Concept of Dynamical Traps and Action Points

Within the paradigm of intermittent human control, we propose a novel model of dynamical traps to describe the balancing of a dynamical system near its unstable equilibrium. The core of this model is probabilistic, featuring alternating transitions between two behavioral modes of the subject – active and passive phases – in regulating the dynamics of the controlled object. These modes are delineated by action points, which represent the moments when the subject decides to switch between modes. This switching behavior is modeled using an original stochastic differential equation. Within this approach, action points are conceptualized as stepwise transitions in a special variable, \(\zeta \), which switches between two boundary values: \(\zeta = 0\) and \(\zeta = 1\). The introduced trap function, \(\Omega (\Delta )\), quantifies the subject’s perception of the object’s deviation from the unstable equilibrium or a desired state, thereby determining the current priority of the two behavioral modes. Crucially, these transitions— action points—occur before the trap function reaches its extreme values, \(\Omega (\Delta ) = 0\) or \(\Omega (\Delta ) = 1\), underscoring the probabilistic nature of intermittent human control.