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\centerline{\bf Point Mapping-Based Analysis of Nonlinear Systems}
\centerline{\bf Using Symbolic Algebra}

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\centerline{Ramesh S. Guttalu and Henryk Flashner}
\centerline{Department of Mechanical Engineering}
\centerline{University of Southern California}
\centerline{Los Angeles, CA 90089-1453, USA}
\centerline{E-mail: guttalu@almaak.usc.edu}

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Symbolic algebra based analysis of dynamical systems has increased
significantly in recent years. In this paper, we present an approach for the
analysis of stability and bifurcations of periodic solutions of
multivariable, multiparameter nonlinear systems described by coupled
ordinary differential equations. The formulation is based on the concept of
{\em point mapping} (or period to period mapping). Using computer algebra,
symbolic expressions for the point mapping and its dependence on system's
parameters are evaluated using a recursive procedure. The symbolic
truncation algorithm described here yields coefficients of a multinomial
expansion of the point mapping in terms of the state variables and
parameters. In this algorithm, the truncation order is arbitrary, and the
evaluation of coefficients of the lower order terms is not affected by those
of the higher order terms. The symbolic point mapping expression is then
used to express the stability and bifurcation conditions as explicit
functional relations between system's parameters.  The approach is first
demonstrated by carrying out a perturbation type analysis for determining
harmonic solutions and stability of the Duffing's oscillator. Then the
method is used to study the parametric stability and bifurcation of a
nonlinear Mathieu's equation.  It is shown that, unlike perturbation
analysis, the proposed symbolic point mapping method provides a very
accurate solution even for {\em large values} of the parameters.  Moreover,
unlike existing symbolic computation approaches, the proposed method can
be applied to multiple degree-of-freedom systems with many design
parameters.

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