1,276 research outputs found
Time-transformations for the event location in discontinuous ODEs
In this paper, we consider numerical methods for the location of events of ordinary differential equations. These methods are based on particular changes of the independent variable, called time-transformations. Such a time-transformation reduces the integration of an equation up to the unknown point, where an event occurs, to the integration of another equation up to a known point. This known point corresponds to the unknown point by means of the time-transformation. This approach extends the one proposed by Dieci and Lopez [BIT 55 (2015), no. 4, 987-1003], but our generalization permits, amongst other things, to deal with situations where the solution approaches the event in a tangential way. Moreover, we also propose to use this approach in a different manner with respect to that of Dieci and Lopez
Dynamical Systems on Networks: A Tutorial
We give a tutorial for the study of dynamical systems on networks. We focus
especially on "simple" situations that are tractable analytically, because they
can be very insightful and provide useful springboards for the study of more
complicated scenarios. We briefly motivate why examining dynamical systems on
networks is interesting and important, and we then give several fascinating
examples and discuss some theoretical results. We also briefly discuss
dynamical systems on dynamical (i.e., time-dependent) networks, overview
software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than
original version, some reorganization and also more pointers to interesting
direction
Gravitational wave signatures of the absence of an event horizon. I. Nonradial oscillations of a thin-shell gravastar
Gravitational waves from compact objects provide information about their
structure, probing deep into strong-gravity regions. Here we illustrate how the
presence or absence of an event horizon can produce qualitative differences in
the gravitational waves emitted by ultra-compact objects. In order to set up a
straw-man ultra-compact object with no event horizon, but which is otherwise
almost identical to a black hole, we consider a nonrotating thin-shell model
inspired by Mazur and Mottola's gravastar, which has a Schwarzschild exterior,
a de Sitter interior and an infinitely thin shell with finite tension
separating the two regions. As viewed from the external space-time, the shell
can be located arbitrarily close to the Schwarzschild radius, so a gravastar
might seem indistinguishable from a black hole when tests are only performed on
its external metric. We study the linearized dynamics of the system, and in
particular the junction conditions connecting internal and external
gravitational perturbations. As a first application of the formalism we compute
polar and axial oscillation modes of a thin-shell gravastar. We show that the
quasinormal mode spectrum is completely different from that of a black hole,
even in the limit when the surface redshift becomes infinite. Polar QNMs depend
on the equation of state of matter on the shell and can be used to distinguish
between different gravastar models. Our calculations suggest that
low-compactness gravastars could be unstable when the sound speed on the shell
vs/c>0.92.Comment: 19 pages, 8 figures. In press in Physical Review D. We found a new
family of modes and improved the discussion of nonradial instabilit
A Brief Survey on Non-standard Constraints: Simulation and Optimal Control
In terms of simulation and control holonomic constraints are well documented and thus termed standard. As non-standard constraints, we understand non-holonomic and unilateral constraints. We limit this survey to mechanical systems with a finite number of degrees of freedom. The long-term behavior of non-
holonomic integrators as compared to structure-preserving integrators for holonomically constrained systems is briefly discussed. Some recent research regarding the treatment of unilaterally constrained systems by event-driven or time-stepping schemes for time integration and in the context of optimal control problems is outlined
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