Quadratic Map Research Articles

Order preserving maps on quantum measurements

We study the partially ordered set of equivalence classes of quantum measurements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement outcomes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps.

Constructing keyed strong S-Box with higher nonlinearity based on 2D hyper chaotic map and algebraic operation

As the only nonlinear component, S-Box plays an important role in cryptography. To overcome the problems in some 1D chaotic maps such as poor randomness and lacking ergodicity, we constructed a 2D exponential quadratic chaotic map (2D-EQCM), and analyzed its dynamic behavior through phase diagram, Lyapunov exponent, Kolmogorov entropy, correlation dimension and randomness testing. The results demonstrated that the 2D-EQCM with ergodicity and better randomness can be served as pseudo-random number generator (PRNG). Furthermore, to generate a large number of S-Boxes with higher nonlinearity, we designed a keyed strong S-Box construction algorithm using the 2D-EQCM and algebraic operation based on seed S-Boxes. Experimental results verified the effectiveness of the proposed keyed strong S-Box construction algorithm.

On CR maps from the sphere into the tube over the future light cone

We determine all local smooth or formal CR maps from the unit sphere S3⊂C2 into the tube T:=C×iR3⊂C3 over the future light cone C:={x∈R3:x12+x22=x32,x3>0}. This result leads to a complete classification of proper holomorphic maps from the unit ball in C2 into Cartan's classical domain of type IV in C3 that extend smoothly to some boundary point. Up to composing with CR automorphisms of the source and target, the classification consists of four algebraic maps. Two maps among them were known earlier in the literature. They can be generalized to higher dimensional cases and were shown to be “rigid” when the source dimension is at least 4 in a recent paper by Xiao and Yuan. Two newly discovered quadratic polynomial maps provide counterexamples to a conjecture appeared in the same paper for the case of dimension two.

Renormalization of the Two-Dimensional Border-Collision Normal Form

All nondegenerate, continuous, piecewise-linear maps on [Formula: see text] with two pieces are equivalent to a member of a four-parameter family of maps known as the two-dimensional border-collision normal form. This paper shows how the powerful technique of renormalization can be applied to this family and reveals previously undescribed bifurcation structure in a succinct way. We partition a parameter region where the family is known to exhibit chaos robustly into infinitely many subregions in an explicit way. We then show the chaotic attractor has different numbers of connected components in different subregions. The results rely on a careful analysis of the global dynamics of the renormalization operator. This is challenging because the operator is essentially a quadratic map on [Formula: see text].

A cluster of 1D quadratic chaotic map and its applications in image encryption

Chaotic systems are widely used in designing encryption algorithms for their ideal dynamical performances. One-dimensional (1D) chaotic maps have the highest efficiency in implementation and have achieved great attention. However, 1D chaotic maps have a common security weakness, which is that their key space is relatively small. Therefore, in this paper, a cluster of 1D quadratic chaotic maps is proposed according to the topological conjugate theory. The 1D chaotic map has three tunable parameters which have a significant expansion in parameter space than the traditional 1D chaotic maps. The 1D map is proved to be chaotic theoretically since it is topologically conjugated with a logistic chaotic map. An example of a 1D quadratic chaotic map is provided in this paper, and several numerical simulation results indicate that this 1D quadratic map has ideal chaotic characteristics, which are consistent with the theoretical analysis. To verify the effectiveness of this 1D quadratic map, a novel image encryption algorithm is proposed. The security of this image encryption algorithm is completely dependent on the properties of the 1D quadratic map. Security experimental test results show that this image encryption algorithm has a high-security level, and is quite competitive with other chaos-based image encryption algorithms.

Quasi-adelic measures and equidistribution on

AbstractBaker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

Quadratic rational maps with integer multipliers

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps that have an integer multiplier at each cycle.

Quadratic and cubic Newton maps of rational functions

The dynamics of all quadratic Newton maps of rational functions is completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three of any rational function is conformally conjugate to a unicritical polynomial (i.e., with exactly one finite critical point). However, there are cubic Newton maps which are conformally conjugate to other polynomials. The Julia set of such a Newton map is shown to be a closed curve. It is a Jordan curve whenever the Newton map has two attracting fixed points.

Simultaneous rational periodic points of degree-2 rational maps

Let S be the collection of quadratic polynomial maps, and degree 2-rational maps whose automorphism groups are isomorphic to C2 defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in S, we give a complete description of triples (f1,f2,p) such that p is a rational periodic point for both fi∈S, i=1,2. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers a,b are periodic points of the map ϕt1,t2(z)=t1z+t2/z for infinitely many nonzero rational pairs (t1,t2) if and only if a2=b2.

Quadratic Rational Maps with a 2-Cycle of Siegel Disks

For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.

A brief note on fractal dynamics of fractional Mandelbrot sets

This paper preliminary examines a kind of Mandelbrot set generated by a fractional difference quadratic map involving Caputo-like fractional h-difference operators. A connectivity index is proposed based on numerical methods, which avoids difficulties in discussion at the topological level. The dynamics of those sets in two kinds of noise environments are considered involving connectivity, symmetry and dimension. Several typical cases are visualized to illustrate the main conclusions.

Constructing a 3D Exponential Hyperchaotic Map with Application to PRNG

Some weaknesses of 1D chaotic maps, such as lacking of ergodicity, multiple bifurcations, dense periodic windows, and short iteration period, limit their practical applications in cryptography. A higher-dimensional chaotic map with ergodicity can solve these problems. Based on 1D quadratic map, a 3D exponential hyperchaotic map (3D-EHCM) is constructed, and its dynamic behaviors, such as phase diagram, Lyapunov exponent spectrum, Kolmogorov entropy (KE), correlation dimension, approximate entropy and randomness, are analyzed and tested. The results demonstrate that the 3D-EHCM has ergodicity in a larger range of control parameter, and its state points have a longer period. To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.

