Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.

For dynamical systems governed by feedback laws, time delays arise naturally in the feedback loop to represent effects due to communication, transmission, transportation or iternia effects. The introduction of time delays in a system of differential equations results in an infinite dimensional state space. We give an introduction to the basic theory of delay equations. After this introduction we illustrate the theory with a local analysis of the Lang-Kobayashi equations describing a laser with delayed optical feedback.

Systems with infinite dimensional state space: The Hilbert space approach

Inventory Management with Stochastic Lead Times

This article analyzes a continuous time back-ordered inventory system with stochastic demand and stochastic delivery lags for placed orders. This problem in general has an infinite dimensional state space and is hence intractable. We first obtain the set of minimal conditions for reducing such a system's state space to one dimension and show how this reduction is done. Next, by modeling demand as a diffusion process, we reformulate the inventory control problem as an impulse control problem. We simplify the impulse control problem to a Quasi-Variation Inequality (QVI). Based on the QVI formulation, we obtain the optimality of the (s, S) policy and the limiting distribution of the inventory level. We also obtain the long run average cost of such an inventory system. Finally, we provide a method to solve the QVI formulation. Using a set of computational experiments, we show that significant losses are incurred in approximating a stochastic lead-time system with a fixed lead-time system, thereby highlighting the need for such stochastic lead-time models. We also provide insights into the dependence of this value loss on various problem parameters.

In this paper we examine optimization problems involving multidimensional nonsmooth integral functionals defined on Sobolev spaces. We obtain necessary and sufficient conditions for optimality in convex, finite dimensional problems using techniques from convex analysis and in nonconvex, finite dimensional problems, using the subdifferential of Clarke. We also consider problems with infinite dimensional state space and we finally present two examples.

In the literature concerning Dirichlet forms and its applications, closability plays a crucial role. In fact, closedness is one of the defining properties of a Dirichlet form. According to this, a number of closability criterions are known in particular cases. An important question is under which conditions closability is kept after changing the reference measure. M. Fukushima, K. Sato, and S. Taniguchi [5] treated this problem for a regular Dirichlet form (E (E )) which is defined on a locally compact separable metric state space. Under technical conditions on some core C ⊆ (E ), they presented a complete solution if the Dirichlet form is either irreducible or transient. An earlier paper dealing with this subject is M. Rockner and N. Wielens [13]. Related results on Lusinean separable metric spaces were published in I. Shigekawa and S. Taniguchi [16]. The aim of this paper is to give general analytical conditions in order to keep closability when turning to a new reference measure. One particular purpose is to present an extension of an assertions in [5] (namely, Corollary 4.2) within a purely measure theoretic framework, i.e., the state space ( B) is just a measurable space. In particular, the set C is defined exclusively in terms of the initial form (E (E )) on 2( μ). The main results are Theorems 2.3, 2.4, and 2.5. We proceed to give some basic definitions.

The numerical solution of a ordinary differential equation (ODE) basically describes any physical phenomenon of interest. Traditionally, the numerical solution of ODE is solved by a discretization process using the classical finite difference approximation. Recently, many studies have used deep neural networks (DNN) to approximate ODE solutions without discretization. However, without the optimization process, DNN cannot run independently. In this work, we proposed an alternative method that combines DNN with a genetic algorithm (GA) as an optimization method. GA initiates a population of DNN trainable parameters, making the DNN training process more streamlined on a single-generation GA. We focused on DNN-GA hybrid solution for linear and nonlinear ordinary differential equations (ODE‘s). Our method is achieved by solving several test problems and demonstrating a comparison with stochastic gradient descent (SGD) optimizers. In most ODE problems, using the same number of iterations and architecture, DNN-GA produces a better loss function than using DNN-SGD.

Here we turn to consideration of systems with time delayed feedback corresponding to models of Cha. 10, which have been implemented as electronic laboratory devices and studied in experiments described in (Kuznetsov and Ponomarenko, 2008; Baranov et al., 2010). In a frame of the hyperbolic theory the status of dynamics observed in these systems is not so well defined because the classic formulation of the theory relates to finite-dimensional systems, while for the timedelay systems the stale space is formally infinite-dimensional. Nevertheless, physically operation principle and observed dynamical phenomena look very similar to those in systems of alternately excited oscillators, for which the hyperbolic nature of the chaotic attractor is well established (see Chaps. 4, 7 and 12). On one hand, it implies a challenging problem for mathematicians to give rigorous definitions and foundation for existence of uniformly hyperbolic chaotic attractors embedded in the infinite dimensional state space of the time-delay systems. On the other hand, accounting transparent principle of operation and evident analogy with the alternately excited oscillators, on physical level of reasoning one can confidently suggest that the chaotic dynamics in time-delay systems of the considered class will have the same features relevant to technical applications like structural stability (robustness).

Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time-steps in a classical time integrator. This intrinsic randomization allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean-square convergence analysis are derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean-square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.

The question of power and asymptotic stability of infinite dimensional discrete-time state space systems is investigated. It is shown that every balanced realization is asymptotically stable. Conditions are given for balanced, input normal, or output normal realization to be asymptotically and/or power stable. >

We give a simple proof that for classical Dirichlet forms on infinite dimensional linear state spaces the intrinsic closure of a set of full measure has full capacity. Furthermore, we show that the C1,q-capacity of a set, enlarged by adding the linear span of a basis in the generalized CameronMartin space remains zero if it was zero for slightly bigger capacities a priori.

Time-delayed dynamical systems are defined in an infinite dimensional state-space. When the system has multiple stable steady-state responses, the global analysis of the system such as finding the domains of attraction and boundary basin has to be done in the infinite dimensional state-space. This paper examines this issue and shows that there are rich opportunities for global analysis research of nonlinear time-delayed dynamical systems.