Circle Map Research Articles

Thematic World Maps in the News: How Effective are They on Smartphones?

ABSTRACT Thematic world maps are commonly used in online news media, but rarely optimized for mobile devices. Scaling down world maps to fit small screens makes many smaller countries difficult to identify, introducing readability issues and an area bias when reporting global data. We introduce and evaluate map designs addressing two common use cases in this context: A ‘glorified list’ use case, where the map mainly displays data about many countries and a ‘spatial distribution’ use case, where the spatial distribution of population-related phenomena is the focus. For the former, we propose a choropleth map split by continent and scales-up continents with many smaller countries and for the latter, we propose a graduated circle map split into a Western and Eastern hemisphere. In a user study with 333 readers of the Neue Zürcher Zeitung (NZZ), a daily newspaper in Switzerland, we evaluate these novel map designs.

New comparative approach to multi-level thresholding: chaotically initialized adaptive meta-heuristic optimization methods

Abstract One method aimed at enhancing the performance of meta-heuristic optimization techniques is the incorporation of chaotic systems. Instead of irregular distributions in the search space, chaotic distributions are employed in the initial population of optimization algorithms to improve the efficiency of the search process. This approach enables search agents distributed in a chaotic manner to effectively explore the search space. The initial populations of both the well-established PSO algorithm and the enhanced WSO algorithm, which incorporates advanced search techniques, are distributed in the search space according to the characteristics of Logistic, Chebyshev, Circle, Sine, and Piecewise chaotic maps in this study. The original PSO and WSO algorithms, as well as the resulting chaotically initialized PSO and chaotically initialized WSO algorithms, were tested using 23 benchmark functions. Subsequently, the Otsu method was integrated into the tested optimization algorithms to obtain multi-level thresholding values. These algorithms were applied to five different test images with a manually determined number of thresholds. The results obtained were presented in the study and evaluated using statistical tests.

A High‐Security Chaotic Encryption Model for Biomedical Image Protection

ABSTRACTWith the rapid advancements in artificial intelligence (AI), ensuring the privacy and security of patient medical images has emerged as a pressing concern in the field of image privacy protection. Traditional medical image encryption methods, however, have often been criticized for their lack of flexibility and insufficient security measures. To address these challenges, proposed a new image encryption algorithm that uses the R3 (rotate, rescale, randomize) Model to apply the concepts of chaotic and hyperchaotic systems to new image encryption approaches. Based on the hyperchaotic behavior of the circle map system and the unpredictable behavior of the Chua system, the proposed algorithm performs scaling, rotation, and randomization on the target image. Basically, the input medical image is scaled and rotated using a chaotic system to minimize the association between neighboring pixels. To create an encrypted image, the permuted image is next subjected to a diffusion operation with the assistance of S‐box. By using these chaotic systems unpredictable behavior and sensitivity to beginning conditions, an encryption technique with a huge key space of 25208 is created, improving security overall and fortifying its resistance to brute‐force attacks. The speed at which the algorithm processes data and the minimal number of resources it consumes demonstrate its effectiveness, making it appropriate for real‐time applications. To evaluate the security and computational efficiency of the suggested encryption system, a series of grayscale medical photographs were subjected to extensive testing. In addition to having a large key space of about 25208, it also showed resistance to differential cryptanalysis, with NPCR and UACI values above 99.60% and 30.20%, respectively. The encrypted images' entropy was close to eight, suggesting improved system security. The technique is ideal for real‐time applications, with encryption speeds of 0.80 s for 512 × 512 images and 0.6 s for 256 × 256 images. The scheme's capacity to provide strong security with little computing overhead and complexity is demonstrated by experimental results.

Optimal linear response for expanding circle maps

Abstract We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation T ˙ of a circle map T we consider (i) the response of the expectation of an observation function and (ii) the response of isolated spectral points of the transfer operator of T. In each case, under mild conditions on the set of feasible perturbations T ˙ we show there is a unique optimal feasible infinitesimal perturbation perspective, we remark that even a movemen T ˙ optimal , maximising the increase of the expectation of the given observation function or maximising the increase of the spectral gap of the transfer operator associated to the system. We derive expressions for the unique maximiser T ˙ optimal in terms of its Fourier coefficients. We also devise a Fourier-based computational scheme and apply it to illustrate our theory.

Proportions of incommensurate, resonant, and chaotic orbits for torus maps.

This paper focuses on distinguishing classes of dynamical behavior for one- and two-dimensional torus maps, in particular, between orbits that are incommensurate, resonant, periodic, or chaotic. We first consider Arnold's circle map, for which there is a universal power law for the fraction of nonresonant orbits as a function of the amplitude of the nonlinearity. Our methods give a more precise calculation of the coefficients for this power law. For two-dimensional torus maps, we show that there is no such universal law for any of the classes of orbits. However, we find different categories of maps with qualitatively similar behavior. Our results are obtained using three fast and high precision numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders.

