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

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 (FF) 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. FF 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.