Stochastic basins of attraction for uncertain initial conditions

Outcomes of nonlinear dynamical systems strongly depend on the initial conditions. However, it is common knowledge that it is not possible to fix the initial conditions in real experiments, with uncertainties being inherent to it. In this work, the problem of uncertain initial conditions in dynamical systems is investigated. These uncertainties are represented through four postulates, and a new definition of basin of attraction is deduced to quantify their effects on the global dynamics. It is shown that the new definition is a convolution between the uncertainty distribution and the deterministic basin of attraction, and a reduced-cost strategy to compute them is defined. A Helmholtz nonlinear oscillator is investigated by considering initial conditions with uniform and normal distributions, and a three-dimensional Hénon map is analyzed, too. Finally, the influence of the uncertainty level and the effectiveness of new basin definition are verified. •The influence of random initial conditions on global dynamics is verified.•Stochastic basin of attraction gA(x0) is generalized for random initial conditions.•It is shown that gA(x0) is the convolution of a random field and a classical basin.•Computations are significantly cheaper than other probabilistic strategies.