Riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space. An example is the synchronized motion of two identical chaotic oscillators. The paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps. We determine the regio...

The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi’s polynomial maps and groups of degree 2 is established and used to study the structure of the latter. Introduction. Polynomial maps appear in nilpotent group theory...

Iteration of randomly chosen quadratic maps defines a Markov process: Xn+1 = εn+1Xn(1−Xn), where εn are i.i.d. with values in the parameter space [0,4] of quadratic maps Fθ(x) = θx(1− x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Ha...

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a ...

Orbit portraits were introduced by Goldberg and Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical antiholomorphic polynomials, and give an explicit description of the orbit ...

The assumption of Gaussianity of the primordial perturbations plays an important role in modern cosmology. The most direct test of this hypothesis consists in testing Gaussianity of the CMB maps. Counting the pixels with the temperatures in given ranges and thus estimating the one point probability function of the field is the simplest of all the tests. Other usually more complex tests of Gauss...

The complex dynamical analysis of the cubic-order iterative family is proposed to draw fractal images via M\"{o}bius conjugacy map applied a quadratic polynomial $(z-A) (z-B)$. resulting dynamics clearly visualized through various stability surfaces and parameter spaces using Mathematica.

In this paper we describe the dynamics of singularly perturbed complex polynomials. That is, we start with a complex polynomial whose dynamics are well understood. Then we perturb this map by adding a pole, i.e., by adding in a term of the form λ/(z − a)d where the parameter λ is complex. This changes the polynomial into a rational map of higher degree and, as we shall see, the dynamical behavi...

The main idea of this chapter is to try to measure to which extent a surface S is different from a plane, in other words, how “curved” is a surface. The idea of doing this is by assigning to each point P on S a unit normal vector N(P ) (that is, a vector perpendicular to the tangent plane at P ). We are measuring to which extent is the map from S to R given by P 7→ N(P ) (called the Gauss map) ...

We propose a randomized relax-and-round inference algorithm that samples near-MAP configurations of a binary pairwise Markov random field. We experiment on MAP inference tasks in several restricted Boltzmann machines. We also use our underlying sampler to estimate the log-partition function of restricted Boltzmann machines and compare against other sampling-based methods. 1. Background and setu...