Class Of Ordinary Differential Equations Research Articles

On oscillations in a system with a piecewise smooth coefficient

In this paper a straightforward perturbation procedure will be presented for a class of ordinary differential equations with piecewise smooth terms in the equations. From the constructed approximations of the solutions a map can be derived from which the existence and the stability of time-periodic solutions can be determined.

Stabilization of Diffusion-Induced Chaotic Processes

In this paper, it is shown that diffusion-inducedchaos in a distributed system can be suppressed byfeedback-free parametric action.The possibility of stabilizing chaotic oscillationswas described for the first time by, most likely, Alek-seev and Loskutov [1, 2], who considered a class ofordinary differential equations with nonpolynomialright sides. Later, a feedback-free method for suppress-ing chaos in systems of a certain class was analyticallysubstantiated by Loskutov and Shishmarev [3, 4] (seealso Loskutov [5] and references therein). This subjectbecame popular after publication of a paper by Ott

NONCONVEX CONSERVATION LAWS AND ORDINARY DIFFERENTIAL EQUATIONS

The paper deals with the well posedness of a class of ordinary differential equations. The vector field depends on the solution to a scalar conservation law, whose flux function is assumed to have a single inflection point (from whence ‘nonconvex’ is derived). Filippov solutions to the ordinary differential equations are considered, and Hölder continuous dependence on the initial data is proved. The motivation for the problem is a model of traffic flow.

Numerical Integration of a Class of Ordinary Differential Equations on the General Linear Group of Matrices

This paper concerns with the numerical solution of a class of ordinary differential equations on Gl(n), the set of all n×n nonsingular real matrices. In particular, we consider matrix dynamical systems of the form Y′=Y−TF(Y). The presence of the inverse of the solution and of possible escape times make the numerical solution of this kind of differential equations somewhat worrisome. Here, we suggest some numerical techniques to avoid some problems arising in its numerical solution.

Three-dimensional Cauchy–Riemann structures and second-order ordinary differential equations

The equivalence problem for second-order ordinary differential equations (ODEs) given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate three-dimensional Cauchy–Riemann structures. This approach enables an analogue of the Fefferman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second-order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.

A Hölder continuous ODE related to traffic flow

This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C0 of the solution to the ODE from the initial data of the conservation law in L1.This problem is motivated by a model of traffic flow.

Control Parametrization Enhancing Technique for Solving a Special ODE Class with State Dependent Switch

In this paper, the control parametrization enhancing technique (CPET) is applied to obtain numerical solution to a special class of ordinary differential equations where the right-hand side has state dependent discontinuities. Switching locations corresponding to these discontinuities can be calculated exactly and conveniently. Illustrative examples are provided to demonstrate the novel technique.

Concerning the action of an invertible point transformation on a class of ordinary differential equations

In the category of the group theoretic methods (direct methods) using invertible point transformation, we give a useful recursive relation from one step to the next, which is performed without any further computations. This approach shows that the method is straightforward to implement. Also, the applications of the method for the general nth order differential equations are given.

On a Class of Ordinary Differential Equations of Various Orders on a Graph

On a Class of Ordinary Differential Equations of Various Orders on a Graph

A new class of scale free solutions to linear ordinary differential equations and the universality of the golden mean [formula omitted

A new class of finitely differentiable scale free solutions to the simplest class of ordinary differential equations is presented. Consequently, the real number set gets replaced by an extended physical set, each element of which is endowed with an equivalence class of infinitesimally separated neighbours in the form of random fluctuations. We show how a sense of time and evolution is intrinsically defined by the infinite continued fraction of the golden mean irrational number ( 5 −1)/2 , which plays a key role in this extended SL(2,R) formalism of calculus analogous to El Naschie’s theory of E (∞) spacetime manifold. Time may thereby undergo random inversions generating well defined random scales, thus allowing a dynamical system to evolve self similarly over the set of multiple scales. The late time stochastic fluctuations of a dynamical system enjoys the generic 1/ f spectrum. A universal form of the related probability density is also derived. We prove that the golden mean number is intrinsically random, letting all measurements in the physical universe fundamentally uncertain. The present analysis offers an explanation of the universal occurrence of the golden mean in diverse natural and biological processes as well as the mass spectrum of high energy particle physics.

