Variational electrodynamics of Atoms

We generalize Wheeler-Feynman electrodynamics by the minimization of a finite action functional defined for variational trajectories that are required to merge continuously into given past and future boundary segments. We prove that the boundary-value problem is well-posed for two classes of boundary data and show that the well-posed solution in general has velocity discontinuities, henceforth broken extrema. Along regular segments, broken extrema satisfy the Euler-Lagrange neutral differential delay equations with state-dependent deviating arguments. At points where velocities are discontinuous, broken extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta are continuous. The electromagnetic fields of the variational trajectories are derived quantities that can be extended only to a bounded region B of space-time. For extrema with a finite number of velocity discontinuities, extended fields are defined for all point in B with the exception of sets of zero measure. The extended fields satisfy the integral laws of classical electrodynamics for most surfaces and curves inside B. As an application, we study globally bounded trajectories with vanishing far-fields for the hydrogenoid atomic models of hydrogen, muonium and positronium. Our model uses solutions of the neutral differential delay equations along regular segments and a variational approximation for the collisional segments. Each hydrogenoid model predicts a discrete set of finitely measured neighbourhoods of orbits with vanishing far-fields at the correct atomic magnitude and in quantitative and qualitative agreement with experiment and quantum mechanics, i.e., the spacings between consecutive discrete angular momenta agree with Planck's constant within thirty-percent, while orbital frequencies agree with a corresponding spectroscopic line within a few percent.Comment: Full re-write using same equations and back to original title (version 18 compiled with the wrong figure 5). A few commas introduced and all paragraphs broken into smaller ones whenever possibl

Dynamics of Simple Balancing Models with State Dependent Switching Control

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.Comment: 36 pages, 12 figure

Discrete time piecewise affine models of genetic regulatory networks

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. When compared to other models of regulatory networks, these models have an additional parameter which is identified as quantifying interaction delays. In spite of their simplicity, their dynamics presents a rich variety of behaviours. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle -- with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.Comment: 34 page

Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise

Switch-like behaviour in dynamical systems may be modelled by highly nonlinear functions, such as Hill functions or sigmoid functions, or alternatively by piecewise-smooth functions, such as step functions. Consistent modelling requires that piecewise-smooth and smooth dynamical systems have similar dynamics, but the conditions for such similarity are not well understood. Here we show that by smoothing out a piecewise-smooth system one may obtain dynamics that is inconsistent with the accepted wisdom --- so-called Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit. By subjecting the system to white noise, we show that these discrepancies can be understood in terms of potential wells that allow solutions to dwell at the discontinuity for long times. Moreover we show that spurious dynamics will revert to Filippov dynamics, with a small degree of stochasticity, when the noise magnitude is sufficiently large compared to the order of smoothing. We apply the results to a model of a dry-friction oscillator, where spurious dynamics (inconsistent with Filippov's convention or with Coulomb's model of friction) can account for different coefficients of static and kinetic friction, but under sufficient noise the system reverts to dynamics consistent with Filippov's convention (and with Coulomb-like friction).Comment: submitted to: Nonlinear Dynamic

Accurate difference methods for linear ordinary differential systems subject to linear constraints

We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).