Density-Colored Bifurcation Diagrams — A Complementary Tool for Chaotic Map Analysis

Moysis, Lazaros; Lawnik, Marcin; Volos, Christos

doi:10.1142/s0218127423300367

Cited by 3 publications

(1 citation statement)

References 33 publications

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“…To construct a 1D bifurcation diagram, we use a modified version of the density-color bifurcation analysis tool proposed in [33]. Our modification consists of calculating the relative density of points for each parameter value (for each vertical line) separately.…”

Section: Bifurcation Analysismentioning

confidence: 99%

Fractal Tent Map with Application to Surrogate Testing

Kopets,

Rybin,

Vasilchenko

et al. 2024

Fractal Fract

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Discrete chaotic maps are a mathematical basis for many useful applications. One of the most common is chaos-based pseudorandom number generators (PRNGs), which should be computationally cheap and controllable and possess necessary statistical properties, such as mixing and diffusion. However, chaotic PRNGs have several known shortcomings, e.g., being prone to chaos degeneration, falling in short periods, and having a relatively narrow parameter range. Therefore, it is reasonable to design novel simple chaotic maps to overcome these drawbacks. In this study, we propose a novel fractal chaotic tent map, which is a generalization of the well-known tent map with a fractal function introduced into the right-hand side. We construct and investigate a PRNG based on the proposed map, showing its high level of randomness by applying the NIST statistical test suite. The application of the proposed PRNG to the task of generating surrogate data and a surrogate testing procedure is shown. The experimental results demonstrate that our approach possesses superior accuracy in surrogate testing across three distinct signal types—linear, chaotic, and biological signals—compared to the MATLAB built-in randn() function and PRNGs based on the logistic map and the conventional tent map. Along with surrogate testing, the proposed fractal tent map can be efficiently used in chaos-based communications and data encryption tasks.

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Section: Bifurcation Analysismentioning

confidence: 99%

Fractal Tent Map with Application to Surrogate Testing

Kopets,

Rybin,

Vasilchenko

et al. 2024

Fractal Fract

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Matryoshka multistability: Coexistence of an infinite number of exactly self-similar nested attractors in a fractal phase space

Karimov,

Babkin,

Rybin

et al. 2024

Chaos, Solitons & Fractals

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Revealing Hidden Features of Chaotic Systems Using High-Performance Bifurcation Analysis Tools Based on CUDA Technology

Rybin,

Butusov,

Shirnin

et al. 2024

Int. J. Bifurcation Chaos

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Bifurcation analysis is an essential tool in nonlinear dynamics. Bifurcation diagrams help to discover subtle features of investigated dynamics such as chaotic and periodic regimes, hidden attractors and fixed points. However, plotting high-resolution bifurcation diagrams can be a computationally challenging task, especially in multiparametric evaluation. It should be noted that the bifurcation analysis is a task with natural parallelism and thus can be efficiently solved using hybrid central-graphics processing architectures. In this paper, we propose an advanced algorithm and special software for plotting bifurcation diagrams using calculations accelerated by graphics processing unit in combination with a highly efficient semi-implicit ordinary differential equation solver. Time series processing is based on the extraction of amplitude and phase features for the density-based spatial clustering of applications with noise to determine oscillation periodicity. We showcase the features of the application of proposed solutions on a set of test chaotic systems. The performance of the analysis algorithms is investigated in comparison with conventional solutions based on central processing unit and several approaches known from the literature. We explicitly show that the proposed algorithm outperforms known solutions in both calculation speed and precision.

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Density-Colored Bifurcation Diagrams — A Complementary Tool for Chaotic Map Analysis

Cited by 3 publications

References 33 publications

Fractal Tent Map with Application to Surrogate Testing

Fractal Tent Map with Application to Surrogate Testing

Matryoshka multistability: Coexistence of an infinite number of exactly self-similar nested attractors in a fractal phase space

Revealing Hidden Features of Chaotic Systems Using High-Performance Bifurcation Analysis Tools Based on CUDA Technology

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