1Petersburg Nuclear Physics Institute named by B.P.Konstantinov of NRC ``Kurchatov Institute'' , Russia

In complex multi-element systems, feedback loops between the order parameter and the control parameter can significantly affect collective dynamics. We examine the role of feedback in a specific multi-element system where an avalanche-like change in a threshold characteristic of the elements (stress) triggers a switch in their binary state (opinion). Examples of such behavior include a two-way voting process in a community of individuals influenced by their social environment, and decision-making in a group of traders responding to market pressure. To model the system, we use a modified noisy voter model on a scale free network proposed in [1]. The control parameter is the activity of the elements — the probability per unit time that an element interacts with its neighborhood. The average opinion serves as the order parameter. At constant activity, two stationary dynamical modes occur in the system. Near unity activity, the system switches between positive and negative consensus (ordered mode). At low activity, opposing opinions coexist, and the average opinion remains near zero (disordered mode). We introduce feedback as a decreasing linear dependence of activity on the squared average opinion. Analytically and numerically, we show that the feedback produces a new dynamical mode. In this regime, the system oscillates between two majority-consensus states, wherein the majority of elements share the same opinion while a persistent minority holds the opposite one. Moreover, feedback stabilizes the system dynamics. This manifests as a decrease in the standard deviation of the averaged opinion as the feedback becomes stronger. [1] N.E.Savitskaya, T.A.Fedorova, Stabilization of the Multi-element System Response to the Avalanche-like Perturbation due to the Assortativity of the Inter-element Interaction Network , Nonlinear Phenomena in Complex Systems, vol.28, No 2, 2025, pp.175-184