Topological textures in externally driven nonequilibrium systems have attracted increasing interest because they provide a route to generating and manipulating nontrivial vector configurations without relying on static magnetic or ferroelectric ordering. Here, we investigate the resonant dynamics of a monolayer square graphene membrane under periodic out-of-plane driving, with particular emphasis on the emergence, spatial organization, and phase-dependent evolution of localized topological textures in the full three-dimensional displacement and velocity fields. A continuum tensioned-membrane model with fixed boundaries is first established to derive the eigenmodes and eigenfrequency relation. The Föppl-von Kármán geometric nonlinearity is then introduced to clarify how finite out-of-plane deformation induces in-plane strain and displacement components, thereby allowing an initially flexural response to develop into a three-dimensional vector field.
Molecular dynamics simulations are performed for a monolayer square graphene membrane containing 11040 carbon atoms and having a side length of 169.74 Å. The membrane is subjected to a uniform equibiaxial prestrain of 5%, fully relaxed, thermally equilibrated at 300 K, and fixed along finite-width boundary regions. A spatially uniform harmonic force is subsequently applied to the movable atoms in the out-of-plane direction. The effective two-dimensional pretension extracted from the equilibrated in-plane stress is approximately 11.62 N/m. Substituting this value into the continuum model yields a fundamental frequency of approximately 0.163 THz, in close agreement with the value of 0.16 THz obtained from fast Fourier transform analysis of the molecular dynamics trajectories. By tuning the driving frequency to successive resonances, clear standing-wave patterns from low- to high-order modes are obtained. As the resonance order increases, the number of nodal lines grows and the membrane is progressively divided into regularly arranged localized vibration cells.
To determine whether these localized cells possess nontrivial topology, the complete atomic displacement and velocity vectors are reconstructed from the molecular dynamics trajectories, interpolated onto a two-dimensional grid, normalized, and mapped onto the unit sphere. The corresponding local skyrmion density and skyrmion number are then evaluated from the spatial derivatives of the normalized vector fields. At selected resonance phases, both displacement and velocity fields exhibit a continuous core-to-background reversal of the out-of-plane component together with predominantly radial in-plane components, forming Néel-type skyrmion-like textures. These localized units are spatially locked to the high-order standing-wave pattern rather than being ordinary vibration extrema. For the velocity field under the sixth-order resonance, the average local skyrmion number of several cores reaches approximately N_sk = 0.9923, demonstrating a nearly integer real-space winding.
Phase-resolved analysis over one oscillation period further reveals that the localized topological units are intrinsically dynamical. Their vector configurations evolve continuously with the driving phase, remain close to N_sk = +1 or -1 over most phase intervals, and pass through noninteger transitional states during polarity reversal. The term “stable” therefore refers to reproducible recurrence under sustained resonant driving rather than to a time-independent or intrinsically protected state. These results establish a direct connection among flexural resonance, geometric nonlinearity, and real-space topology in an atomically thin membrane, and demonstrate that high-order mechanical resonance can generate periodically recurring dynamical topological textures. The proposed mechanism also provides a physical basis for frequencyaddressable topological encoding, topology-assisted mode recognition, and multimodal mechanical sensing in two-dimensional nanoelectromechanical systems.