Tangent spaces in non-linear dynamical systems are state dependent. Hence, it is generally not possible to exactly represent a non-linear dynamical system by a linear one over "nite segments of the evolving trajectories in the phase space. It is known from the well-known theorem of Hartman and Grobman that a non-linear oscillator admits a linearization within a neighbourhood of any hyperbolic "xed point in the sense that the linearized #ow bears a topological conjugacy to the non-linear #ow within the same neighbourhood. This linearization is based on the construction of the tangent space to the non-linear vector "eld at the hyperbolic "xed point. Even though useful in bifurcation analysis, such a linearization procedure cannot be successfully applied to identically simulate the non-linear #ow over the phase space because the functional form of the topological conjugacy is not known. With this in mind, an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non-linear dynamical system at that point in the state space where the solution vector is desired. Several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results. It is shown that the locally transversal linearization (LTL) procedure "nally reduces the given set of non-linear ordinary di!erential equations (ODEs) to a set of transcendental algebraic equations with the desired solution vector as the unknown. The variables in the state space appear as unknown quantities in this approach. Finally, one arrives at non-linear Poincare maps for the non-linear oscillator. In this scheme, the characteristic time interval for the map is arbitrarily "xed. The methodology is found to be quite versatile for handling non-linear dynamical systems. In particular, it is veri"ed that these schemes are capable of accurately predicting a wide spectrum of typically non-linear response characteristics, such as limit cycles, multi-periodicity, almost periodicity and chaos. For a limited class of response patterns, the principle proposed has another advantage in that the time step for integrating the non-linear ODEs need not be small. An improved and a more general higher order version of the LTL method is also considered. A distinct advantage of this higher version is in its improved ability for a closer simulation of phase-dependent time histories where a di!erence in the initial conditions leads to a phase di!erence in the realized solution history.