Dynamics in high-dimensional model gene networks

We consider dynamics in a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of products of other genes by a logical function. Thus, the concentration of a particular gene's product is driven either up or down depending on the particular combination of activity states (active or inactive, i.e., gene product above or below threshold) of a set of relevant ‘input’ genes. The interactions are based on binary functions but the evolution of gene product concentrations is continuous in time, unlike the discrete-time Boolean networks of Kauffman and others. Numerical methods allow rapid and accurate integration of the model equations. Also, theory now allows analytic confirmation of numerically observed attractors. This enables us to determine changes in the distribution of dynamical properties (attractor types) of random networks as the size of the networks increase and as the number of inputs to each gene increases.

Nonlinear eigenvalue problems for a class of ordinary differential equations

We consider the class of nonlinear eigenvalue problems where yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

Two Asymptotic Forms of the Rotational Solution for Wave Propagation Inside Viscous Channels with Transpiring Walls

Summary In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel’s head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves with oscillations matching the natural frequencies of the enclosure are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyse. The mean flow is of the Berman type and can be obtained from the Navier–Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ordinary differential equations. Due to the oscillatory behaviour of the problem, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem’s characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem’s solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.

A Rigorous ODE Solver and Smale’s 14th Problem

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the twenty-first century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations.

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations.

Global attractors for degenerate partial differential equations from nonlinear viscoelasticity

We study several problems for the forced motion of light, uniform, nonlinearly viscoelastic bodies carrying heavy attachments. A ‘reduced’ problem for such motions is obtained by setting the ratio of the inertia of the viscoelastic body to the inertia of the attachment equal to zero. Using methods from infinite-dimensional dynamical systems theory, we prove that the degenerate partial differential equation of this reduced problem has an attractor and that this attractor is contained in an invariant two-dimensional manifold on which solutions are governed by the classical ordinary differential equation for the forced motion of a particle on a massless spring.

Symbolic dynamics and computation in model gene networks.

We analyze a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of other genes by a logical function. The dynamics in these equations are represented by a directed graph on an n-dimensional hypercube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the boundary between neighboring orthants, the equation is integrated until the boundary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time that same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequence of orthants that arise from integration of the equation starting and ending on the same boundary. The union of the words defines the language. Letters and words correspond to analytically computable Poincare maps of the equation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of dynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation. (c) 2001 American Institute of Physics.

Integration of case studies on Global Change by means of qualitative differential equations

We present a novel methodology to integrate qualitative knowledge from different case studies on Global Change related issues into a single framework. The method is based on the concept of qualitative differential equations (QDEs) which represents a mathematically well-defined approach to investigate classes of ordinary differential equations (ODEs) used in conventional modeling exercises. These classes are defined by common qualitative features, e.g., monotonicity, signs, etc. Using the QSIM-algorithm it is possible to derive the set of possible solutions of all ODEs in the class. By this one can formulate a common, qualitatively specified cause–effect scheme valid for all case studies. The scheme is validated by testing it against the actually observed histories in the study regions with respect to their reconstruction by the corresponding QDE. The method is outlined theoretically and exemplary applied to the problem of land-use changes due to smallholder agriculture in developing countries. It is shown that the seven case-studies used can be described by a single cause–effect scheme which thus constitutes a pattern of Global Change. As a generally valid prerequisite for sustainability of this kind of land-use the presence of wage labor is shown to represent a decisive factor.

Unicité dans le problème de Cauchy pour un dériveur sans accrétivité

We prove a uniqueness property for a class of ordinary differential equations governed by continuous derivors (i.e., the opposite is continuous and quasimonotone with respect to ( R +) N ) which do not satisfy any accretivity condition or any Nagumo–Osgood–Kamke's condition.

Time-Splitting Errors in the Numerical Integration of Semilinear Systems of Ordinary Differential Equations

In this paper the authors analyze splitting errors in numerical schemes for a semilinear system of ordinary differential equations (ODEs). It is well known that errors occur even when splitting the continuous fully linear system analytically, consequently splitting numerical schemes introduces additional errors. A general approach to delineate and avoid such errors for the above-mentioned class of ODEs is proposed.