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Systems of rays in the presence of distribution of hyperplanes

100%

Janeczko S.

Banach Center Publications

1995

tom 32

nr 1

245-260

Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems $v_H$ on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields $v_H$ are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered.

Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems

88%

Freihold M., Hofer E.

International Journal of Applied Mathematics and Computer Science

2009

tom 19

nr 3

485-499

Interval arithmetic techniques such as VALENCIA-IVP allow calculating guaranteed enclosures of all reachable states of continuous-time dynamical systems with bounded uncertainties of both initial conditions and system parameters. Considering the fact that, in naive implementations of interval algorithms, overestimation might lead to unnecessarily conservative results, suitable consistency tests are essential to obtain the tightest possible enclosures. In this contribution, a general framework for the use of constraints based on physically motivated conservation properties is presented. The use of these constraints in verified simulations of dynamical systems provides a computationally efficient procedure which restricts the state enclosures to regions that are physically meaningful. A branch and prune algorithm is modified to a consistency test, which is based on these constraints. Two application scenarios are studied in detail. First, the total energy is employed as a conservation property for the analysis of mechanical systems. It is shown that conservation properties, such as the energy, are applicable to any Hamiltonian system. The second scenario is based on constraints that are derived from decoupling properties, which are considered for a high-dimensional compartment model of granulopoiesis in human blood cell dynamics.

Ograniczanie wyników

1 Banach Center Publications

1 International Journal of Applied Mathematics and Computer Science

1 Freihold M.

1 Hofer E.

1 Janeczko S.

1 2009

1 1995

Wyniki wyszukiwania

Systems of rays in the presence of distribution of hyperplanes

Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems