In view of their unique shape morphing behaviour, dielectric elastomer-based minimum energy structures (DEMES) have received an increasing attention in the technology of electroactive soft transduction. Because several of them undergo a time-dependent motion during their operation, understanding their nonlinear dynamic behaviour is crucial to their effective design. Additionally, in the recent past, there has been a growing scientific interest in imparting anisotropy to the material behaviour of dielectric elastomers in view of ameliorating their actuation performance. Spurred with these ongoing efforts, this paper presents an analytical framework for investigating the nonlinear dynamic behaviour of aniso-visco-hyperelastic DEMES actuator with an elementary rectangular geometry. We use a rheological model comprising two Maxwell elements connected in parallel with two single spring elements for modelling the material behaviour of the DE membrane. The governing equations of motion for the underlying non-conservative system are then derived using the Euler–Lagrange equation. The proposed model is used for building insights into the attainable equilibrium states, periodicity of the response as well as the resonant behaviour of the DEMES actuator over a feasible range of anisotropy and viscosity parameters. Our results reveal that the DEMES with hyperelastic material properties exhibits a supercritical pitchfork bifurcation of equilibrium state which is further accelerated in terms of attained equilibrium angle due to membrane anisotropy. A significant enhancement in the equilibrium angle attained by the structure with the extent of membrane anisotropy parameter is observed, indicating a favourable impact of material anisotropy. Poincare maps and phase-portraits are presented for assessing the periodicity of the nonlinear oscillations. The frequency response of the actuator for a combined DC and AC load indicates an upsurge in the resonant frequency with an increase in anisotropy parameter. The underlying analytical model and the trends presented in this study can find their potential use in the design and development of the futuristic anisotropic DEMES actuators subjected to time-dependent actuation. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.