Simulation and Machine Learning-Based Dynamical Analysis of a Happiness Model with Hedonic and Eudaimonic Effects Using Singular and Non-Singular Fractal–Fractional Differential Operators

Abstract

Hedonic and eudaimonic well-being are core components of happiness, as understood in positive psychology. This study develops a discrete-time, non-integer order framework using fractal–fractional ([Formula: see text]) differentials with both singular and non-singular kernels, integrated with a stochastic Levenberg–Marquardt backpropagation (LMB) neural network (NN) architecture. The algorithm effectively captures memory-dependent, nonlinear behavioral dynamics across a range of psychometric conditions. [Formula: see text] differential equations (FF-DEs), employing Mittag-Leffler kernels, enable chaotic and hyperchaotic behavior under both disturbed and undisturbed conditions. Simulations explore three cases: varying fractional order, fractal dimension, or both. Empirical validation is performed using Spanish happiness data from 2009–2023, with performance evaluated via [Formula: see text], MSE, AE, and regression analysis. The LMB-NN framework is trained on 78% of the dataset, with the remaining 22% split equally for validation and testing, achieving accuracy up to 5–7 decimal places. Error histograms, state transition metrics, and strong correlation validate the model’s precision and robustness. Furthermore, emotional regulation and synchronization are examined, positioning this hybrid FF-LMB-NN framework as a powerful tool for psychotherapeutic modeling and personalized well-being analysis. This research integrates systems theory, control strategies, and psychological insight, offering valuable implications for psychotherapy and mental wellness.