A semi-analytical locally transversal linearization method for non-linear dynamical systems

An numeric-analytical, implicit and local linearization methodology, called the locally transversal linearization (LTL), is developed in the present paper for analyses and simulations of non-linear oscillators. The LTL principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non-linear dynamical system at that particular point in the state space where the solution vector is sought. For purposes of numerical implementation, two di erent numerical schemes, namely LTL-1 and LTL-2 schemes, based on the LTL methodology are presented. Both LTL-1 and LTL-2 procedures nally reduce the given set of non-linear ordinary di erential equations (ODEs) to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space. While in the LTL-1 scheme the desired solution vector at a forward time point enters the linearized di erential equations as an unknown parameter, in the LTL-2 scheme a set of unknown residues enters the linearized system as parameters. A limited set of examples involving a few wellknown single-degree-of-freedom (SDOF) non-linear oscillators indicate that the LTL methodology is capable of accurately predicting many complicated non-linear response patterns, including limit cycles, quasi-periodic orbits and even strange attractors