Analysis of discrete dynamical system models for competing species

A discrete version of the Lotka-Volterra (LV) differential equations for competing population species is analyzed in detail, much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. Another related system, namely, the Exponentially Self Regulating (ESR) population model, is also thoroughly analyzed. It is found that in addition to logistic dynamics - ranging from the very simple to manifestly chaotic regimes in terms of the governing parameters - the discrete LV model and the ESR model exhibit their own brands of bifurcation and chaos that are essentially two dimensional in nature. In particular, it is shown that both systems exhibit twisted horseshoe dynamics associated to a strange invariant set for certain parameter ranges

Spatial chaos of an extensible conducting rod in a uniform magnetic field

The equilibrium equations for the isotropic Kirchhoff rod are known to form an integrable system. It is also known that the effects of extensibility and shearability of the rod do not break the integrable structure. Nor, as we have shown in a previous paper does the effect of a magnetic field on a conducting rod. Here we show, by means of Mel'nikov analysis, that, remarkably, the combined effects do destroy integrability; that is, the governing equations for an extensible current-carrying rod in a uniform magnetic field are nonintegrable. This result has implications for possible configurations of electrodynamic space tethers and may be relevant for electromechanical devices

Attempts to relate the Navier-Stokes equations to turbulence

The present talk is designed as a survey, is slanted to my personal tastes, but I hope it is still representative. My intention is to keep the whole discussion pretty elementary by touching large numbers of topics and avoiding details as well as technical difficulties in any one of them. Subsequent talks will go deeper into some of the subjects we discuss today. The main goal is to link up the statistics, entropy, correlation functions, etc., in the engineering side with a "nice" mathematical model of turbulence

Stabilization of heterodimensional cycles

We consider diffeomorphisms $f$ with heteroclinic cycles associated to saddles $P$ and $Q$ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $P$ and $Q$. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.Comment: 31 pages, 9 figure

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Distribution of resonances for open quantum maps

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.Comment: 60 pages, no figures (numerical results are shown in other references

Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits

The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic and heteroclinic orbits of saddle points. These orbits are most readily computed and studied as intersections of unstable and stable manifolds comprising homoclinic or heteroclinic tangles in the surface. We show how to compute a map of a one-dimensional space similar to a train-track which represents the isotopy-stable dynamics of the surface diffeomorphism relative to a tangle. All orbits of this one-dimensional representative are globally shadowed by orbits of the surface diffeomorphism, and periodic, homoclinic and heteroclinic orbits of the one-dimensional representative are shadowed by similar orbits in the surface.By constructing suitable surface diffeomorphisms, we prove that these results are optimal in the sense that the topological entropy of the one-dimensional representative is the greatest lower bound for the entropies of diffeomorphisms in the isotopy class.Comment: Version submitted to "Dynamical Systems: An International Journal" Section 7 has been further revised; the method for pA maps is new. Notation has been standardised throughou