Some Results About Global Asymptotic Stability

Álvaro Castañeda,Víctor Guíñez

doi:10.1007/s12346-013-0102-8

Álvaro Castañeda, Víctor Guíñez

Open Access

https://doi.org/10.1007/s12346-013-0102-8

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Abstract

We study the global asymptotic stability of the origin for the continuous and discrete dynamical system associated to polynomial maps in R n (especially when n = 3)oftheform F = λ I + H,with F(0) = 0,where λisarealnumber,Itheidentity map, and H a map with nilpotent Jacobian matrix JH . We distinguish the cases when the rows of JH are linearly dependent over R and when they are linearly independent over R. In the linearly dependent case we find non-linearly triangularizable vector fields F for which the origin is globally asymptotically stable singularity (respectively fixed point) for continuous (respectively discrete) systems generated by F.I n the independent continuous case, we present a family of maps that have orbits escaping to infinity. Finally, in the independent discrete case, we show a large family of vector fields that have a periodic point of period 3.

Highlights

Let F : Rn → Rn be a C1-map with F(0) = 0
The origin is a singular point of the differential system x = F(x), (1)
We call the continuous dynamical system generated by F to the dynamical system associated to (1) (resp. (2))

Summary

Introduction

We give examples of vector fields in N (λ, 3) which are linearly trianguralizable (that is, triangular after a linear change of coordinates) in [6] We study the global asymptotic stability of the origin for the continuous and discrete dynamical system generated by maps F = λ I + H ∈ N (λ, 3) which are linearly dependent. In the case f (t) is a polynomial of degree one, we show the global asymptotic stability of the origin for the continuous and discrete cases (see Theorems 2.5 and 2.6). The section concludes showing that, for a linearly dependent map F ∈ N (λ, 3), in order for the origin not to be a globally asymptotically stable singularity The origin is not a globally asymptotically stable fixed point

The Linearly Dependent Case

The Linearly Independent Case

The Continuous Case

A2 25 A2

The Discrete Case