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

  • *Corresponding author

    *Corresponding author 

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

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  • 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.

    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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