Nonlinear Dynamical Model and Analysis of Emotional Propagation Based on Caputo Derivative - Liang Hong; Lipu Zhang (Mathematics, 2025)
Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria (<semantics><msubsup>Wij*</msubsup>=<mstyle scriptlevel="0" displaystyle="true">1δ</mstyle>sech2(∥Ei−Ej∥)</semantics>, e.g., <semantics>W*≈1.40</semantics> under typical parameters <semantics>η=0.5</semantics>, <semantics>δ=0.3</semantics>). We further derive closed-form expressions for the steady-state variance under stochastic perturbations (<semantics>Var(Wij)=<mstyle scriptlevel="0" displaystyle="true"><msubsup>σζ2</msubsup>2ηδ</mstyle></semantics>) and demonstrate a less than 6% deviation between simulated and theoretical values when <semantics>σζ=0.1</semantics>. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for <semantics>μ=0.5</semantics>. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research.