Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map

It is well-known that the dynamics of the Arnold circle map is phase-locked in regions of the parameter space called Arnold tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map changes dramatically. Inside the Arnold tongues open regions of multistability exist, and the parameter dependency of the dynamics becomes rather complex. This paper discusses the bifurcation structure inside the Arnold tongue with zero rotation number and includes a study of nonsmooth bifurcations that happen for large nonlinearity in the region with strange nonchaotic attractors.Comment: 25 pages, 22 colored figures in reduced quality, submitted to Int. J. of Bifurcation and Chaos, a supplementary website (http://www.mpipks-dresden.mpg.de/eprint/jwiersig/0004003/) is provide

Adjoint methods for computing sensitivities in local volatility surfaces

In this paper we present the adjoint method of computing sensitivities of option prices with respect to nodes in the local volatility surface. We first introduce the concept of algorithmic differentiation and how it relates to\ud path-wise sensitivity computations within a Monte Carlo framework. We explain the two approaches available: forward mode and adjoint mode. We illustrate these concepts on the simple example of a model with a geometric Brownian motion driving the underlying price process, for which\ud we compute the Delta and Vega in forward and adjoint mode. We then go on to explain in full detail how to apply these ideas to a model where the underlying has a volatility term defined by a local volatility surface. We provide source codes for both the simple and the more complex case and\ud analyze numerical results to show the strengths of the adjoint approach

Micro-meteoroid seismic uplift and regolith concentration on kilometric scale asteroids

Seismic shaking is an attractive mechanism to explain the destabilisation of regolith slopes and the regolith migration found on the surfaces of asteroids (Richardson et al. 2004; Miyamoto et al. 2007). Here, we use a continuum mechanics method to simulate the seismic wave propagation in an asteroid. Assuming that asteroids can be described by a cohesive core surrounded by a thin non-cohesive regolith layer, our numerical simulations of vibrations induced by micro-meteoroids suggest that the surface peak ground accelerations induced by micro-meteoroid impacts may have been previously under-estimated. Our lower bound estimate of vertical accelerations induced by seismic waves is about 50 times larger than previous estimates. It suggests that impact events triggering seismic activity are more frequent than previously assumed for asteroids in the kilometric and sub-kilometric size range. The regolith lofting is also estimated by a first order ballistic approximation. Vertical displacements are small, but lofting times are long compared to the duration of the seismic signals. The regolith movement has a non-linear dependence on the distance to the impact source which is induced by the type of seismic wave generating the first movement. The implications of regolith concentration in lows of surface acceleration potential are also discussed. We suggest that the resulting surface thermal inertia variations of small fast rotators may induce an increased sensitivity of these objects to the Yarkovsky effect.Comment: Accepted for publication in Icaru

High-fidelity Multidisciplinary Sensitivity Analysis and Design Optimization for Rotorcraft Applications

A multidisciplinary sensitivity analysis of rotorcraft simulations involving tightly coupled high-fidelity computational fluid dynamics and comprehensive analysis solvers is presented and evaluated. A sensitivity-enabled fluid dynamics solver and a nonlinear flexible multibody dynamics solver are coupled to predict aerodynamic loads and structural responses of helicopter rotor blades. A discretely consistent adjoint-based sensitivity analysis available in the fluid dynamics solver provides sensitivities arising from unsteady turbulent flows and unstructured dynamic overset meshes, while a complex-variable approach is used to compute structural sensitivities with respect to aerodynamic loads. The multidisciplinary sensitivity analysis is conducted through integrating the sensitivity components from each discipline of the coupled system. Accuracy of the coupled system is validated by conducting simulations for a benchmark rotorcraft model and comparing solutions with established analyses and experimental data. Sensitivities of lift computed by the multidisciplinary sensitivity analysis are verified by comparison with the sensitivities obtained by complex-variable simulations. Finally the multidisciplinary sensitivity analysis is applied to a constrained gradient-based design optimization for a HART-II rotorcraft configuration

Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3)

Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces and moments that arise from a potential and from control forces and moments. The key features of this paper are its use of computational procedures that are guaranteed to preserve the geometry of the optimal solutions. The theoretical basis for the computational procedures is summarized, and examples of optimal spacecraft maneuvers are presented.Comment: IEEE Conference on Decision and Control, 2006. 6 pages, 19 figure