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It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z² −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers. Full article

The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations. Full article