Search Results (46)

This paper investigates the dynamics of a three-dimensional nonlinear model of the financial system and the conditions for the emergence of chaotic behavior. The well-known chaotic system with given parameters and initial conditions is considered as a basis. For the initial model, critical points are analyzed, two-dimensional and three-dimensional phase portraits are constructed, and Lyapunov exponents are calculated, which allow confirming the presence of chaos and assessing the degree of sensitivity to initial data. Next, a modification of the system is proposed, consisting of changing the degree of the variable in the second equation. For the group of models obtained, we considered the generalized form of the system, found its critical points, and classified them. At the next stage, a bifurcation analysis was performed: by changing the key parameters of the modified systems, bifurcation diagrams were constructed, and parameter regions corresponding to critical points, periodicity, quasi-periodicity, and chaos were identified. The results demonstrate that the nature of the dynamics depends significantly on both the parameters and the degree of nonlinearity and allow conclusions to be drawn about the mechanisms of chaos in the financial model under consideration. Full article

Financial markets often display nonlinear and turbulent dynamics during periods of stress, and crude-oil and global equity systems frequently demonstrate closely connected forms of instability. Earlier studies report multifractality, chaotic features and regime-dependent spillovers across commodities and equities, yet existing approaches rarely succeed in capturing both the intrinsic complexity of oil-market behavior and the changing structure of cross-asset dependence. This limitation reduces the ability to distinguish calm from turbulent regimes and weakens short-horizon risk assessment. The present study introduces a unified framework that quantifies and predicts systemic instability within the coupled oil–equity system. The analysis constructs a crude-oil complexity index based on multifractal fluctuation analysis, permutation and approximate entropy, and Lyapunov-based indicators of chaotic dynamics. At the same time, it develops an information-theoretic network of global equity and energy-sector returns and summarizes its instability through measures of edge turnover, spectral radius, degree entropy and strength dispersion. These components are combined to form the Coupled Multiscale Chaos Index (CMCI), a scalar state variable that distinguishes calm, transitional and chaotic market regimes. Empirical results indicate that Brent and WTI exhibit pronounced multifractality, elevated entropy and positive Lyapunov exponents, while the dependence network becomes more centralized, more clustered and more capable of shock amplification during high-CMCI states. The CMCI moves closely with realized volatility and provides significant predictive content for five-day variance across major global equity benchmarks, with performance superior to models that rely only on macro-financial controls. Out-of-sample evaluation shows that forecasts incorporating measures of complexity record substantially lower MSE and QLIKE losses. The findings indicate that systemic instability reflects the interaction between local chaotic dynamics in crude-oil markets and turbulence in the global dependence network. The CMCI offers a practical early-warning indicator that supports risk management, forecasting and macroprudential supervision. Full article

In this manuscript, we investigate a fractional-order conformable three-dimensional chaotic financial model with interest rate, investment demand, and price index compartments. On the application of fixed-point theorems and nonlinear analysis, we establish theoretical results regarding the existence and uniqueness of a solution and also study Ulam–Hyers criteria for the stability of the solution of the considered system. Further, we use the fractional-order Runge–Kutta (RK-4) method to approximate the solution of our problem. Also, deep neural network (DNN) techniques are applied to investigate the model from artificial intelligence (AI) perspectives. Numerical simulation shows that it reproduces accurately the qualitative dynamics and confirms the theoretical stability results of the mentioned system. Subsequently, for the DNN analysis, we follow the Levenberg–Marquardt algorithm using Matlab 2023. Different quantities like the root-mean-square error (RMSE), mean squared error (MSE), and regression coefficient and a comparison with numerical data are presented graphically. Also, absolute errors between numerical values and those predicted by DNNs corresponding to different fractional orders are presented. Full article

Financial stability in interconnected markets is increasingly challenged by nonlinear interactions that amplify local disturbances into systemic crises. This study models a financial system as a complex network of coupled chaotic nodes, where each node represents a nonlinear macroeconomic subsystem governed by endogenous feedback dynamics. In contrast to traditional centralized interventions, a pinning control strategy is proposed to stabilize a network through selective control of a small subset of influential nodes. Numerical simulations show how local crises propagate through coupling links, generating systemic instability, and how the proposed impulsive control scheme effectively suppresses chaos and restores synchronization across an entire network. Results highlight the efficiency of localized interventions for achieving global stability, offering new theoretical insights into mechanisms of financial correlation and design of control-based resilience strategies for complex economic systems. Full article

