Hubertus F. von Bremen

An efficient QR based method for the computation of Lyapunov exponents

Abstract An efficient and numerically stable method to determine all the Lyapunov characteristic exponents of a dynamical system is developed. Numerical experiments are presented highlighting some aspects of convergence, accuracy and efficiency in the computation of the Lyapunov characteristic exponents.

An efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems

This paper proposes a new approach for computing the Lyapunov Characteristic Exponents (LCEs) for continuous dynamical systems in an efficient and numerically stable fashion. The method is adapted to systems with small dimensions. Numerical examples illustrating the accuracy of method are presented.

Analytical and Experimental Study of Free Vibration Response of Soft-core Sandwich Beams

The natural frequencies and corresponding vibration modes of a cantilever sandwich beam with a soft polymer foam core are predicted using the higher-order theory for sandwich panels (HSAPT), a two-dimensional finite element analysis, and classical sandwich theory. The predictions of the higher-order theory are shown to be in good agreement with experimental measurements made with a simple experimental setup, as well as with finite element analysis. Experimental observations and analytical predictions show that the classical sandwich theory is not capable of accurately predicting the free vibration response of soft-core sandwich beams. It is shown that the vibration response of the cantilever soft-core sandwich beam consists of a group of five lower frequency shear (antisymmetric) modes that are followed by a group of four thickness-stretch (symmetric) modes. For the higher frequency range, the vibration modes alternate between groups of one-two antisymmetric and symmetric modes. For very high frequencies, interactive vibration response is observed. Experiments show that the damping properties of the foam core are manifest most noticeably in the case of thickness-stretch vibration modes, whereas the influence of damping on the anti-symmetric modes is insignificant.

A New Approach to Cycling in a 2-locus 2-allele Genetic Model

In the reproductive process new genetic types arise due to crossing over and recombination at the meiotic stage. A simplified biological model will be developed which incorporates this effect and the effect of selection. Although a chromosome may contain thousands of genes we will consider a simplified model consisting of two genetic loci, each containing two alleles of some gene. The model will be then turned into a difference equation or mapping model x* = G(x,r) where x represents the frequency distribution of genotypes in a certain infinite population, x* is this distribution one generation later and r is the recombination parameter. For a certain choice of fitness and recombination parameters the mapping exhibits several fixed points. As r is varied one of the fixed points of the mapping loses its stability due to a conjugate pair of eigenvalues of the linearized mapping leaving the unit disk. It is shown that the required non-resonance conditions and “nonlinear damping” condition are satisfied and thus the fixed point undergoes a Neimark–Sacker bifurcation to a cycling or oscillatory state. Once a cycling orbit is established one can conclude that genetic variation (over time) of the population can be maintained. This work reformulates and proves earlier observations of Alan Hastings in a way that makes the treatment of chromosomes with more genetic loci more straightforward.

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the ‘reduced’ equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping 𝒯 of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping 𝒯 is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.

A note on the computation of the largest p LCEs of discrete dynamical systems

Two efficient and numerically stable methods for the computation of the largest p Lyapunov characteristic exponents of an n dimensional discrete dynamical systems are presented. The efficiencies of the proposed methods are compared with the efficiencies of other methods through an operation count, and illustrated with an example.

Implementation of approach to compute the Lyapunov characteristic exponents for continuous dynamical systems to higher dimensions

A general method for accurately computing the Lyapunov characteristic exponents (LCEs) of continuous dynamical systems has been developed in [10]. In this paper, this method is extended to implementations on systems of arbitrary dimensions. An accurate implementation of the extension of the method to compute LCEs that uses available numerical techniques is presented for systems of arbitrary dimensions. Previously in [10], the original approach had only been implemented for general systems of up to dimension three. Closed form expressions for the exponential of skew symmetric matrices of order 4 and 5 will be presented. These formulas are used to obtain an accurate and efficient implementation of the method for dynamical systems of dimensions 4 and 5. Numerical examples illustrating the accuracy of the implementations are presented.

Global Asymptotic Stability in the Jia Li Model for Genetically Altered mosquitoes

Malaria remains a major killer with more than 1 million deaths each year in sub-Saharan Africa alone while yellow fever, dengue fever, West Nile virus, encephalitis and filariasis continue to have an impact on populations worldwide. The Anopheles strains of mosquitoes are largely responsible for the transmission of Plasmodium or malaria, the Culex tarsalis accounts largely for West Nile virus, encephalitis and filariasis and the Aedes aegypti is associated with yellow fever and dengue.

Machine-augmented composites

We have recently demonstrated that composites with unique properties can be manufactured by embedding many small simple machines in a matrix instead of fibers. We have been referring to these as Machine Augmented Composites (MAC). The simple machines modify the forces inside the material in a manner chosen by the material designer. When these machines are densely packed, the MAC takes on the properties of the machines as a fiber-reinforced composite takes on the properties of the fibers. In this paper we describe the Machine Augmented Composite concept and give the results of both theoretical and experimental studies. Applications for the material in clamping mechanisms, fasteners, gaskets and seals are presented. In addition, manufacturing issues are discussed showing how the material can be produced inexpensively.

Computational explorations into the dynamics of rings of coupled oscillators

This paper deals with the response of homogeneous and inhomogeneous rings of coupled oscillators where each individual oscillator, when uncoupled from the others, is chaotic. It is shown that coupling can bring about a wide variety of global responses, and that there is a significant range of coupling values when the response of the ring is periodic despite the fact that each oscillator is chaotic. In fact numerous periodic solutions can be found depending on the initial conditions. The response of a coupled set of homogeneous and non-homogeneous rings is also investigated showing that the behavior of such coupled compartmental models can be quite counterintuitive and sensitive to the parameters that describe the extent and nature of coupling.