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A semi‐analytical locally transversal linearization method for non‐linear dynamical systems
Author(s) -
Roy D.,
Ramachandra L. S.
Publication year - 2001
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.154
Subject(s) - mathematics , linearization , dynamical systems theory , transcendental equation , ordinary differential equation , mathematical analysis , attractor , ode , transversal (combinatorics) , dynamical system (definition) , interval (graph theory) , differential equation , tangent vector , tangent , nonlinear system , geometry , physics , quantum mechanics , combinatorics
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 different numerical schemes, namely LTL‐1 and LTL‐2 schemes, based on the LTL methodology are presented. Both LTL‐1 and LTL‐2 procedures finally reduce the given set of non‐linear ordinary differential 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 differential 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 well‐known 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. Copyright © 2001 John Wiley & Sons, Ltd.