Our main interest lies in studying the dynamical behaviour of time discretizations for nonlinear initial-boundary value problems of parabolic type. For Runge-Kutta and linear multistep methods with variable stepsizes, we specify results on the long-term behaviour nearby a hyperbolic equilibrium point. We employ an abstract formulation of partial differential equations as ordinary differential equations on Banach spaces. As we extend ideas from the theory of semilinear equations to the fully nonlinear case, our main tools are a modification of a discrete variation-of-constants formula and Banach's fixed point theorem. Commencing with a model problem describing a nonlinear heat conduction process, we expose the analytical framework of sectorial operators and Banach space valued Hölder continuous functions and we illustrate the theoretical results by means of numerical experiments.