We give complete answers to the open problem that concerns the robustness of the Palmer-Sacker-Sell trichotomy of variational systems exposed to perturbations, both in nonuniform and uniform settings. We provide a new method based on the input-output criteria for trichotomy from [48]. Considering a discrete variational system on \(\Theta \times X\) that has a nonuniform exponential trichotomy, for perturbations from \(\mathcal {L}^1(\Theta , \mathcal B(X))\) which satisfy an estimate relative to the original exponent and function, we demonstrate that the perturbed system has a nonuniform exponential trichotomy with the same exponent and an explicit function. As a consequence, we obtain a robustness criterion for the uniform exponential trichotomy of variational dynamics, which generalizes the main result in [21]. Furthermore, we prove that this trichotomy does not persist under perturbations from \(\mathcal {L}^p(\Theta , \mathcal {B}(X)) \setminus \mathcal {L}^1(\Theta , \mathcal {B}(X))\) , with \(p\in (1, \infty ]\) . Our results apply to general variational systems without assuming any additional hypotheses on their coefficients.
