Qualitative analysis by piecewise linear approximation

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Abstract

This paper describes a computer program called PLR that derives the qualitative behavior of ordinary differential equations. Current qualitative reasoning programs derive the abstract behaviour of a system by simulating hand-crafted ‘qualitative’ versions of the differential equations that characterize the system and summarizing the results. PLR infers more detailed information by constructing and analysing piecewise linear approximations of the original equations. The analysis employs the phase space representation of dynamic systems theory. PLR constructs a phase diagram for a system of piecewise linear equations by partitioning phase space into regions in which the system is linear, analysing the linear systems, and combining the results. It pastes together the local analyses into a global phase diagram by determining which sequences of regions the trajectories can traverse. The current implementation of PLR only handles second-order systems, but the method extends to higher-order systems. As an example of PLR's performance, I present its analyses of the Lienard and van der Pol equations.

References (8)

  • Morris W. Hirsch et al.

    Differential Equations, Dynamical Systems, and Linear Algebra

    (1974)
  • Fred Brauer et al.

    The Qualitative Theory of Ordinary Differential Equations

    (1969)
  • P. Sacks Elisha

    Hierarchical reasoning about inequalities

There are more references available in the full text version of this article.

Cited by (4)

This research was supported in part by National Institutes of Health Grant No. R01 LM04493 from the National Library of Medicine.

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