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Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows
No full textInfinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow. We show that spirals associated with periodicity hubs emerge/accumulate
at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor
HydraMP: Exploiting shared memory parallelism in HYDRA with OpenMP
No full textMulticore CPUs are now found in desktops, servers and supercomputers but many existing parallel performance analysis tools were designed for the single-core distributed-memory world. We investigate the practicality of taking an existing tool, namely the HYDRA response time analyser, and parallelising it with OpenMP to produce a multithreaded implementation suitable for execution on multicore shared-memory machines. We discuss the amount of software engineering work required and show that only a small number of lines of code need to be added to achieve dramatic speed-ups over the serial version. We also compare the run-times of our OpenMP-parallelised version with existing MPI-parallelised code on the same hardware
On the stability of Hamiltonian relative equilibria with non-trivial isotropy
No full textWe consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu and by Lerman and Singer. In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of the Lie algebra of the symmetry group, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more easily computed criterion for stability. The result is also extended to apply to systems whose momentum map is not coadjoint equivariant
Iterative Fault Tolerant Control Based on Stochastic Distribution
Get PDFA new design of a fault tolerant control (FTC)-
based an adaptive, �xed-structure PI controller, with constraints
on the state vector for nonlinear discrete-time system subject to
stochastic non-Gaussian disturbance is studied. The objective
of the reliable control algorithm scheme is to design a control
signal such that the actual probability density function (PDF)
of the system is made as close as possible to a desired PDF, and
make the tracking performance converge to zero, not only when
all components are functional but also in case of admissible
faults. A Linear Matrix Inequality (LMI)-based FTC method
is presented to ensure that the fault can be estimated and
compensated for. A radial basis function (RBF) neural network
is used to approximate the output PDF of the system. Thus, the
aim of the output PDF control will be a RBF weight control with
an adaptive tuning of the basis function parameters. The key
issue here is to divide the control horizon into a number of equal
time intervals called batches. Within each interval, there are a
�xed number of sample points. The design procedure is divided
into two main algorithms, within each batch, and between any
two adjacent batches. A P-type ILC law is employed to tune the
parameters of the RBF neural network so that the PDF tracking
error decreases along with the batches. Suf�cient conditions for
the proposed fault tolerance are expressed as LMIs. An analysis
of the ILC convergence is carried out. Finally, the effectiveness
of the proposed method is demonstrated with an illustrated
example
iGen 0.1: the automated generation of a parameterisation of entrainment in marine stratocumulus
Get PDFIn a previous paper we described a new technique for automatically generating parameterisations using a program called iGen. iGen generates parameterisations by analysing the source code of a~high resolution model that resolves the physics to be parameterised. In order to demonstrate that this technique scales up to deal with models of realistic complexity we have used iGen to generate a parameterisation of entrainment in marine stratocumulus. We describe how iGen was used to analyse the source code of an eddy resolving model (ERM) and generate a parameterisation of entrainment velocity in marine stratocumulus in terms of the large-scale state of the boundary layer. The parameterisation was tested against results from the DYCOMS-II intercomparison of ERM models and iGen's parameterisation of mean entrainment velocity was found to be 5.27 � 10�3 ± 0.62 � 10�3 m s�1 compared to 5.2 � 10�3 ± 0.8 � 10�3 m s�1 for the DYCOMS-II ensemble of large eddy simulation (LES) models
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
No full textWe develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials.
It incorporates scaling of
the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable
conditioning and backward stability properties,
and a preprocessing step that
reveals and deflates the zero and infinite eigenvalues contributed by singular
leading and trailing matrix coefficients.
The algorithm is backward stable for quadratics that are not too heavily damped.
Numerical experiments show that
our MATLAB implementation of the algorithm,
quadeig, outperforms the MATLAB
function polyeig in terms of both stability and efficiency
Gaussian Elimination
Get PDFAs the standard method for solving systems of linear equations, Gaussian elimination (GE) is one of the most important and ubiquitous numerical algorithms. However, its successful use relies on understanding its numerical stability properties and how to organize its computations for efficient execution on modern computers. We give an overview of GE, ranging from theory to computation. We explain why GE computes an LU factorization and the various benefits of this matrix factorization viewpoint. Pivoting strategies for ensuring numerical stability are described. Special properties of GE for certain classes of structured matrices are summarized. How to implement GE in a way that efficiently exploits the hierarchical memories of modern computers is discussed. We also describe block LU factorization, corresponding to the use of pivot blocks instead of pivot elements, and explain how iterative refinement can be used to improve a solution computed by GE. Other topics are GE for sparse matrices and the role GE plays in the TOP500 ranking of the world's fastest computers
Hysteretic regime switching diffusions and resource extraction
Get PDFWe calculate the probability that an extraction project will be abandoned, directly from a real options
model closely related the seminal work of Brennan and Schwartz (1985). We assume that the resource is
extracted at two alternative rates, with a capital cost for switching, and with an option to abandon due
to unsatisfactory market prices. The abandonment probability is expressed as a hitting probability for a
regime switching diffusion with hysteresis, which is shown to be the unique solution of a system of coupled
boundary value problems. Our work lends itself to use as a quantitative and easily interpreted measure of
risk in the planning of extraction projects. Numerical results show that the abandonment probability may
be non-monotone with respect to the volatility of the price process, in contrast with project valuations. In
the one-dimensional stationary case, the stochastic process is a hysteretic system with noise in the sense of
Freidlin et al. (2000), and we obtain a closed-form expression for the hitting or abandonment probability in
this case
iGen: A program for the automated generation of models and parameterisations
No full textComplex physical systems can often be simulated using very high-resolution models but this is not always practical because of computational restrictions. In this case the model must be simplified or parameterised, but this is a notoriously difficult process that often requires the introduction of `model assumptions' that are hard or impossible to justify. Here we introduce a new approach to parameterising models. The approach makes use of a newly developed computer program, which we call iGen, that analyses the source code of a high-resolution model and formally derives a much faster parameterisation that closely approximates the original, reporting bounds on the error introduced by any approximations. These error bounds can be used to formally justify use of the parameterised model in subsequent numerical experiments. Using increasingly complex physical systems as examples we illustrate that iGen has the ability to produce parameterisations that run typically orders of magnitude faster than the underlying, high-resolution models from which they are derived and show that iGen has the potential to become an important tool in model development
Stable topological transitivity properties of Rn-extensions of hyperbolic transformations
No full textWe consider Rn skew-products of a class of hyperbolic dynamical
systems. It was proved by Nit¸ic�a and Pollicott [NP] that for an Anosov
diffeomorphism of an infranilmanifold there is (subject avoiding
natural obstructions) an open and dense set f : ! RN for which the
skew-product f (x, s) = ((x), s+f(x)) on �RN has a dense orbit.
We prove a similar result in the context of an Axiom A hyperbolic flow
on an attractor