Improving financial time series forecasting presents challenges because models often struggle to identify diverse fault patterns in unseen data. This issue is critical in fintech, where accurate and reliable forecasting of financial data is essential for effective risk management and informed investment strategies. This work addresses these challenges by initializing the weights and biases of two proposed models, Gated Recurrent Units (GRUs) and the Echo State Network (ESN), with different chaotic sequences to enhance prediction accuracy and capabilities. We compare reservoir computing (RC) and recurrent neural network (RNN) models with and without the integration of chaotic systems, utilizing standard initialization. The models are validated on six different datasets, including the 500 largest publicly traded companies in the US (S&P500), the Irish Stock Exchange Quotient (ISEQ) dataset, the XAU and USD forex pair (XAU/USD), the USD and JPY forex pair with respect to the currency exchange rate (USD/JPY), Chinese daily stock prices, and the top 100 index of UK companies (FTSE 100). The ESN model, combined with the Lorenz system, achieves the lowest error among other models, reinforcing the effectiveness of chaos-trained models for prediction. The proposed ESN model, accelerated by the Kintex-Ultrascale KCU105 FPGA board, achieves a maximum frequency of $83.5$ MHz and a power consumption of $0.677$ W. The results of the hardware simulation align with MATLAB R2025b fixed-point analysis. Full article

Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision that is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate errors, often undetectable. We investigate the possibility of reliable (error-free) time series of chaos. We prove that this is feasible for two well-studied isomorphic chaotic maps, namely the Tent map and the Logistic map. The generated chaotic time series have an unlimited horizon of predictability. A new linear formula for the horizon of predictability of the Analytic Computation of the Logistic map, for any given precision and acceptable error, is obtained. Reliable (error-free) time series of chaos serve as the “gold standard” for chaos applications. The practical significance of our findings include: (i) the ability to compare the performance of neural networks that predict chaotic time series; (ii) the reliability and numerical accuracy of chaotic orbit computations in encryption, maintaining high cryptographic strength; and (iii) the reliable forecasting of future prices in chaotic economic and financial models. Full article

We develop a four-dimensional nonlinear model of a reflexive financial system by extending the Xin–Zhang system with a self-reinforcing sentiment channel. The model comprises four interacting variables—interest rate, investment demand, price index, and market confidence—and incorporates reflexivity to capture feedback between economic fundamentals and investor sentiment. A Lyapunov function shows that the system is well-posed and dissipative, ensuring bounded trajectories. We then analyse the dynamics using standard nonlinear-dynamics tools. Reflexive confidence sustains chaotic motion, inhibits convergence to equilibria, and produces irregular, aperiodic bifurcation patterns; sentiment-driven feedback destabilises a dissipative macroeconomic model and sustains volatility, as evidenced by a positive largest Lyapunov exponent and Kolmogorov–Sinai entropy greater than zero. Using U.S. monthly consumer sentiment and the S&P 500, we observe co-movement, a medium-horizon lead of sentiment, and a nonlinear persistence map $w_{t+1}=f\left(w_t\right)$—stylised facts consistent with the model’s self-reinforcing confidence channel. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article

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

To address the growing threat of quantum computing to classical cryptographic primitives, this study introduces NTRU-MCF, a novel lattice-based signature scheme that integrates multidimensional lattice structures with fractional-order chaotic systems. By extending the NTRU framework to multidimensional polynomial rings, NTRU-MCF exponentially expands the private key search space, achieving a key space size $\ge 2^{256}$ for dimensions $m\ge 2$ and rendering brute-force attacks infeasible. By incorporating fractional-order chaotic masks generated via a hyperchaotic Lü system, the scheme introduces nonlinear randomness and robust resistance to physical attacks. Fractional-order chaotic masks, generated via a hyperchaotic Lü system validated through NIST SP 800-22 randomness tests, replace conventional pseudorandom number generators (PRNGs). The sensitivity to initial conditions ensures cryptographic unpredictability, while the use of a fractional-order L hyperchaotic system—instead of conventional pseudorandom number generators (PRNGs)—leverages multiple Lyapunov exponents and initial value sensitivity to embed physically unclonable properties into key generation, effectively mitigating side-channel analysis. Theoretical analysis shows that NTRU-MCF’s security reduces to the Ring Learning with Errors (RLWE) problem, offering superior quantum resistance compared to existing NTRU variants. While its computational and storage complexity suits high-security applications like military and financial systems, it is less suitable for resource-constrained devices. NTRU-MCF provides robust quantum resistance and side-channel defense, advancing PQC for classical computing environments. Full article

Financial time series in volatile markets often exhibit non-stationary behavior and signatures of stochastic chaos, challenging traditional forecasting methods based on stationarity assumptions. In this paper, we introduce a novel multi-expert forecasting system (MES) that leverages ensemble machine learning techniques—including bagging, boosting, and stacking—to enhance prediction accuracy and support robust risk management decisions. The proposed framework integrates diverse “weak learner” models, ranging from linear extrapolation and multidimensional regression to sentiment-based text analytics, into a unified decision-making architecture. Each expert is designed to capture distinct aspects of the underlying market dynamics, while the supervisory module aggregates their outputs using adaptive weighting schemes that account for evolving error characteristics. Empirical evaluations using high-frequency currency data, notably for the EUR/USD pair, demonstrate that the ensemble approach significantly improves forecast reliability, as evidenced by higher winning probabilities and better net trading results compared to individual forecasting models. These findings contribute both to the theoretical understanding of ensemble forecasting under chaotic market conditions and to its practical application in financial risk management, offering a reproducible methodology for managing uncertainty in highly dynamic environments. Full article

