Intertwined basins of attraction of dynamical systems

In this paper, we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results.     

Non-Autonomous basins of attraction and their boundaries

We study whether the basin of attraction of a sequence of automorphisms of ℂ ^{k is biholomorphic to} ℂ ^{k. In particular, we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to} ℂ ^{k, if every map is iterated sufficiently many times. We also construct Fatou-Bieberbach domains in} ℂ² whose boundaries are four-dimensional.     

The structure and distribution of prime ends

Summary In this paper, we study the prime ends of a simply connected, plane domain. We use the metric concept of the separation of subsets of a domain to define the lateral points of a prims end (Section 3). The idea of right and left side-chains of rankn then leads to the concept of right and left lateral points of rankn (n = 1, 2, ..., ) (Section 5). Thus we achieve an analysis which goes beyond Carathéodory's classification (sea Section 2) of the points of a prime end into principal and subsidiary points (Hauptpunkte andNebenpunkte). Each subsidiary point of a prime endP is a lateral point ofP; but even a principal point ofP can be a right and left lateral point of infinite rank ofP.The need for the new concepts and for the extended analysis arises from the following fact: There exist very simple domains with boundary points that are principal points of a prime end but which, at the same time, possess some of the properties of subsidiary points, relative to the same prime end. In other words, a boundary point may not only belong to two or more prime ends; but in each of the prime ends, it may play a multiple role.It is known [2, p. 349] that, in the space of the prime ends of a fixed domain, the set of prime ends having subsidiary points is a set of first category. In Section 3, we show that the analogue of this proposition for lateral points is false; all the prime ends of certain highly laminar domains have lateral points.A theorem of Lindelöf and others ([4, p. 28]; see [2, p. 347] for further references) asserts that if the function maps the unit disk ¦ ¦ < 1 conformally onto a simply connected domain, andL is a Stolz path approaching the point = 1, then the closure of the set (L) contains no subsidiary points of the prime end corresponding to the point = 1. In Section 4, we formulate and prove an analogous theorem for lateral points.In Section 6, we consider a special phenomenon which at first sight might be expected to re exceptional, but which we show can affect a residual set of prime ends. Section 7 deals with a theorem of Frankl.G. Piranian's contribution to this paper was made under Contract DA 20-018-ORD-13585, Office of Ordnance Research, U. S. Army.     

Prey-predator systems with intertwined basins of attraction

In this article, we show the existence of intertwined basins of attraction for a class of prey-predator systems, which improves the extant results by deleting some crucial and hard testing conditions.     

Two characteristics of planar intertwined basins of attraction

In this paper, we investigate the intertwined basins of attraction for planar dynamical systems. We prove that the intertwining property is preserved by topologically equivalent systems. Two necessary and sufficient conditions for a planar system having intertwined basins are given.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

Smoothly intertwined basins of attraction in a prey-predator model

No fractals, no chaos, yet hard to predict. Using an easy version of the straddle-orbit procedure introduced by Grebogi and co-workers, we investigate the basins of attraction of a system of ODE in the plane of interest for the biological control of parasites in agriculture.All the phase space, except for a smooth curve separating the basins, consists of points that asymptotically evolve to one of two possible equilibria. The prediction of which final equilibrium will be attained by the system is nevertheless obstructed by the intertwining of the two basins, that indefinitely accumulate on each other in a region bounded by the coordinate axis.Research partially supported by Ministero della Università della Ricerca Scienufka e Tecnologica (MURST) and GNFM-CNR.     

A comparison between iterative methods by using the basins of attraction

There exists a real competition between authors to construct improved iterative methods for solving nonlinear equations. In this paper, by using computer experiment, we study the basins of attraction for some of the iterative methods for solving the equation P(z) = 0, where P:C→C is a complex coefficients polynomial, and this allows us to compare their performances (the area of convergence and theirs speed). The beauty fractal pictures generated by these methods are presented too.     

Perfect codes from the dual point of view I

The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.     

Convergence acceleration from the point of view of linear programming

The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Euler's transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.     

The theory of manifolds from the non-deterministic point of view

Summary Using the methods of non-deterministic analysis it is possible to define in a differentiable manifold a special structure called Gauss structure which allows one to study differentiable maps from one manifold into another without the use of local charts. The main theorem in this paper shows how it is possible to locally recover the jacobian of such maps by using the Gauss structures defined in the manifolds in question.     

Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

Structured continua from a lagrangian point of view

Sunto In questo lavoro si studiano le ipersuperfici di bigrado (p, 2)in P ¹×P ^m ,m3m dispari p1.Tali ipersuperfici per p=1si presentanto in modo naturale nello studio del problema di Torelli per le intersezioni complete di due quadriche trasverse in P ^m .Viene analizzata la loro struttura e dimostrato che per p2il Teorema di Torelli non vale per tali ipersuperfici mentre vale per p=1 (Turin).

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Partially supported by a C.N.R. fellowship at the University of Georgia, Athens, Georgia.     

Generalized quadrangles from a local point of view

In this lecture, I will survey several recent results in the local theory of generalized quadrangles. Starting with a short introduction to the global automorphism theory, I will motivate as such the local viewpoint, and overview some of the most important local properties which are investigated nowadays. Recent results on skew translation quadrangles and forms will be described, including a solution of a question of Payne which generalizes work of Havas et?al. (Finite geometries, groups, and computation, 2006; Adv Geom 26:389?C396, 2006), and then I will mention parts of a classification of skew translation quadrangles which is being prepared by the author. Finally, I will consider conditions which are both global and local.     

Displacement interpolations from a Hamiltonian point of view

One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the Finsler manifolds and manifolds with a Ricci flow background. In this paper, we study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results.