The transition from simple to fractal basin boundary structure displayed by unimodal maps

One-dimensional unimodal maps of the form X_{n + 1} = F (r, x_n) exhibit periodic orbits, or p-cycles, confined within specific values of the control parameter r. For any p-cycle almost all initial conditions x_O will iterate to one of attracting fixed points of the map under iteration of the pth iterate of the map F^(p). The basin boundaries which confine points attracted to the fixed points are simple or fractal depending upon whether the cycle is a member of the period doubling cascade or is a cycle in the aperiodic region beyond the period doubling accumulation point. The topology of these basin structures and the transition from simple to fractal behaviour is analyzed as a function of control parameter in terms of a scenario which parallels that of the progression from period doubling to the aperiodic regime.     

Fractal basin boundary of a two-dimensional cubic map

The fractal basin boundary of a two-dimensional cubic map and the origin of the fractal boundary are investigated. The fractal dimensions of the boundary and of the set of homoclinic points are computed.     

Structure and crises of fractal basin boundaries

We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon.     

Diffusion in a closed basin and boundary reflection

Summary In this paper the diffusion equation is solved with the eigenvalue method for a rectangular and a circular basin. Peculiar phenomena of accumulation of pollutant are observed in the corresponding numerical investigation. These phenomena are interpreted in terms of multiple reflection on the boundary, in a suitable generalization of the image source method in the two-dimensional case.

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Riassunto In questo lavoro l'equazione di diffusione è risolta per un bacino rettangolare ed uno circolare con il metodo degli autovalori. Particolari fenomeni di accumulazione si sono osservati nel corrispondente studio numerico. Questi fenomeni sono interpretati in termini di diffusione multipla sul contorno in un'opportuna generalizzazione del metodo della sorgente immagine al caso bidimensionale.

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Резюме В зтой статье решается уравнение диффузии с помощью метода собственных знаачений для прямоугольного и круглого бассейнов. При соответствующем численном исследовании наблюдаются специфические явления накапливания загряз-няющих веществ. Эти явления интерпретируются в терминах многократного отражения от границ, при соответствующем обобщении метода мнимых источников в двумерном случае.

     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

Phase transitions on fractal lattices with long-range interactions

A fractal latticeF is defined here to comprise all points of the forma + ma+ m² a+ ... +m^qa^(q), whereq is a nonnegative integer anda, a,..., a^(q)A, whereA is a finite set of points in some Euclidean space. Providedm is not too small (in particular,m must be at least 2), the dimension ofF is shown to beD = log n/logm, wheren is the number of points inA. It is shown further that an Ising model onF, with a ferromagnetic pair interaction r^– between spins separated by a distancer, has a phase transition ifD < < 2D. On the other hand, for > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.     

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.     

Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.     

Fluctuational dissipative electromagnetic interaction between a nanoprobe and a surface

In the framework of fluctuational electromagnetic theory, analytical expressions are obtained for the dynamic dissipative damping forces acting on the probe of an atomic-force microscope (AFM), as well as between two plane surfaces at their contact. The contacts between materials typical of AFM, quartz-microbalance, and surface-force apparatus experiments are considered. The conditions for nondissipative slide are discussed. A comparison between the calculated oscillator quality factor associated with fluctuational dissipative forces and its values obtained in AFM experiments with a silicon probe and a mica sample shows that they are of the same order of magnitude; therefore, an experimental investigation of such forces is feasible.     

Grain boundary transitions in binary alloys

A thermodynamic diffuse interface analysis predicts that grain boundary transitions in solute absorption are coupled to localized structural order-disorder transitions. An example calculation of a planar grain boundary using a symmetric binary alloy shows that first-order boundary transitions can be predicted as a function of the crystallographic grain boundary misorientation and empirical gradient coefficients. The predictions are compared to published experimental observations.     

Transmission of quantum particles through a one-dimensional fractal potential barrier

We consider scattering of a nonrelativistic quantum particle by a one-dimensional fractal potential barrier carried by a generalized Cantor set. We obtain recurrence relations for the reflection coefficient and examine the scaling properties as functions of the wave number.V. D. Kuznetsov Siberian Physicotechnical Institute, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–127, July, 1993.     

Fluctuational escape from a quasi-hyperbolic attractor in the Lorenz system

Noise-induced escape from the basin of attraction of a quasi-hyperbolic chaotic attractor in the Lorenz system is considered. The investigation is carried out in terms of the theory of large fluctuations by experimentally analyzing the escape prehistory. The optimal escape trajectory is shown to be unique and determined by the saddle-point manifolds of the Lorenz system. We established that the escape process consists of three stages and that noise plays a fundamentally different role at each of these stages. The dynamics of fluctuational escape from a quasi-hyperbolic attractor is shown to differ fundamentally from the dynamics of escape from a nonhyperbolic attractor considered previously [1]. We discuss the possibility of analytically describing large noise-induced deviations from a quasi-hyperbolic chaotic attractor and outline the range of outstanding problems in this field.     

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD)method,we study bifurcations governing the escape of periodically forced oscillators in a potential well,in which a chaotic saddle plays an extremely important role.Int this paper,we find the chaotic saddle,and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property,that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins.The chaotic saddle in the Wada fractal boundary,by colliding with a chaotic attractor,leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system.We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary particularly concentrating on its discontinuous bifurcations(metamorphoses),We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries.After a final escape bifurcation,there only exists the attractor at infinity;a chaotic saddle with a beautiful pattern is left behind in phase space.     

Orientational transitions in a ferronematic layer with bistable anchoring at the boundary

A possibility of the first-order transition, as well as reentrant transitions, induced by an external magnetic field between the homeotropic phase and the hybrid homeotropically planar phase in a ferronematic liquid crystal (ferronematic) with bistable anchoring at the layer boundary is demonstrated in the framework of a continuum theory. The critical values of the material parameters of the ferronematic, the anchoring energy, the thickness of the layer, and the magnetic field strength, for which this transition is possible, are determined. The cases of positive and negative diamagnetic anisotropy of the ferronematic are considered.