Papers by William F Langford
A new energy balance model (EBM) is presented and is used to study paleoclimate transitions. Whil... more A new energy balance model (EBM) is presented and is used to study paleoclimate transitions. While most previous EBMs only dealt with the globally averaged climate, this new EBM has three variants: Arctic, Antarctic and tropical climates. The EBM incorporates the greenhouse warming effects of both carbon dioxide and water vapour, and also includes ice-albedo feedback and evapotranspiration. The main conclusion to be inferred from this EBM is that the climate system may possess multiple equilibrium states, both warm and frozen, which coexist mathematically. While the actual climate can exist in only one of these states at any given time, the EBM suggests that climate can undergo transitions between the states via mathematical saddle-node bifurcations. This paper proposes that such bifurcations have actually occurred in Paleoclimate transitions. The EBM is applied to the study of the Pliocene paradox, the glaciation of Antarctica and the so-called warm, equable climate problem of both the mid-Cretaceous Period and the Eocene Epoch. In all cases, the EBM is in qualitative agreement with the geological record.
CRC Press eBooks, Nov 25, 2020
The referee quotes several sentences from the Introduction to make this point. The authors will r... more The referee quotes several sentences from the Introduction to make this point. The authors will rewrite those introductory paragraphs, to better position this contribution in the discipline. The referee then points out that the interest of an EBM bifurcation analysis is not to provide an accurate prediction that would supersede the current state-of-the-art. It can provide a closer examination of the conditions that would generate a bifurcation.
... CMS CONFERENCE PROCEEDINGS Editorial Board Frederick V. Atkinson Bernhard Banaschewski Colin ... more ... CMS CONFERENCE PROCEEDINGS Editorial Board Frederick V. Atkinson Bernhard Banaschewski Colin W. Clark Erwin O. Kreyszig (Chairman) John B ... and LOUISE A. RAPHAEL 221 Distributional solutions of ordinary differential equations by ALLAN M. KRALL, RP KANWAL ...
These files are MATLAB software used to generate the figures in "Anthropocene Climate Change... more These files are MATLAB software used to generate the figures in "Anthropocene Climate Change", to be published in Nonlinear Processes in Geophysics, 2020.
Contemporary Mathematics, 1986
Journal of Physics: Conference Series, 2021
Climate models predict that the climate of the Earth is warming and will continue to warm in comi... more Climate models predict that the climate of the Earth is warming and will continue to warm in coming centuries, if there is no mitigation. A recent energy balance model [Kypke et al., Nonlin. Process. Geophys. 27 (2020) 391–409] forecasts that, if the current increase of carbon dioxide in the atmosphere continues unabated, then in the next century the climate of the Earth will not only get warmer, but will transition abruptly via a bifurcation, to a warm equable climate unlike any climate seen on Earth since the Pliocene. This transition to a new climate state is a topological change. That model includes the effects of water vapour feedback and ice albedo feedback, as well as ocean and atmospheric heat transport. This paper adds to that model further amplification by permafrost feedback. That is, as the Arctic warms, permafrost will thaw, releasing large amounts of the greenhouse gases carbon dioxide and methane, which cause further warming. Since knowledge of permafrost stores and r...
In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam... more In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather
In the Pliocene Epoch (5.3–2.6 million years ago), there was an abrupt cooling of the Arctic, fro... more In the Pliocene Epoch (5.3–2.6 million years ago), there was an abrupt cooling of the Arctic, from an ice-free to an ice-covered climate state. A simple conceptual mathematical model of Arctic climate is used to explore the potential role of forcing factors, such as \(\mathrm {CO}_2\) concentration and ocean heat transport to the Arctic, as well as nonlinear feedback mechanisms, such as ice-albedo feedback and water vapour feedback, in the climate change of the Pliocene Arctic. The mathematical model provides a plausible explanation for this abrupt climate change, involving both of these forcing factors and both of the nonlinear feedback mechanisms. The model also sheds light on the fact that modern general circulation models have been unable to reproduce this dramatic change in Arctic climate.
The theory of bifurcation of solutions to two-point boundary value problems is developed for a sy... more The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered. The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point ...
Bifurcation Problems and their Numerical Solution, 1980
Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation ca... more Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation can produce secondary bifurcations of periodic solutions, and tertiary bifurcations of periodic or aperiodic solutions lying on an invariant torus. A complete classification of the resulting bifurcation diagrams is presented, with emphasis on the cases which exhibit tertiary bifurcation. Calculations involving successive transformations to polar normal forms lead to existence theorems for the secondary and tertiary solutions and asymptotic formulae for the invariant torus.
Numerical Methods for Bifurcation Problems, 1984
The normal form equations for the interactions of a Hopf bifurcation and a hysteresis bifurcation... more The normal form equations for the interactions of a Hopf bifurcation and a hysteresis bifurcation of stationary states can give rise to an axisym-metric attracting invariant torus. NonaxiSymmetrie perturbations are found to produce phase locking, period doubling, bistability, and a family of strange attractors.
... Volume 4 William F. Langford and Wayne Nagata, Editors Normal forms and homoclinic chaos 1995... more ... Volume 4 William F. Langford and Wayne Nagata, Editors Normal forms and homoclinic chaos 1995 3 Anthony Bloch, Editor Hamiltonian and gradient flows, algorithms and controls 1994 2 KA Morris, Editor Control of flexible structures 1993 1 Michael J. Enos, Editor Dynamics ...
Dynamical Systems and Their Applications in Biology, 2003
Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms
The paper deals with the problem of secure communication with “synchronized chaos” as a masking s... more The paper deals with the problem of secure communication with “synchronized chaos” as a masking scheme. An algorithm is suggested that has no drawbacks of the previously reported methods. Namely, the hidden signals in conventional schemes can be unmasked using the phase space reconstruction technique. The presented communication scheme, that works through a modular arithmetic filter, is shown not to have this drawback.
Canadian Applied Mathematics Quarterly
Reanalyses of climate data for recent decades have indicated that the Hadley cells of the atmosph... more Reanalyses of climate data for recent decades have indicated that the Hadley cells of the atmospheric circulation are expanding toward the poles as well as slowing in their circulation velocity. Similarly, recent meteorological data show a poleward movement of the jet streams that affect midlatitude weather. Although the precise mechanism of these changes in Hadley cells and jet streams is not fully understood, it is believed to be linked with global warming. In this paper we study a simple mathematical model to investigate whether such qualitative changes could be induced purely by the physics of rotating spherical convection. The model consists of the Navier-Stokes equations for a Boussinesq fluid, rotating in a spherical shell, differentially heated with a latitudinal surface temperature gradient. Many other factors that influence the atmospheric circulation are excluded from this model. A decrease in the pole-to-equator temperature gradient in the model leads to an expansion and...
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Papers by William F Langford