Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor

Alexandre N. Carvalho; José A. Langa; James C. Robinson

doi:10.3934/cpaa.2020088

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2020, Volume 19, Issue 4: 1997-2013. Doi: 10.3934/cpaa.2020088

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Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor

1.
Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain
3.
Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK

^*Corresponding author
^*Corresponding author

Dedicated to professor Tomás Caraballo on the occasion of his 60th birthday

Received: August 2019

Revised: October 2019

Published: January 2020

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Abstract

We investigate the forwards asymptotic dynamics of non-autonomous differential equations. Our approach is centred on those models for which the vector field is only defined for non-negative times, that is, the laws of evolution are not given, or simply not known, for times before a given time (say time $ t = 0 $). We will be interested in the cases for which the 'driving' (time shift) semigroup has a global attractor in which backwards solutions are not necessarily unique. Considering vector fields in the global attractor of the driving semigroup allows for a natural way to extend vector fields, defined only for non-negative times, to the whole real line. These objects play a crucial role in the description of the asymptotic dynamics of our non-autonomous differential equation. We will study, in some particular cases, the isolated invariant sets of the associated skew-product semigroup with the aim of characterising the global attractor. We develop an example for which we derive decomposition for the global attractor of skew-product semigroup from the characterisation of the attractor of the associated driving semigroup.

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Mathematics Subject Classification: Primary: 37B55, 37L30, 35B40, 35B41, 37L05.

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Figure 2.2. Global attractor for vector feld (2.4) in its positive hull Σ⁺.

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References

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