David G. Retzloff

Delay equations and radiation damping

Abstract Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.

On the ionization of a Keplerian binary system by periodic gravitational radiation

The gravitational ionization of a Keplerian binary system via normally incident periodic gravitational radiation of definite helicity is discussed. The periodic orbits of the planar tidal equation are investigated on the basis of degenerate continuation theory. The relevance of the Kolmogorov‐Arnold‐Moser theory to the question of gravitational ionization is elucidated, and it is conjectured that the process of ionization is closely related to the Arnold diffusion of the perturbed system.

Dynamics of the CR equations modeling a constant flow stirred tank reactor

concentration of species A [mol I-‘] concentration of species B [mol I-‘] dimensionless heat transfer coefficient concentration of species C [mol I-‘] specific heat [cal g-’ K-‘1 concentration of species D [mol 1-i) concentration of species E [mol l-l] activation energy of species I [cal mol-‘1 concentration of species F [mol I-‘] volumetric flow rate [1 s-l] concentration of species G [mol l-i] heat transfer coefficient [cal mm2 s-l K-l] concentration of species H [mol i-i] heat of reaction for reacting species I [cal mol-‘1 concentration of species I [mol I-‘] concentration of species J [mol I-‘] concentration of species K [mol I-‘] frequency factor for reacting species I [s-r] concentration of species L Imol I-‘] concentration of species M [mol I-‘] dimensionless feed concentration ratio for species i concentration of species N [mol l-i] concentration of species 0 [mol I-‘] universal gas constant [cal mall’ K-‘1 heat transfer area [m*] time [s] dimensionless time temperature [K] reactor volume [I] dimensionless concentration of species i dimensionless temperature dimensionless frequency factor for reacting species i dimensionless heat of reaction for reacting species i dimensionless activation energy for reacting species i density [g m-‘1 residence time in the reactor [s]

Gravitational ionization: a chaotic net in the Kepler system

The long-term nonlinear dynamics of a Keplerian binary system under the combined influences of gravitational radiation damping and external tidal perturbations is analysed. Gravitational radiation reaction leads the binary system towards eventual collapse, while the external periodic perturbations could lead to the ionization of the system via Arnold diffusion. When these two opposing tendencies nearly balance each other, interesting chaotic behaviour occurs which is briefly studied in this paper. It is possible to show that periodic orbits can exist in this system for sufficiently small damping. Moreover, we employ the method of averaging to investigate the phenomenon of capture into resonance.

Evolutionary dynamics while trapped in resonance: a Keplerian binary system perturbed by gravitational radiation

The method of averaging is used to investigate the phenomenon of capture into resonance for a model that describes a Keplerian binary system influenced by radiation damping and external normally incident periodic gravitational radiation. The dynamical evolution of the binary orbit while trapped in resonance is elucidated using a second-order partially averaged system. This method provides a theoretical framework which can be used to explain the main evolutionary dynamics of a physical system that has been trapped in resonance.

Chaotic behavior in the dynamical system of a continuous stirred tank reactor

Abstract The dynamical system describing a continuous stirred tank reactor (CSTR) for the reactions A→B→C and A→C, B→D is considered. A circulating attractor with accompanying circulating orbits is shown to exist when the critical point of the system is unique and unstable. The orbit structure has been numerically found to consist of periodic orbits and chaotic behavior.

Chaos in the Kepler system

The long-term dynamical evolution of a Keplerian binary orbit due to the emission and absorption of gravitational radiation is investigated. This work extends our previous results on transient chaos in the planar case to the three-dimensional Kepler system. Specifically, we consider the nonlinear evolution of the relative orbit due to gravitational radiation damping as well as external gravitational radiation that is obliquely incident on the initial orbital plane. The variation of orbital inclination, especially during resonance capture, turns out to be very sensitive to the initial conditions. Moreover, we discuss the novel phenomenon of chaotic transition.

A new mathematical formulation of accelerated observers in general relativity. II

The observation of a general vector field based on exp* is employed to obtain formulas for the coordinate velocity and coordinate acceleration of a geodesic particle. Our results are shown to reduce to those based on a parallel transport definition of observation in special relativity. In general relativity the difference between the expressions for the coordinate velocity and coordinate acceleration derived from the two definitions of observation is given in terms of the Riemann curvature tensor.

Cotangent bundle approach to noninertial frames

The most general possible noninertial acceleration in special relativity is formulated with differential forms in the cotangent bundle. We show that the Lie derivative plays the same role in the cotangent bundle that the covariant derivative plays in the tangent bundle. We also show that a cotangent bundle analog of Fermi–Walker transport can be based upon the, ’’cotangent‐geodesic’’ equation, Luω=0. This gives a generalization of the work by Kiehn on classical Hamiltonian mechanics to special relativity.

Sustained resonance: a binary system perturbed by gravitational radiation

The general phenomena associated with sustained resonance are studied in this paper in connection with relativistic binary pulsars. We represent such a system by two point masses in a Keplerian binary system that evolves via gravitational radiation damping as well as an external tidal perturbation. For further simplification, we assume that the external tidal perturbation is caused by a normally incident circularly polarized monochromatic gravitational wave. In this case, the second-order partially averaged equations are studied and a theorem of C Robinson is employed to prove that for certain values of the physical parameters resonance capture followed by sustained resonance is possible in the averaged system. We conjecture that sustained resonance can occur in the physical system when the perturbing influences nearly balance each other.