Discrete Nonlinear Planar Systems and Applications to Biological Population Models

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates

Discrete Breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm

The metron model: Towards a unified deterministic theory of fields and particles

A summary is given of the principal concepts of a unified deterministic theory of fields and particles that have been developed in more detail in a pre- vious comprehensive four-part paper (Hasselmann, 1996a,b, 1997a,b). The model is based on the Einstein vacuum equations, Ricci tensor RLM = 0, in a higher-dimensional space. A space of at least eight dimensions is re- quired to incorporate all other forces as well as gravity in Einstein’s gen- eral relativistic formalism. It is hypothesized that the equations support soliton-type solutions (”metrons”) that are localized in physical space and are periodic in extra (”harmonic”) space and time. The solitons represent waves propagating in harmonic space that are locally trapped in physical space within a wave guide produced by a distortion of the background met- ric. The metric distortion, in turn, is generated by nonlinear interactions (radiation stresses) of the wave field. (The mutual interaction mechanism has been demonstrated for a simplified Lagrangian in Part 1 of the previous paper). In addition to electromagnetic and gravitational fields, the metron solutions carry periodic far fields that satisfy de Broglie’s dispersion relation. These give rise to wave-like interference phenomena when particles interact with other matter, thereby resolving the wave-particle duality paradox. The metron solutions and all particle interactions on the microphysical scale (with the exception of the kaon system) satisfy strict time-reversal symmetry, an arrow of time arising only at the macrophysical level through the introduc- tion of time-asymmetrical statistical assumptions. Thus Bell’s theorem on the non-existence of deterministic (hidden variable) theories, which depends crucially on an arrow-of-time, is not applicable. Similarly, the periodic de Broglie far fields of the particles do not lead to unstable radiative damping, the time-asymmetrical outgoing radiation condition being replaced by the time-symmetrical condition of zero net radiation. Assuming suitable polarization properties of the metron solutions, it can be shown that the coupled field equations of the Maxwell-Dirac-Einstein sys- tem as well as the Lagrangian of the Standard Model can be derived to low- est interaction order from the Einstein vacuum equations. Moreover, since Einstein’s vacuum equations contain no physical constants (apart from the introduction of units, namely the velocity of light and a similar scale for the harmonic dimensions, in the definition of the flat background metric), all physical properties of the elementary particles (mass, charge, spin) and all universal physical constants (Planck’s constant, the gravitational constant, and the coupling constants of the electroweak and strong forces) must fol- low from the properties of the metron solutions. A preliminary inspection of the structure of the solutions suggests that the extremely small ratio of gravitational to electromagnetic forces can be explained as a higher-order nonlinearity of the gravitational forces within the interior metron core. The gauge symmetries of the Standard Model follow from geometrical symme- tries of the metron solutions. Similarly, the parity violation of the weak interactions is attributed to a reflexion asymmetry of the metron solutions (in analogy to molecules with left- and right-rotational symmetry), rather than to a property of the basic Lagrangian. The metron model also yields further interaction fields not contained in the Standard Model, suggesting that the Standard Model represents only a first-order description of elemen- tary particle interactions. While the Einstein vacuum equations reproduce the basic structure of the fields and lowest-order interactions of quantum field theory, the particle content of the metron model has no correspondence in quantum field theory. This leads to an interesting interpretation of atomic spectra in the metron model. The basic atomic eigenmodes of quantum electrodynamics appear in the metron model as the scattered fields generated by the interaction of the orbiting electron with the atomic nucleus. For certain orbits, the eigenmodes are in resonance with the orbiting electron. In this case, the eigenmode and orbiting electron represent a stable self-supporting configuration. For circular orbits, the resonance condition is identical to the integer-action condition of the Bohr orbital model. Thus the metron interpretation of atomic spectra yields an interesting amalgam of quantum electrodynamics and the original Bohr model. However, it remains to be investigated whether higher-order computations of the metron model are able to reproduce atomic spectra to the same high degree of agreement with experiment as QED. On a more fundamental level, the basic questions of the existence, structure, stability and discreteness of the postulated metron solutions still need to be addressed. However, it is encouraging that, already on the present exploratory level, the basic properties of elementary particles and fields, including the origins of particle properties and the physical constants, can be explained within a unified classical picture based on a straightforward Kaluza-Klein extension to a higher dimensional space of the simplest vacuum form of Einstein’s gravitational equations

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Dynamics of a rational system of difference equations in the plane

We consider a rational system of first order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions and the stability of equilibria