Invariant Manifolds, Zero Dynamics and Stability

  • Hans W. Knobloch
  • Dietrich Flockerzi
Part of the Progress in Systems and Control Theory book series (PSCT, volume 9)


We consider ordinary differential equations which can be written as coupled pairs
$$\dot x = g(t,x,y),\dot y = h(t,x,y)$$
with x ∈ ℝn and y ∈ ℝm. It will be tacitly assumed throughout this paper that g, h and all mappings — as s and w — which appear in the sequel are everywhere defined smooth CN-functions of their respective variables for some appropriate integer N ≥ 1. That solutions of an ordinary differential equation exist on a given (finite) time interval will also be taken for granted. Concerning (1.1) our basic assumption is
$$h(t,x,0) = 0$$
so that y = 0 represents a global invariant manifold for the system (1.1). The differential equation
$$\dot x = g(t,x,0)$$
Then describes the dynamics which prevail within this basic invariant man-ifold. For shortness we refer to (1.3) as to the differential equation of the “zero dynamics” for (1.1) with (1.2).


Ordinary Differential Equation Invariant Manifold Polynomial System Background Material Respective Variable 
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  1. 1.
    L. Cesari, “Asymptotic Behavior and Stability Problems in Ordinary Differential Equations,” 2nd edition, Springer Verlag, Berlin, 1963.CrossRefGoogle Scholar
  2. 2.
    W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations,“Heath Math. Monographs, D.C. Heath and Company, Boston, 1965.Google Scholar
  3. 3.
    H.W. Knobloch, Invariant Manifolds for Ordinary Differential Equations, to appear in: Proceedings of the UAB International Conference on Differential Equations and Mathematical Physics, March 15–21 (1990).Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Hans W. Knobloch
    • 1
  • Dietrich Flockerzi
    • 1
  1. 1.Mathematisches InstitutUniversität WürzburgWürzburgGermany

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