A novel transmuted lomax exponential distribution: Properties and applications

Statistical lifetime distributions play a very important role in modeling data sets in various fields. Extending the existing distributions is of great interest in statistical research. The modification of the distributions provides more flexible model as compared to existing one. In this article, we propose a new probability model using Quadratic Rank Transmutation Map technique, named as Transmuted Lomax Exponential Distribution (TLED). The new distribution can model data sets with increasing, decreasing and bathtub shape hazard rates. Various statistical properties of the proposed distribution such as moments, order statistics, quantile function, mean residual life function and characteristic function are derived. Further, the parameter estimates are obtained through Maximum Likelihood method along with asymptotic confidence intervals. The utility of the new model is evaluated by analyzing two real data sets. In order to access the performance of the new model, several goodness of fit measures is used. The results indicate that the new model best fits the data as compared to the other extensions of the Lomax distribution.

Type-V intermittency from Markov binary block visibility graph perspective

In this work, type-V intermittency is studied from Markov binary block visibility graphs perspective. We considered a piecewise quadratic Poincare map that is a simple model to exhibit this type of intermittency. The mechanism of type-V intermittency is collision of a stable fixed point with a point of discontinuity of the Poincare map. We study the behavior of a dynamical system in the vicinity of the discontinuous or non-differentiable points (NDP) using networks language. Numerical results showed that there is a logarithmic scaling law logε for the average laminar length of the type-V intermittency. We also described their properties based on statistical tools such as the length between reinjection points and the average laminar length. For further investigation, we verified the degree distribution of the complex network generated by type-V intermittency time series and finally, predicted the behavior of type-V intermittency by the proposed theoretical degree distributions.

The Transmuted Marshall-Olkin Extended Topp-Leone Distribution

In this paper, we introduced an extension of the Marshall-Olkin Extended Topp-Leone distribution using the quadratic rank transmutation map (QRTM). Statistical properties of the proposed distribution are examined and its parameter estimates are obtained using the maximum likelihood method. A real data set defined on a unit interval is employed to illustrate the usefulness of the proposed distribution among existing distributions with bounded support.

Transmuted shifted exponential distribution and applications

In this paper, we used the quadratic transmutation map approach to propose a new distribution model called the transmuted shifted exponential distribution (TSED). We derived some of its useful statistical properties such as the moment, Characteristic function, Quantile function, Shannon Entropy, relative entropy, Renyi entropy, moment generating function, mean inactivity function, hazard rate function, survival rate function, and the order statistics. The simulation results showed that as the sample size increases, the biases, variances, and root mean square errors become smaller for the censored sample. We used two life data sets to evaluate the numerical applications of the proposed TSED. The results obtained are compared with some existing probability distribution models. It is observed that the TSED gave a better fit to all the data sets under consideration than other models.

Anti-integrability for Three-Dimensional Quadratic Maps

We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced 30 years ago for the Frenkel--Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations to obtain orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node and period-doubling bifurcations have symbol sequences that differ in exactly one position.

Iterated nth order nonlinear quantum dynamics with mixed initial states

Nonlinear quantum transformations play an important role in quantum state distillation, aiding quantum communication schemes. The simplest of these involve quadratic rational maps for qubits, higher order versions have not been explored in detail. We consider such iterated nonlinear protocols for qubits, involving a generalized CNOT, a measurement, and a single-qubit Hadamard gate processing n inputs in identical quantum states. To characterize the global properties of these protocols, we determine the fixed points and relevant fixed cycles. We show that a repelling mixed fixed point exists for all orders, marking a phase transition related to the disappearance of the fractalness of borders of different basins of attraction.

Development of an open framework for a qualitative and quantitative comparison of power system and electricity grid models for Europe

The ongoing needs to develop power systems towards more environmentally friendly technologies with respect to climate change in conjunction with the continuous evolution of the respective market conditions is leading to a transition away from the traditional system operation. The upcoming challenges have motivated the development of an increasing number of models for transmission grids. Nevertheless, the high complexity of such models renders it exceedingly difficult to compare their results as well as any corresponding conclusions.In this paper, we develop an open framework to compare a variety of pan-European transmission grid models with a strong focus on the German power system. The comparison is performed in both a qualitative and quantitative manner, depending on the investigated modeling aspect including input data, methods, system boundaries and results. The quantitative model comparison is done by performing harmonized model experiments, one for 2016 as back testing and one for 2030 for analyzing the future system.Core elements of our comparison framework are:We proved that our comparison framework is suitable to make similarities and differences between the different model results visible, e.g. using quadratic heat maps. To ensure transparency and to support the open modeling community, the fact sheets with the model specifications and the database with selected model results are uploaded on the open energy platform.

On elastic graph spines associated to quadratic Thurston maps

For a quadratic Thurston map having two distinct critical points and $n$ postcritical points, we count the number of possible dynamical portraits. We associate elastic graph spines to several hyperbolic quadratic Thurston rational functions. These functions have four postcritical points, real coefficients, and invariant real intervals. The elastic graph spines are constructed such that each has embedding energy less than one. These are supporting examples to Dylan Thurston's recent positive characterization of rational maps. Using the same characterization, we prove that with a combinatorial restriction on the branched covering and a cycle condition on the dynamical portrait, a quadratic Thurston map with finite postcritical set of order $n$ is combinatorially equivalent to a rational map. This is a special case of the Bernstein-Levy theorem.