Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle's invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle's invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times.     

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.     

The performance of a Tau preconditioner on systems of ODEs

In this work, we investigate the new preconditioner of the Tau method, introduced by Ghoreishi and Mohammad Hosseini [13], on systems of ODEs. With a different and relatively straightforward view on the Tau formulation of systems of ODEs this preconditioner is reformulated and explained. An error analysis of the method is also addressed. Some numerical comparisons with the standard Chebyshev-Tau method and the preconditioned method of Cabos confirm the overall efficiency of this preconditioner for systems of ODEs as well.     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

Asymptotic behavior of multistep runge-kutta methods for systems of delay differential equations

1. IntroductionIn order to assess the asymptotic behavior of numerical methods for DDEs, much attention has been given in the literature to the scalar case (cL [1-6]). UP to now) only partialresults (of. [7-10]) have dealt with the delay systemswhere y(t) = (yi(t), so(t),' ) yp(t))" E Cd, which is unknown for t > 0, L and M areconstat complex p x Hmatrices, T > 0 is a constat delay and W(t) 6 CP is a specifiedinitial function.In [111, C.J. Zhang and S.Z. Zhou made an investigation on…     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

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.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

A verified inexact implicit Runge–Kutta method for nonsmooth ODEs

Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science and a scholarship from Egyptian Government.     

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler-Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler-Lagrange ODEs that admit Noether symmetry.     

Generalized notions of symmetry of ODEs and reduction procedures

This paper describes the notion of σ‐symmetry, which extends the one of λ‐symmetry, and its application to reduction procedures of systems of ordinary differential equations (ODEs) and of dynamical systems (DS) as well. We also consider orbital symmetries, which give rise to a different form of reduction of DS. Finally, we discuss how DS can be transformed into higher order ODEs, and how these symmetry properties of the DS can be transferred into reduction properties of the corresponding ODEs. Many examples illustrate the various situations. Copyright © 2013 John Wiley & Sons, Ltd.     

数值离散化对环面分支结构的影响

Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.     

Some new additive Runge–Kutta methods and their applications

We propose some new additive Runge–Kutta methods of orders ranging from 2 to 4 that may be used for solving some nonlinear system of ODEs, especially for the temporal discretization of some nonlinear systems of PDEs with constraints. Only linear ODEs or PDEs need to be solved at each time step with these new methods.     

WAVEFORM RELAXATION METHODS AND ACCURACY INCREASE

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Discrete QMR and BCG in the numerical solution of linear systems of ODEs

In this paper we consider the use of QMR and BCG in the numerical solution of linear systems of ODEs reformulated as second kind integral equations. The implementation of the methods is performed in practice by applying them on suitably discretized equations. An important issue in this contest is to compare the behaviour of the discrete and continuous iteration. Here we consider the case of linear systems of ODEs. The discretization is performed using a continuous Runge-Kutta method. A numerical experiment concludes the paper.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

The Waveform Relaxation method for systems of differential/algebraic equations

This paper reports efforts towards establishing a parallel numerical algorithm known as Waveform Relaxation (WR) for simulating large systems of differential/algebraic equations. The WR algorithm was established as a relaxation based iterative method for the numerical integration of systems of ODEs over a finite time interval. In the WR approach, the system is broken into subsystems which are solved independently, with each subsystem using the previous iterate waveform as “guesses” about the behavior of the state variables in other subsystems. Waveforms are then exchanged between subsystems, and the subsystems are then resolved repeatedly with this improved information about the other subsystems until convergence is achieved.

In this paper, a WR algorithm is introduced for the simulation of generalized high-index DAE systems. As with ODEs, DAE systems often exhibit a multirate behavior in which the states vary as differing speeds. This can be exploited by partitioning the system into subsystems as in the WR for ODEs. One additional benefit of partitioning the DAE system into subsystems is that some of the resulting subsystems may be of lower index and, therefore, do not suffer from the numerical complications that high-index systems do. These lower index subsystems may therefore be solved by less specialized simulations. This increases the efficiency of the simulation since only a portion of the problem must be solved with specially tailored code. In addition, this paper established solvability requirements and convergence theorems for varying index DAE systems for WR simulation.     

An improved class of generalized Runge–Kutta methods for stiff problems. Part I: The scalar case

A new family of p-stage methods for the numerical integration of some scalar equations and systems of ODEs is proposed. These methods can be seen as a generalization of the explicit p-stage Runge–Kutta ones, while providing better order and stability results. We will show in this first part that, at the cost of losing linearity in the formulas, it is possible to obtain explicit A-stable and L-stable methods for the numerical integration of scalar autonomous ODEs. Scalar autonomous ODEs are of very little interest in current applications. However, be begin studying this kind of problems because most of the work can be easily extended to a more general situation. In fact, we will show in a second part (entitled ‘The separated system case'), that it is possible to generalize our methods so that they can be applied to some non-autonomous scalar ODEs and systems. We will obtain linearly implicit L-stable methods which do not require Jacobian evaluations. In both parts, some numerical examples are discussed in order to show the good performance of the new schemes.     

Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type – Part II: Linear multistep methods

Summary. It is shown that appropriate linear multi-step methods (LMMs) applied to singularly perturbed systems of ODEs preserve the geometric properties of the underlying ODE. If the ODE admits an attractive invariant manifold so does the LMM. The continuous as well as the discrete dynamical system restricted to their invariant manifolds are no longer stiff and the dynamics of the full systems is essentially described by the dynamics of the systems reduced to the manifolds. These results may be used to transfer properties of the reduced system to the full system. As an example global error bounds of LMM-approximations to singularly perturbed ODEs are given. Received May 5, 1995 / Revised version received August 18, 1995