On the Boussinesq/Full dispersion systems and Boussinesq/Boussinesq systems for internal waves

In this paper, we consider some asymptotic models for internal waves in the small amplitude/small amplitude regime, which were derived recently by Bona, Lannes and Saut. We first prove that the Boussinesq/Full dispersion systems and the Boussinesq/Boussinesq systems can be derived from the Full dispersion/Full dispersion systems. Then using a contraction-mapping argument and the energy method, we will prove that the derived systems that are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. In particular, we improve and extend some known results on the well-posedness of Boussinesq systems for surface waves.     

Models of p-adic mechanics

We segregate the class of ultrametric (p-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincaré recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum p-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.     

A note on random walks with absorbing barriers and sequential Monte Carlo methods

In this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.     

Continuous Limits of Classical Repeated Interaction Systems

We consider the physical model of a classical mechanical system (called “small system”) undergoing repeated interactions with a chain of identical small pieces (called “environment”). This physical setup constitutes an advantageous way of implementing dissipation for classical systems; it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions (Attal and Pautrat in Ann Henri Poincaré 7:59–104, 2006), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations (SDEs) in the limit. In particular, we recover the usual Langevin equations associated with the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton’s equations and the repeated interactions. We embed it into a continuous-time dynamical system and compute the limit when the time step goes to 0. This way, we obtain a discrete-time approximation of SDE, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the L ^p and almost sure convergence of this scheme. We end up with applications to concrete physical examples such as a charged particle in a uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.     

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

Binding of atoms and stability of molecules in hartree and thomas—fermi type theories

This paper is the sequel of a previous work where we showed a general necessary and sufficient condition for the stability of an arbitrary molecular system (possibly ionized) in the framework of Hartree or Thomas-Fermi type theories. This condition, roughly speaking, meant that certain particular subsystems have to be bound. We show here in particular that this condition reduces for general molecular system with nonnegative excess charge to the binding of all subsystems with the same property. For neutral inolecular systems, this reduces to the binding of all neutral subsystems. In both cases, all other subsystems can be bound. We also show that, for the Hartree-Fock and Hartree models, this condition involves only “physical” sulxystems We use these reduced conditions to conclude allout the stability or the binding in some particular cases. This work 1s also the second of a series devoted to these equations and we shall come back on the binding of neutral systems in Part 3.     

Upper bound for the Hausdorff dimension of invariant sets of evolution variational inequalities

We consider the method of determining observations for obtaining an upper bound for the fractal dimension and the Hausdorff dimension of invariant sets of variational inequalities. We suggest a process for constructing determining observations, in particular, for dissipativity, with the use of frequency theorems for evolution systems (the Likhtarnikov–Yakubovich theorem). As an example, we consider a viscoelasticity problem in mechanics.     

Computation of the Delta in Multidimensional Jump-Diffusion Setting with Applications to Stochastic Volatility Models

We study the robustness of options prices to model variation in a multidimensional jump-diffusion framework. In particular, we consider price dynamics in which small variations are modeled either by a Poisson random measure with infinite activity or by a Brownian motion. We consider both European and Exotic options and we study their deltas using two approaches: the Malliavin method and the Fourier method. We prove robustness of the deltas to model variation. We apply these results to the study of stochastic volatility models for the underlying and the corresponding options.     

Mathematical modeling of seepage flows in coastal pressure water-bearing strata

We study mathematical models of certain two-dimensional steady-state seepage flows of sweet groundwater. We assume that the flows go out of a rectangular pressure water-bearing stratum and enter a sea with saline water, which has a water sheet above its surface. On the base of these models we propose algorithms, calculating the intrusion of sea saline water into shoaling sweet water strata in cases, when flows of groundwater enter the sea sidewise (the scheme of P. Ya. Polubarinova-Kochina and M. K. Mikhailov) or from below (the scheme of J. Bear and G. Dagan). Using exact analytic dependencies established with the help of the Polubarinova-Kochina method and numerical computations, we study the influence of all physical parameters of the models onto the type of the intrusion and its degree. In a particular case, with no fresh water stratum over the sea surface, we compare the results for both inflow schemes and consider the intrusion specificity, depending on the initial location of fluids interface. We study the limit cases of flows. We prove that a certain particular choice of values of unknown parameters of the conformal mapping, which enter in a solution, leads to the well-known results of P. Ya. Polubarinova-Kochina for the classical filtration problem in a rectangular dam.     

Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems

We consider nearly-integrable systems under a relatively small dissipation. In particular we investigate two specific models: the discrete dissipative standard map and the continuous dissipative spin-orbit model. With reference to such samples, we review some analytical and numerical results about the persistence of invariant attractors and of periodic attractors.      

