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The dichotomy spectrum approach for a global nonuniform asymptotic stability problem: Triangular case via uniformization

Part of: Stability theory Topological dynamics Qualitative theory

Published online by Cambridge University Press:  12 September 2025

Álvaro Castañeda Open the ORCID record for Álvaro Castañeda [Opens in a new window] ,
Ignacio Huerta  and
Gonzalo Robledo
Álvaro Castañeda*
Affiliation:
Departamento de Matemáticas, https://ror.org/047gc3g35 Universidad de Chile , Casilla 653, Santiago 7800024, Chile
Ignacio Huerta
Affiliation:
Departamento de Matemática, https://ror.org/05510vn56 Universidad Técnica Federico Santa María , Casilla 110-V, Valparaíso 2340000, Chile e-mail: ignacio.huertan@usm.cl grobledo@uchile.cl
Gonzalo Robledo
Affiliation:
Departamento de Matemáticas, https://ror.org/047gc3g35 Universidad de Chile , Casilla 653, Santiago 7800024, Chile
*
e-mail: castaneda@uchile.cl
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Abstract

We introduce a new conjecture of global asymptotic stability for nonautonomous systems, which is fashioned along the nonuniform exponential dichotomy spectrum and whose restriction to the autonomous case is related to the classical Markus–Yamabe Conjecture: we prove that the conjecture is fulfilled for a family of triangular systems of nonautonomous differential equations satisfying boundedness assumptions. An essential tool to carry out the proof is a necessary and sufficient condition ensuring the property of nonuniform exponential dichotomy for upper block triangular linear differential systems. We also obtain some byproducts having interest on itself, such as the diagonal significance property in terms of the above-mentioned spectrum.

Keywords

Nonuniform bounded growthnonuniform exponential dichotomynonuniform spectrumdiagonal significanceMarkus–Yamabe problem

MSC classification

Primary: 37B55: Nonautonomous dynamical systems
Secondary: 34D09: Dichotomy, trichotomy 34C11: Growth, boundedness

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was funded by FONDECYT Regular Grant 1240361. The second author was funded by FONDECYT POSTDOCTORAL Grant 3210132. The third author was funded by FONDECYT Regular Grant 1210733.

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