Nonlinear Synthesis pp 132-140 | Cite as

# Invariant Manifolds, Zero Dynamics and Stability

Chapter

## Abstract

We consider ordinary differential equations which can be written as coupled pairs
with x ∈ ℝ
so that
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).

$$\dot x = g(t,x,y),\dot y = h(t,x,y)$$

(1.1)

^{n}and y ∈ ℝ^{m}. 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$$h(t,x,0) = 0$$

(1.2)

*y*= 0 represents a global invariant manifold for the system (1.1). The differential equation$$\dot x = g(t,x,0)$$

(1.3)

## Keywords

Ordinary Differential Equation Invariant Manifold Polynomial System Background Material Respective Variable
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## References

- 1.L. Cesari, “Asymptotic Behavior and Stability Problems in Ordinary Differential Equations,” 2nd edition, Springer Verlag, Berlin, 1963.CrossRefGoogle Scholar
- 2.W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations,“Heath Math. Monographs, D.C. Heath and Company, Boston, 1965.Google Scholar
- 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