This paper introduces a novel three-dimensional financial risk system that exhibits complex dynamical behaviors, including chaos, multistability, and a butterfly attractor. The proposed system is an extension of the Zhang financial risk model (ZFRM), with modifications that enhance its applicability to real-world economic stability assessments. Through numerical simulations, we confirm the system’s chaotic nature using Lyapunov exponents (LE), with values calculated as $L_1=3.5547,$ $L_2=0,$ $L_3=-22.5642$, indicating a positive Maximal Lyapunov Exponent (MLE) that confirms chaos. The Kaplan–Yorke Dimension (KYD) is determined as D_k = 2.1575, reflecting the system’s fractal characteristics. Bifurcation analysis (BA) reveals parameter ranges where transitions between periodic, chaotic, and multistable states occur. Additionally, the system demonstrates coexisting attractors, where different initial conditions lead to distinct long-term behaviors, emphasizing its sensitivity to market fluctuations. Offset Boosting Control (OBC) is implemented to manipulate the chaotic attractor, shifting its amplitude without altering the underlying system dynamics. These findings provide deeper insights into financial risk modeling and economic stability, with potential applications in financial forecasting, risk assessment, and secure economic data transmission. Full article

Standard correlation analysis is one of the frequently used methods in financial markets. However, this matrix can give erroneous results in the conditions of chaos, fractional systems, entropy, and complexity for the variables. In this study, we employed the time-dependent correlation matrix based on isospectral flow using the Lie group method to assess the price of Bitcoin and gold from 19 July 2010 to 31 December 2024. Firstly, we showed that the variables have a chaotic and fractional structure. Lo’s rescaled range (R/S) and the Mandelbrot–Wallis method were used to determine fractionality and long-term dependence. We estimated and tested the d parameter using GPH and Phillips’ estimators. Renyi, Shannon, Tsallis, and HCT tests determined entropy. The KSC determined the evidence of the complexity of the variables. Hurst exponents determined mean reversion, chaos, and Brownian motion. Largest Lyapunov and Hurst exponents and entropy methods and KSC found evidence of chaos, mean reversion, Brownian motion, entropy, and complexity. The BDS test determined nonlinearity, and later, the time-dependent correlation matrix was obtained by using the stochastic SO(2) Lie group. Finally, we obtained robustness check results. Our results showed that the time-dependent correlation matrix obtained by using the stochastic SO(2) Lie group method yielded more successful results than the ordinary correlation and covariance matrix and the Spearman correlation and covariance matrix. If policymakers, financial managers, risk managers, etc., use the standard correlation method for economy or financial policies, risk management, and financial decisions, the effects of nonlinearity, fractionality, entropy, and chaotic structures may not be fully evaluated or measured. In such cases, this can lead to erroneous investment decisions, bad portfolio decisions, and wrong policy recommendations. Full article

Randomness plays a crucial role in numerous applications, with cryptography being one of the most significant areas where its importance is evident. A major challenge in cryptographic applications is designing a reliable key generator that meets stringent security requirements. Existing methods often suffer from predictability and fail to provide robust randomness, necessitating novel mathematical approaches. In this study, we propose an innovative mathematical framework that integrates quantum wave functions with chaotic systems to enhance the unpredictability and security of random number generation. The proposed approach leverages the inherent uncertainty of quantum mechanics and the dynamic behavior of chaos to generate statistically strong random sequences. The analysis results confirm that the proposed generator successfully passes all standard statistical randomness tests, demonstrating its effectiveness in cryptographic applications. Additionally, we present a practical implementation of the proposed method as an image encryption algorithm, showcasing its potential for real-world information security solutions. The findings suggest that this approach can contribute significantly to secure communication systems, financial transactions, and other domains requiring high-level cryptographic security. Full article

This paper introduces a novel chaotic finance system derived by incorporating a modeling uncertainty with an absolute function nonlinearity into existing financial systems. The new system, based on the works of Gao and Ma, and Vaidyanathan et al., demonstrates enhanced chaotic behavior with a maximal Lyapunov exponent (MLE) of 0.1355 and a fractal Lyapunov dimension of 2.3197. These values surpass those of the Gao-Ma system (MLE = 0.0904, Lyapunov dimension = 2.2296) and the Vaidyanathan system (MLE = 0.1266, Lyapunov dimension = 2.2997), signifying greater complexity and unpredictability. Through parameter analysis, the system transitions between periodic and chaotic regimes, as confirmed by bifurcation diagrams and Lyapunov exponent spectra. Furthermore, multistability is demonstrated with coexisting chaotic attractors for p = 0.442 and periodic attractors for p = 0.48. The effects of offset boosting control are explored, with attractor positions adjustable by varying a control parameter k, enabling transitions between bipolar and unipolar chaotic signals. These findings underline the system’s potential for advanced applications in secure communications and engineering, providing a deeper understanding of chaotic finance models. Full article