Cone of non-linear dynamical system and group preserving schemes

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The Lie-group shooting method for solving the Bratu equation (2011, methodology): “The present paper, provides a LGSM for the Bratu problem, which is based on the group preserving scheme (GPS), developed by Liu [19] for the integration of initial value problems.”
Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method (2013, methodology): “Recently, Liu and his coworkers [36–38] have successfully applied the Lie-group integration method, namely the mixed group-preserving scheme (MGPS), which is a combination of the forward group-preserving scheme [34] and the backward group-preserving scheme [39], together with the Lie-group”
The Lie-Group Shooting Method for Singularly Perturbed Two-Point Boundary Value Problems (2006, methodology): “(23) Accordingly, Liu (2001) has developed a grouppreserving scheme (GPS) given as follows: Xk+1 = G(k)Xk, (24) where Xk denotes the numerical value of X at the discrete tk, and G(k) ∈ SOo(2,1) satisfies GTgG = g, (25) det G = 1, (26) G00 > 0, (27) where G00 is the 00th component of G. 184…”
The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions (2006, background): “(6) and (7) for the Dirichlet problem, Eqs.”