This study investigates the influence of a non-ideal external force on the complexity of attraction basins in a system of coupled Van der Pol oscillators. By introducing a phase-modulated force \(\varvec{\Phi (t) = f}_{\varvec{0}}\, \varvec{\cos [\omega t + a}_{\varvec{0}}\, \varvec{\sin (b}_{\varvec{0}} \varvec{\omega t)]}\) , we analyze how parameters \(\varvec{a}_{\varvec{0}}\) , \(\varvec{b}_{\varvec{0}}\) , and \(\varvec{\omega }\) affect the system’s dynamics, particularly the structure and fractal properties of attraction basins. Using numerical simulations, we compute the topological entropy ( \(\varvec{h}_{\varvec{top}}\) ) and uncertainty coefficient ( \(\varvec{\alpha }\) ) to quantify boundary complexity and sensitivity to initial conditions. Our results reveal that variations in \(\varvec{\omega }\) induce transitions between regular and chaotic regimes, with peak entropy values ( \(\varvec{h}_{\varvec{top}} \varvec{\approx 5.95}\) for \(\varvec{a}_{\varvec{0}} \varvec{= b}_{\varvec{0}} \varvec{= 0.5}\) ) corresponding to the emergence of multiple attractors and fractal basin boundaries. These findings highlight the critical role of external forcing in controlling synchronization and bifurcations, with direct implications for applications such as cardiac pacemakers and robust control systems. The proposed metrics ( \(\varvec{h}_{\varvec{top}}\) , \(\varvec{\alpha }\) ) provide a robust framework for predicting dynamical transitions in nonlinear coupled oscillators.
