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.     

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.     

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.     

Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems

We consider nearly-integrable systems under a relatively small dissipation. In particular we investigate two specific models: the discrete dissipative standard map and the continuous dissipative spin-orbit model. With reference to such samples, we review some analytical and numerical results about the persistence of invariant attractors and of periodic attractors.      

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.     

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.     

Dynamics with riddled basins of attraction in models of interacting populations

Recently it has been shown that when there are chaotic attractors whose basins are such that every point in the basin has pieces of another attractors's basin arbitrarily nearby, the basins are said to be riddled. A key requirement for the occurrence of a riddled basin is the loss of transverse stability of an invariant subspace, of dimension less than the full space, containing a chaotic attractor. This type of complex dynamics has been found in simple models of interacting populations for which the invariant subspace is defined by the extinction of one species. The characterizations and implications of these behaviors for population ecology are discussed.     

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.     

Oscillation types,multistability, and basins of attractors in symmetrically coupled period-doubling systems

Symmetrically coupled nonlinear oscillator systems demonstrating transition to chaos via a sequence of period-doubling bifurcations under variation of the control parameter exhibit various types of mutual synchronization. For these coupled systems, with dissipatively coupled logistic maps, we consider a hierarchy of possible oscillation types using the value of the time shift between oscillations of the subsystems as a basis for the classification of multistable states. For oscillation states and their basins of attraction the ways of evolution are studied under variation of the parameters of nonlinearity and coupling. The obtained results are compared with those of physical experiment with a system of coupled, periodically driven nonlinear resonators.     

Bifurcations of basins of attraction from the view point of prime ends

In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Carathéodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.     

Lauwerier吸引子结构和SBR测度

Abstract. In this paper,the Lauwerier map     

Periodic Solutions of Nonlinear Wave Equations with Dissipative Boundary Conditions

Applying Nash-Moser's implicit function theorem, the author proves the existence of periodic solution to nonlinear wave equation u_{tt} - u_{xx} + εg(t, x, u, u_t, u_x, u_{tt}, u_{tx}, u_{xx}) = 0 with a dissipative boundary condition, provided ε is sufficiently small.     

Newton’s method’s basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton’s method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.     

Noninvertibility and the structure of basins of attraction in a model adaptive control system

Summary We present a two-dimensional, nonlinear map, arising from a simple adaptive control problem, which exhibits disconnected boundaries separating the basins of attraction of its coexisting attractors. We perform a detailed study of the relation between this phenomenon and the noninvertible nature of the map and demonstrate how the complex basin structure is caused by a change in the number of preimages of points along a stable manifold.     

Selection of windows, attractors and self-similar patterns in a coupled map lattice

It is shown that a lattice of diffusively coupled logistic maps displays self-similar period-doubling cascades to chaos with all the known stages of pattern formation. The location of the self-similar patterns is determined. The basins of attraction yielding window structures, so far believed to be negligibly small, are shown to cover virtually all initial conditions given a certain maximum amplitude to the random initial conditions. As a consequence a means for selecting attractors in a CML is obtained. A new pattern selection regime at high nonlinearity is reported and the basins of attraction of some attractors of small lattices are investigated.     

Periodic points in the intersection of attracting immediate basins boundaries

We give conditions under which the intersection between two attracting immediate basins boundaries of a rational map contains at least one periodic point.     

Dissipative homoclinic loops of two‐dimensional maps and strange attractors with one direction of instability

We prove that when subjected to periodic forcing of the form certain two‐dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai‐Ruelle‐Bowen measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov and applies the recent theory of rank 1 maps developed by Wang and Young. We prove a general theorem and then apply this theorem to an explicit model: a forced Duffing equation of the form © 2011 Wiley Periodicals, Inc.     

Records in the classical and quantum standard map

Record statistics is the study of how new highs or lows are created and sustained in any dynamical process. The study of the highest or lowest records constitute the study of extreme values. This paper represents an exploration of record statistics for certain aspects of the classical and quantum standard map. For instance the momentum square or energy records is shown to behave like that of records in random walks when the classical standard map is in a regime of hard chaos. However different power laws is observed for the mixed phase space regimes. The presence of accelerator modes are well-known to create anomalous diffusion and we notice here that the record statistics is very sensitive to their presence. We also discuss records in random vectors and use it to analyze the quantum standard map via records in their eigenfunction intensities, reviewing some recent results along the way.