXiv:math.DS/0201239v1, to appear).     

Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)_k?0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk?0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)_k?0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.     

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?_tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?_α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?_α,ε from a proper lifting of ?_α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??_α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

Synchronization of Stochastic Two-Layer Geophysical Flows

In this paper, the two-layer quasigeostrophic flow model under stochastic wind forcing is considered. It is shown that when the layer depth or density differ- ence across the layers tends to zero, the dynamics on both layers synchronizes to an averaged geophysical flow model.     

THE RELATION OF DIMENSION,ENTROPY AND LYAPUNOV EXPONENT IN RANDOM CASE

We consider random systems generated by two-sided compositions of random surface diffeomorphisms, together with an ergodic Borel probability measure μ. Let D（μω） be its dimension of the sample measure, then we prove a formula relating D（μω） to the entropy and Lyapunov exponents of the random system, where D （μω） is dimHμω, dimBμm, or dimBμm.     

Poincaré Normal and Renormalized Forms

We briefly review the main aspects of (Poincaré–Dulac) normal forms; we have a look at the nonuniqueness problem, and discuss one of the proposed ways to further reduce the normal forms. We also mention some convergence results.     

A Cl closing lemma for equivariant endomorphisms under actions of finite groups

A C^l closing lemma is proved for equivariant endomorphisms under actions of finite groups. Our result shows that, for such an endomorphism, a nonwandering orbit together with its symmetric conjugates can be closed up under a Cl-small equivariant perturbation Project partially supported by the National Natural Science Foundation of China (Grant No. 19531070).     

Hypercyclic algebras

We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.     

Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.