In this paper we present a technique for reducing to a minimum the number of commutators required in the practical implementation of Lie group methods for integrating numerically matrix differential equations. This technique is subsequently applied to the linear and nonlinear case for constructing new geometric integrators, optimal with respect to the number of commutators. © 2002 Elsevier Scie...

This is a PhD project proposal for Morten Dahlby, the advisor is Brynjulf Owren. The topic of the project is numerical integration of nonlinear wave models, with special emphasis on structure preserving and geometric integrators. The goal is to study conservative nite di erence schemes, multi-symplectic integrators, exponential integrators and schemes based on Euler's equations. As a part of Wa...

We introduce a family of impulselike methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast part and a slow part. Methods of this family are specified by two weight functions φ, ψ; one is used to average positions and the other to mollify the force. When the fast forces are conservative and φ = ψ, the methods here coincide wit...

We study the largest Liapunov exponent for products of random matrices. The two classes of matrices considered are discrete, d-dimensional Laplacians, with random entries, and symplectic matrices that arise in the study of d-dimensional lattices of coupled, nonlinear oscillators. We derive bounds on this exponent for all dimensions, d, and we show that if d 3, and the randomness is not too stro...

An index theorem is a tool for computing the change of the index (i.e., the number of negative eigenvalues) of a symmetric monotone matrix-valued function when its variable passes through a singularity. In 1995, the first author proved an index theorem in which a certain critical matrix coefficient is constant. In this paper, we generalize the above index theorem to the case when this critical ...

We use analytic methods to define Family Gromov-Witten Invariants for Kähler surfaces. We prove that these are well-defined invariants of the deformation class of the Kähler structure. Gromov-Witten invariants are counts of holomorphic curves in a symplectic manifold X. To define them using the analytic approach one chooses an almost complex structure J compatible with the symplectic structure ...

In this paper, we apply the symplectic integration method using jolt factorization described in an earlier paper to the symplectic map describing the nonlinear pendulum Hamiltonian. We compare results obtained with this method with those obtained using nonsymplectic methods and demonstrate that our results are much better.

We consider the application of a recently developed “conditional integrator” technique for the output regulation of a class of nonlinear systems to process control systems. For a special choice of the controller parameters, our controller reduces to a finely tuned saturated PI/PID type controller with an anti-windup structure. This is of particular significance because while the application of ...

This paper revisits dynamic surface control based adaptive Takagi-Sugeno (TS) fuzzy model i.e. (DSC-TS), for a problem of uncertain nonlinear dynamical systems with bounded disturbance. Dynamic surface control (DSC) methods with the combination of TS-model are used for control and approximation of uncertain nonlinear systems up to a tolerance limit. DSC has the advantage over integrator backpst...