This paper studies properties of homogeneous approximations in a geometric, coordinate-free setting. Specifically, we develop the concept of homogeneity in the 0-limit and \(\infty \) -limit (i.e., homogeneity in the bi-limit) with respect to general dilations induced by semi-Euler vector fields. A key contribution of the paper is a result relating the regularity properties of homogeneous in the bi-limit functions and vector fields to their degree of homogeneity and the local behavior of the associated dilation flow near the equilibrium set and infinity. Building on these concepts, we establish new finite time and fixed time semistability results for a class of dynamical systems possessing a continuum of equilibria that are homogeneous in the bi-limit with respect to a semi-Euler vector field. Finally, we use these results to develop thermodynamically inspired distributed consensus control protocols for multiagent network systems for achieving coordination tasks in fixed time.
