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In a differential approach elaborated, we study the evolution of theparameters of Gaussian, mixed, continuous variable density matrices, whosedynamics are given by Hermitian Hamiltonians expressed as quadratic forms ofthe position and momentum operators or quadrature components. Specifically, weobtain in generic form the differential equations for the covariance matrix,the mean values, and the density matrix parameters of a multipartite Gaussianstate, unitarily evolving according to a Hamiltonian $\hat{H}$. We also presentthe corresponding differential equations which describe the nonunitaryevolution of the subsystems. The resulting nonlinear equations are used tosolve the dynamics of the system instead of the Schr\"odinger equation. Theformalism elaborated allows us to define new specific invariant andquasi-invariant states, as well as states with invariant covariance matrices,i.e., states were only the mean values evolve according to the classicalHamilton equations. By using density matrices in the position and in thetomographic-probability representations, we study examples of these properties.As examples, we present novel invariant states for the two-mode frequencyconverter and quasi-invariant states for the bipartite parametric amplifier.

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States