Gerhard Dangelmayr

Do Fires in Savannas Consume Woody Biomass? A Comment on Approaches to Modeling Savanna Dynamics

Savanna ecosystems have long been fertile ground for mathematical modeling of vegetation structure and the role of resources and disturbance in tree‐grass coexistence. In recent years, several authors have presented models that explore how savanna fires suppress the woody community, alter ecosystem dynamics, and promote grass persistence. We argue, however, that the assumption that fires influence savanna dynamics by consuming woody biomass may be wrong because, in reality, fires kill seedlings and saplings that constitute little biomass relative to adult trees. We present a simple alternative that separates the woody community into a subadult (fire‐sensitive) class and an adult (fire‐resistant) class and explore how this ecologically more realistic, but still simplified, model may provide better simulations of demographic processes and response to fires in savannas.

Steady-state mode interactions in the presence of 0(2)-symmetry

This paper studies the multiple bifuraction phenomenon of steady-state mode interactions in the presence of 0(2)-symmetry. For such problems the flow on the centre manifold is determined by a vector field in or that is equivariant under an action of 0(2). The action is related to the wave numbers of the unstable modes. The unfolded normal forms for these equivariant bifurcation problems admit primary bifurcations to single-mode solutions, secondary bifurcations to mixed-mode solutions and, in some instances, tertiary bifurcations to travelling and standing waves. The bifurcation behaviour depends crucially on the wave numbers. For small wave numbers, the mixed-mode solutions encounter subordinate saddle-node bifurcations.

A model problem in the representation of digital image sequences

Abstract An optimal feature based representation is computed for the characterization of sequences of a family of digital images arising in the animation of a speaking face. The relation between spatial and temporal correlations is considered and two different experiments are presented. A low-dimensional characterization of lip motion is generated in terms of the 20 most significant features. The mathematical framework allows both the synthesis (generation of real and simulated motion) and analysis (classification) of lip motion. Words are represented both as small matrices and as curves in the plane and are shown to have distinct signatures which are apparently robust. Typical compression ratios vary from O (100:1) to O (1000:1).

A codimension three bifurcation for the laser with saturable absorber

We study the five mode equations which are used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance. We show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue. A center manifold reduction is performed to fix the three-dimensional submanifold in parameter space where this degeneracy occurs. The associated Takens-normal form is given. By unfolding the normal form we obtain all structurally stable phase portraits near this bifurcation point and display them in the form of bifurcation diagrams with the laser pumping rate as a distinguished bifurcation parameter. These diagrams allow a unifying analytical and geometrical description of many different numerical solutions of the equations describing a laser with absorber. In particular, they yield the connection of the small amplitude periodic solutions with passiveQ-switching and suggest new bifurcation processes, which one can expect to occur for physical parameters near the critical submanifold. The existence of a codimension four bifurcation is indicated.

Dynamics of Travelling Waves in Finite Containers

A multiscale expansion is used to show that distant sidewalls can cause a travelling wave to reverse periodically its direction of propagation. These reversing states are two-frequency waves and appear via a secondary Hopf bifurcation from a pattern of counterpropagating waves. With increasing Rayleigh number the reversal period diverges and the reversals may become chaotic, before a hysteretic transition to nonreversing waves takes place. The predictions are in qualitative agreement with existing experiments.

Parity-breaking bifurcation in inhomogeneous systems

The steady-state reflection-breaking bifurcation from a circle of nontrivial equilibria in O(2)-equivariant systems results in a pair of travelling waves. When the continuous part of the group O(2) is weakly broken, the corresponding instability may lead to nonsymmetric but steady states. The transition from this state to the travelling wave state with increasing bifurcation parameter is complex, and typically involves sequences of global bifurcations. The possible scenarios are described in detail and the results are related to the dynamics associated with parity-breaking instabilities of spatially periodic patterns in inhomogeneous systems.

Interaction between standing and travelling waves and steady states in magnetoconvection

Abstract A codimension-two bifurcation of Takens-Bogdanov type in Boussinesq magnetoconvection is used to describe the interaction between standing and travelling waves, and steady convection in the nonlinear regime.

On the Hopf Bifurcation with Broken O(2) Symmetry

Translation and reflection symmetries introduce the group 0(2) into bifurcation problems with periodic boundary conditions. The effect on the Hopf bifurcation with 0(2)-symmetry of small terms breaking the translation symmetry is investigated. Two primary branches of standing waves are found. Secondary and tertiary bifurcations involving two different types of modulated waves are analyzed in the neighborhood of secondary Takens-Bogdanov bifurcations. The effects of breaking the phaseshift (in time) and reflection symmetries are briefly considered.

Semiclassical approximation of path integrals on and near caustics in terms of catastrophes

Abstract A semiclassical approximation of Feynmans path integral is derived which, in contra-distinction to conventional formulae, remains finite in conjugate (focal) points and on caustics, exhibits the experimentally observed oscillatory behavior near these and reduces to familiar approximations far away. The approximative path integral is proportional to a generalized Airy integral governed by a Thom catastrophe polynomial whose bifurcation properties correspond to those of the solutions of Euler-Lagrange equations. The result is illustrated by computing the propagator of the quantized anharmonic oscillator.

Optimal chaos control through reinforcement learning

A general purpose chaos control algorithm based on reinforcement learning is introduced and applied to the stabilization of unstable periodic orbits in various chaotic systems and to the targeting problem. The algorithm does not require any information about the dynamical system nor about the location of periodic orbits. Numerical tests demonstrate good and fast performance under noisy and nonstationary conditions. (c) 1999 American Institute of Physics.