Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance

A note on the electrical equivalent of the moment theory

In this short note the relation between the moments of a linear system and the phasors of an electric circuit is discussed. We show that the phasors are a special case of moments and we prove that the components of the solution of a Sylvester equation are the phasors of the currents of the system. We point out several directions in which the phasor theory can be extended using recent generalizations of the moment theory, which can benefit the analysis of circuits and power electronics

A Recipe for State Dependent Distributed Delay Differential Equations

We use the McKendrick equation with variable ageing rate and randomly distributed maturation time to derive a state dependent distributed delay differential equation. We show that the resulting delay differential equation preserves non-negativity of initial conditions and we characterise local stability of equilibria. By specifying the distribution of maturation age, we recover state dependent discrete, uniform and gamma distributed delay differential equations. We show how to reduce the uniform case to a system of state dependent discrete delay equations and the gamma distributed case to a system of ordinary differential equations. To illustrate the benefits of these reductions, we convert previously published transit compartment models into equivalent distributed delay differential equations.Comment: 28 page

Improving LLR Tests of Gravitational Theory

Accurate analysis of precision ranges to the Moon has provided several tests of gravitational theory including the Equivalence Principle, geodetic precession, parameterized post-Newtonian (PPN) parameters $\gamma$ and $\beta$, and the constancy of the gravitational constant {\it G}. Since the beginning of the experiment in 1969, the uncertainties of these tests have decreased considerably as data accuracies have improved and data time span has lengthened. We are exploring the modeling improvements necessary to proceed from cm to mm range accuracies enabled by the new Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) currently under development in New Mexico. This facility will be able to make a significant contribution to the solar system tests of fundamental and gravitational physics. In particular, the Weak and Strong Equivalence Principle tests would have a sensitivity approaching 10$^{-14}$, yielding sensitivity for the SEP violation parameter $\eta$ of $\sim 3\times 10^{-5}$, $v^2/c^2$ general relativistic effects would be tested to better than 0.1%, and measurements of the relative change in the gravitational constant, $\dot{G}/G$, would be $\sim0.1$% the inverse age of the universe. Having this expected accuracy in mind, we discusses the current techniques, methods and existing physical models used to process the LLR data. We also identify the challenges for modeling and data analysis that the LLR community faces today in order to take full advantage of the new APOLLO ranging station.Comment: 15 pages, 3 figures, talk presented at 2003 NASA/JPL Workshop on Fundamental Physics in Space, April 14-16, 2003, Oxnard, C

Model reduction by matching the steady-state response of explicit signal generators

© 2015 IEEE.Model reduction by moment matching for interpolation signals which do not have an implicit model, i.e., they do not satisfy a differential equation, is considered. Particular attention is devoted to discontinuous, possibly periodic, signals. The notion of moment is reformulated using an integral matrix equation. It is shown that, under specific conditions, the new definition and the one based on the Sylvester equation are equivalent. New parameterized families of models achieving moment matching are given. The results are illustrated by means of a numerical example

Lie Point Symmetries and Commuting Flows for Equations on Lattices

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice