this report . . . 73 7.2 Periodic solutions of autonomous systems . . . . . . . . . . . . 74 7.2.1 The Brusselator . . . . . . . . . . . . . . . . . . . . . . 75 7.2.2 The Lorenz system . . . . . . . . . . . . . . . . . . . . 76 7.2.3 The Van der Pol system . . . . . . . . . . . . . . . . . . 80 7.3 Periodic solutions of non-autonomous systems . . . . . . . . . . 82 7.3.1 The forced Brusselator . . . . . . . . . . . . . . . . . . 83 7.4 Solutions of boundary value problems . . . . . . . . . . . . . . 84

Abstract. We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framework unifies several known constructions. We also use the concept of “covariance” of a numerical method with respect to a group of coordinate transformations. The main application is to explore the relationship between spatial symmetries, reversing symmetries, and time symmetry of flows and numerical integrators.

A class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework. It is shown that certain approximations of these equations, such as those arising from spectral or finite element methods in space, one-step time-discretization or a combination of both, also have attractive invariant manifolds. Convergence of the approximate manifolds to the true manifolds is established as the approximation is refined. In this part of the paper applications to the behavior of inertial manifolds under approximation are considered. From this analysis deductions about the structure of the attractor and the flow on the attractor under discretization can be made. 1 Introduction Attractive invariant manifolds for evolution equations are fundamental in understanding long-time dynamics. Import...

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation--dissipative, contractive and gradient systems are analysed in this way.

Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.

Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. Runge-Kutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods; involving only s(s + 1)=2 free parameters for a symplectic s stage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrised coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In this case the order ...

this article we give an overview of the application of theories from dynamical systems to the analysis of numerical methods for initial-value problems. We start by describing the classical viewpoints of numerical analysis and of dynamical systems and then indicate how the two viewpoints can be merged to provide a framework for both the interpretation of data obtained from numerical simulations and the design of effecient numerical methods. This is done in section 2. In addressing the question of how to interpret data, we will show in section 3 how the concept of convergence can be generalized to the numerical approximation of dynamical systems. The main theory is developed for one-step methods for ordinary differential equations and extensions to the study of multistep methods, adaptive time-stepping algorithms and partial differential equations are then outlined. In addressing the question of designing effecient schemes we will show in section 4 how the concept of stability can be generalized to dynamical systems. Stability theory is developed for both one-step and multistep methods for ordinary differential equations; extensions to adaptive time-stepping and to partial differential equations are also outlined. A variety of surveys of this field already exist -- see [13] for a complete study, see [9] for a discussion of convergence in the dynamical systems context, see [12] for a discussion of stability in the dynamical systems context and see [5] for a discussion of both issues in relation to discretization of partial differential equations. Since these surveys contain exhaustive bibliographys we refer to them for detailed references. 1

Rosenbrock methods are popular for solving stiff initial value problems for ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas and implicit Runge-Kutta methods. In this paper, we introduce some trust region techniques to control the time step in the second order Rosenbrock methods for gradient systems. These techniques are different from the local error control schemes. Both the global and local convergence of the new class of trust region Rosenbrock methods for solving the equilibrium points of gradient systems are addressed. Finally some promising numerical results are presented.

This paper discusses the connections between numerical methods for ordinary differential equations, fixed point iterations and sequence transformations.

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size " for an autonomous system can be interpreted as the "-flow of a perturbed nonautonomous system. The perturbation is `rapidly forced' in the sense that it is periodic with respect to time with period ". Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as " tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization. Keywords: Invariant manifolds, center manifolds, nonautonomous systems, one-step methods, centered Euler scheme. AMS subj...