Rigorous computational shadowing of orbits of ordinary differential equations

Summary. The existence of a true orbit near a numerically computed approximate orbit -- shadowing -- of autonomous system of ordinary differential equations is investigated. A general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities, is proved. The practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the Lorenz equations. Received December 15, 1993     

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.     

Periodic orbits of delay differential equations under discretization

This paper deals with the long-time behaviour of numerical solutions of delay differential equations that have asymptotically stable periodic orbits. It is shown that Runge-Kutta discretizations of such equations have attractive invariant curves which approximate the periodic orbit with the order of the method. The research by this author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

A numerical verification method for a periodic solution of a delay differential equation

We describe a numerical method with guaranteed accuracy to enclose a periodic solution for a system of delay differential equations. Using a certain system of equations corresponding to the original system, we derive sufficient conditions for the existence of the solution, the satisfaction of which can be verified computationally. We describe the verification procedure in detail and give a numerical example.     

Numerical approximation of connecting orbits with asymptotic rate

Summary. In this paper we set up and analyze a numerical method for so called {\bf connecting orbits with asymptotic rate } in parameterized dynamical systems. A connecting orbit with asymptotic rate has its initial value in a given submanifold of the phase space (or its cross product with parameter space) and it converges with an exponential rate to a given orbit, e. g. a steady state or a periodic orbit. It is well known that orbits with asymptotic rate can be used to foliate stable or strong stable manifolds of invariant sets. We show that the problem of determining a connecting orbit with asymptotic rate is well-posed if a certain transversality condition is made and a specific relation between the number of stable dimensions and the number of parameters holds. For the proof we employ the implicit function theorem in spaces of exponentially decaying functions. Using asymptotic boundary conditions we truncate the original problem to a finite interval and show that the error decays exponentially. Typically the asymptotic boundary conditions by themselves are the result of a boundary value problem, e. g. if the limiting orbit is periodic. Thus it is expensive to calculate them in a parameter dependent way during the approximation procedure. To avoid this we develop a boundary corrector method which turns out to be nearly optimal after very few steps. Received April 28, 2000 / Revised version received December 18, 2000 / Published online May 30, 2001     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

Error bounds for computed eigenvalues and eigenvectors

Summary On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday     

A Shadowing Theorem for ordinary differential equations

A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then a general theorem which guarantees the existence of shadowing of pseudo orbits in compact hyperbolic sets is proved.Supported in part by the Air Force.Supported in part by NSF grant DMS 9201951.     

Zur existenz periodischer losungen von pseudodifferential gleichungen mit nacheilendem argument

In this paper an existence theorem for periodic solutions of pseudo-differential equations with delayed argument is proved by using an algebra of pseudo-differential operators. The existence of periodic solutions of the heat equation with delayed argument is included.     

The life-span of backward error analysis for numerical integrators

Summary. Backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations. The numerical solution is formally interpreted as the exact solution of a perturbed differential equation, given as a formal and usually divergent series in powers of the step size. For a rigorous analysis, this series has to be truncated. In this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the long-time behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems. Received October 18, 1995 / Revised version received February 28, 1996     

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.Dedicated to Professor Lothar Collatz on the occasion of his 70th birthdayThis paper has been read at the Conference of Numerical Mathematicians, Zeist, The Netherlands, October 12–15, 1979     

Dynamical analysis of toxin producing Phytoplankton–Zooplankton interactions

In the present paper we consider a toxin producing phytoplankton–zooplankton model in which the toxin liberation by phytoplankton species follows a discrete time variation. Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay. Then we establish the existence of local Hopf-bifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena. Explicit results are derived for stability and direction of the bifurcating periodic orbit by using normal form theory and center manifold arguments. Global existence of periodic orbits is also established by using a global Hopf-bifurcation theorem. Finally, the basic outcomes are mentioned along with numerical results to provide some support to the analytical findings.     

Solving neutral delay differential equations with state-dependent delays

In this paper we consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate. After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits.     

On the discretization of a partial differential equation in the neighborhood of a periodic orbit

Sumamry This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed. This answers partly an open problem posed in [4]. Examples of application to dissipative partial equations are provided.     

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.     

Some remarks on an ODE-solver of Kirchgraber

Summary Kirchgraber derived in 1988 an integration procedure (called the LIPS-code) for long-term prediction of the solutions of equations which are perturbations of systems having only periodic solutions. His basic idea is to use the Poincaré map to define a new system which can be integrated with large step-size; the method is specially successful when the period is close to the unperturbed one. Obviously the size of the perturbation modifies the period and therefore affects the precision of the algorithm. In this paper we propose a double modification of Kirchgraber's code: to use a first-order approximation of the perturbed period instead of the unperturbed one, and a scheme specially designed for integration of orbits instead of the Runge-Kutta method. We show that this new code permits a spectacular improvement in accuracy and computation time.     

Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit

We analyze homoclinic orbits near codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions. The main motivation for this study is a self-organized periodic replication process of travelling pulses which has been observed in reaction-diffusion equations. We establish conditions for existence and uniqueness of countably infinite families of curve segments of 1-homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. The main result shows the bifurcation of a number of curves of 1-homoclinic orbits from such codimension-2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points. In addition, a leading order expansion of the associated curves in parameter space is derived. Its coefficients are periodic with one frequency from the imaginary part of the leading stable Floquet exponents of the periodic orbit and one from the winding number.     

A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria

A weak form of the Hartman-Grobman theorem for retarded functional differential equations around hyperbolic equilibria is presented. Orbits on a center-unstable manifold are compared to orbits on a center-unstable subspace of the linearized equation. The result is applied to obtain a conjugacy between the semidynamical system generated by the functional differential equation and its numerical approximation. A version of the Hartman-Grobman theorem around hyperbolic periodic orbits of functional differential equations is also given.     

Periodic orbits of competitive and cooperative systems

In this paper we consider the existence, location and stability type of periodic orbits of competitive and cooperative systems of autonomous ordinary differential equations. Particular attention is given to the existence of invariant manifolds related to periodic orbits and these results are used to improve a result of Hirsch for three dimensional irreducible competitive and cooperative systems. In particular, the Poincaré-Bendixson theorem holds for such three dimensional systems.