A Proposed Nonlinear Dynamical System with Entropy-Modulated Feedback and Change-Dependent Nonlinearity

This work presents a proposed discrete-time nonlinear dynamical system incorporating state persistence, nonlinear feedback, an entropy-like modulation factor, and a change-dependent nonlinear term. The proposed model is defined by the state-update equation: X_(t+1) = (1 − α)X_t + β tanh(WX_t)/(1 + S_t) + r tanh(X_t − X_(t−1)) where the entropy-like modulation factor is defined as: S_t = −[p_t ln(p_t) + (1 − p_t)ln(1 − p_t)]. The model combines three principal mechanisms: persistence of the current state, nonlinear feedback from the current state, and a bounded response to the difference between the current and previous states. The entropy-like term modulates the nonlinear feedback according to the value of p_t. A practical interpretation is developed using time-dependent system demand or online system load as an illustrative example. A Python-based numerical simulation is also provided to examine the resulting state trajectory, entropy-like modulation, and return-map behavior. The work is presented as a proposed mathematical modeling framework. Its mathematical consistency, dynamical behavior, parameter sensitivity, and potential real-world applications require systematic analysis and empirical validation.