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We analyze the formation of transient patterns and spin-spin correlations inquasi-two-dimensional spin-1 homogeneous Bose-Einstein condensates subjected toparametric driving of $s$-wave scattering lengths. The dynamics for an initialferromagnetic phase is identical to that of a scalar condensate. In contrast,intriguing dynamics emerges for an initial polar state. For instance, we showthat competition exists between density patterns and spin-mixing dynamics.Dominant spin-mixing dynamics lead to a gas of polar core vortices andanti-vortices of different spin textures. The density modes of the Bogoliubovspectrum govern the wavenumber selection of Faraday patterns. The spin modesdetermine the vortex density and the spatial dependence of spin-spincorrelation functions. When the density patterns outgrow the spin-mixingdynamics, the spin-spin correlations decay exponentially with a correlationlength of the order a spin healing length; otherwise, they exhibit a Besselfunction dependence. Strikingly, competing instabilities within density andspin modes emerge when both scattering lengths are modulated at differentfrequencies and appropriate modulation amplitudes. The competing instabilityleads to a superposition of density patterns or correlation functions of twodistinct wavelengths. Our studies reveal that fine control over the drivendynamics can be attained by tuning interaction strengths, quadratic Zeemanfield, driving frequencies, and amplitudes.

Patterns, spin-spin correlations and competing instabilities in driven quasi-two-dimensional spin-1 Bose-Einstein condensates