Using Dynamic Geometry Software to Simulate Physical Motion

In this paper we analyse to what extent the computational model of the geometry implemented in a dynamic geometry environment provides models for physical motion, focusing on the continuity issues related to motion. In particular, we go over the utility of dynamic geometry environments to simulate the motion of mechanical linkages, as this activity allows us to compare, by means of dynamic drawings, the computable representation of geometric properties with the real motion of a mechanism. Analysing a simple example, we provide theoretical foundations for particular behaviours observed in the motion of a picture on the screen, which require a subtle interpretation to be understood in a purely physical context. In this way, we reflect on some requirements imposed by the computable representation of knowledge. We consider this work to be a necessary step to determine didactic consequences related to students' perceptions of the moving displays; in particular those concerning the uses of the dragging mode as a tool not only for automatic drawing of many instances of a construction,but also to produce continuous motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized synchronization of strictly different systems: Partial-state synchrony

Generalized synchronization (GS) occurs when the states of one system, through a functional mapping are equal to states of another. Since for many physical systems only some state variables are observable, it seems convenient to extend the theoretical framework of synchronization to consider such situations. In this contribution, we investigate two variants of GS which appear between strictly different chaotic systems. We consider that for both the drive and response systems only one observable is available. For the case when both systems can be taken to a complete triangular form, a GS can be achieved where the functional mapping between drive and response is found directly from their Lie-algebra based transformations. Then, for systems that have dynamics associated to uncontrolled and unobservable states, called internal dynamics, where only a partial triangular form is possible via coordinate transformations, for this situation, a GS is achieved for which the coordinate transformations describe the functional mapping of only a few state variables. As such, we propose definitions for complete and partial-state GS. These particular forms of GS are illustrated with numerical simulations of well-known chaotic benchmark systems.     

Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

We derive several algorithms, including quadratically convergent algorithms, which can be used to calculate the Laplace–Stieltjes transforms of the time taken to return to the initial level in the Markovian stochastic fluid flow model. We give physical interpretations of the algorithms and consider their numerical analysis. The numerical performance of the algorithms, which depends on the physical properties of the process, is discussed and illustrated with simple examples. Besides the powerful algorithms, this paper contributes interesting theoretical results. In particular, the methodology for constructing these algorithms is a valuable contribution to the theory of fluid flow models. Moreover, useful physical interpretations of the algorithms, and related expressions, given in terms of the fluid flow model, can assist in further analysis and help in a better understanding of the model. The authors would like to thank the Australian Research Council for funding this research through Discovery Project DP0770388.     

Systems of partial differential equations of mixed hyperbolic-parabolic type

We prove the global existence and uniqueness of solutions of certain mixed hyperbolic-parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier-Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new.     

Explicit construction of chaotic attractors in Glass networks

Chaotic dynamics have been observed in example piecewise-affine models of gene regulatory networks. Here we show how the underlying Poincaré maps can be explicitly constructed. To do this, we proceed in two steps. First, we consider a limit case, where some parameters tend to ∞, and then consider the case with finite parameters as a perturbation of the previous one. We provide a detailed example of this construction, in 3-d, with several thresholds per variable. This construction is essentially a topological horseshoe map. We show that the limit situation is conjugate to the golden mean shift, and is thus chaotic. Then, we show that chaos is preserved for large parameters, relying on the structural stability of the return map in the limit case. We also describe a method to embed systems with several thresholds into binary systems, of higher dimensions. This shows that all results found for systems having several thresholds remain valid in the binary case.     

Some combinatorial aspects of discrete non-linear population dynamics

Motivated by issues arising in population dynamics, we consider the problem of iterating a given analytic function a number of times. We use the celebrated technique known as Carleman linearization that turns (for a certain class of functions) this problem into simply taking the power of a real number. We expand this method, showing in particular that it can be used for population models with immigration, and we also apply it to the famous logistic map. We also are able to give a number of results for the invariant density of this map, some being related to the Carleman linearization.     

Integrability of Exceptional Hydrodynamic-Type Systems

We consider non-diagonalizable hydrodynamic-type systems integrable by the extended hodograph method. We restrict the analysis to non-diagonalizable hydrodynamic reductions of the three-dimensionalMikhalev equation. We show that families of these hydrodynamictype systems are reducible to the heat hierarchy. Then we construct new particular explicit solutions for the Mikhalev equation.     

On the general nonlinear self-adjoint spectral problem for differential-algebraic systems

We consider a general self-adjoint spectral problem, nonlinear with respect to the spectral parameter, for linear differential-algebraic systems of equations. Under some assumptions, we present a method for reducing such a problem to a general self-adjoint nonlinear spectral problem for a system of differential equations. In turn, this permits one to pass to a problem for a Hamiltonian system of ordinary differential equations. In particular, in this way, one can obtain a method for computing the number of eigenvalues of the original problem lying in a given range of the spectral parameter.     

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L ₁. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations, we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models (parallel case) in three dimensions.