Large coupling asymptotics for the Lyapunov exponents of some Schrödinger cocyles over strongly expanding circle endomorphisms

ABSTRACT We quantify the coupling asymptotics for the Lyapunov exponent of the Schrödinger cocycle over strongly expanding maps x ↦ bx ( mod 1 ) on T , for a large class of potential functions. We refine the previous lower bound results for these Lyapunov exponents, to get asymptotic results for all E ∈ R and for sufficiently expanding circle maps.

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle Maps

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle Maps

The Bowen-Franks type theorems for multivalued maps and impulsive differential equations

The Bowen-Franks type theorems for multivalued maps and impulsive differential equations

Low Complexity of Optimizing Measures Over an Expanding Circle Map

Low Complexity of Optimizing Measures Over an Expanding Circle Map

Monotone Families of Circle Diffeomorphisms Driven by Expanding Circle Maps

We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms x↦bxmod1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\mapsto bx \\mod 1$$\\end{document}, where the integer b is large. We obtain estimates of the fibered Lyapunov exponents and show that in the limit as b tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on b.

Topological Entropy, Sets of Periods, and Transitivity for Circle Maps

Topological Entropy, Sets of Periods, and Transitivity for Circle Maps

Singular Continuous Phase for Schrödinger Operators Over Circle Maps with Breaks

Singular Continuous Phase for Schrödinger Operators Over Circle Maps with Breaks

EDMD for expanding circle maps and their complex perturbations

EDMD for expanding circle maps and their complex perturbations

Parameterized family of annular homeomorphisms with pseudo-circle attractors

Parameterized family of annular homeomorphisms with pseudo-circle attractors

Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents

Rigidity of Fibonacci Circle Maps with a Flat Piece and Different Critical Exponents

How Does the Fractional Derivative Change the Complexity of the Caputo Standard Fractional Map

The impact of power-law memory on the dynamics of the Caputo standard fractional map is investigated in this paper. The definition of a complexity measure for the Caputo standard fractional map is introduced. This measure evaluates both the average algebraic complexity of a trajectory and the distribution of the trajectories of different types in the phase space of the system. The interplay between the small-scale spatial chaos and the large-scale spatial behavior is observed and measured during the transition of the Caputo standard fractional map from the circle map to the classical standard map. It is demonstrated that the impact of the fractional derivative on the complexity of the fractional system is not straightforward and is predetermined by the physical properties governing the dynamics of that system.

$$\rho $$-bounded orbits and minimal sets for generalized quasiperiodically forced circle maps

$$\rho $$-bounded orbits and minimal sets for generalized quasiperiodically forced circle maps

Soil Salinity Inversion in Yellow River Delta by Regularized Extreme Learning Machine Based on ICOA

Soil salinization has seriously affected agricultural production and ecological balance in the Yellow River Delta region. Rapid and accurate monitoring of soil salinity has become an urgent need. Traditional machine learning models tend to fall into local optimal values during the learning process, which reduces their accuracy. This paper introduces Circle map to enhance the crayfish optimization algorithm (COA), which is then integrated with the regularized extreme learning machine (RELM) model, aiming to improve the accuracy of soil salinity content (SSC) inversion in the Yellow River Delta region. We employed Landsat5 TM remote sensing images and measured salinity data to develop spectral indices, such as the band index, salinity index, vegetation index, and comprehensive index, selecting the optimal modeling variable group through Pearson correlation analysis and variable projection importance analysis. The back propagation neural network (BPNN), RELM, and improved crayfish optimization algorithm–regularized extreme learning machine (ICOA-RELM) models were constructed using measured data and selected variable groups for SSC inversion. The results indicate that the ICOA-RELM model enhances the R2 value by an average of about 0.1 compared to other models, particularly those using groups of variables filtered by variable projection importance analysis as input variables, which showed the best inversion effect (test set R2 value of 0.75, MAE of 0.198, RMSE of 0.249). The SSC inversion results indicate a higher salinization degree in the coastal regions of the Yellow River Delta and a lower degree in the inland areas, with moderate saline soil and severe saline soil comprising 48.69% of the total area. These results are consistent with the actual sampling results, which verify the practicability of the model. This paper’s methods and findings introduce an innovative and practical tool for monitoring and managing salinized soils in the Yellow River Delta, offering significant theoretical and practical benefits.

Migration time prediction and assessment of toxic fumes under forced ventilation in underground mines

Migration time prediction and assessment of toxic fumes under forced ventilation in underground mines

Rotation number of 2-interval piecewise affine maps

AbstractWe study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $$f_{\varvec{p}}$$ f p are parametrized by a quintuple $$\varvec{p}$$ p of real numbers satisfying inequations. Viewing $$f_{\varvec{p}}$$ f p as a circle map, we show that it has a rotation number $$\rho (f_{\varvec{p}})$$ ρ ( f p ) and we compute $$\rho (f_{\varvec{p}})$$ ρ ( f p ) as a function of $$\varvec{p}$$ p in terms of Hecke–Mahler series. As a corollary, we prove that $$\rho (f_{\varvec{p}})$$ ρ ( f p ) is a rational number when the components of $$\varvec{p}$$ p are algebraic numbers.