Summary: This paper revisits the problem of estimating the domain of attraction and the nonlinear \(\mathcal{L}_2\) gain for systems with saturation nonlinearities. We construct a virtual input space from the algebraic loop contained in systems, and divide this virtual input space into several regions. In one of these regions, none of the virtual inputs are saturated. In each of the remaining regions, there is a unique virtual input that saturates everywhere with the time-derivative of its saturated signal being zero. These special properties of the virtual inputs in different regions of the virtual input space are combined with an existing piecewise quadratic Lyapunov function that contains the information of virtual input saturation to arrive at a set of less conservative stability and performance conditions, from which we can obtain a larger level set of the piecewise quadratic Lyapunov function as an estimate of the domain of attraction and a tighter scalar function of the bound on the \(\mathcal{L}_2\) norm of the exogenous input as an estimate of the local nonlinear \(\mathcal{L}_2\) gain. Simulation results indicate that the proposed approach has the ability to obtain a significantly larger estimate of the domain of attraction and a significantly tighter estimate of the nonlinear \(\mathcal{L}_2\) gain than the existing methods.