Stability in terms of two measures for set difference equations in space

Abstract

The set of trajectories for discrete dynamical systems (DDS) in the space is investigated. These sets are the solutions for difference equations in a metric space (space of nonempty convex compacts with the Hausdorff metric). On the basis of the comparison principle the general theorems on stability in terms of two measures were established. Applying the Minkowskij theory of mixed volumes, for some classes of nonlinear DDS in space the finite-dimensional comparison systems were constructed. The stability in terms of two measures and Lyapunov stability of fixed points for DDS in space were studied. The examples of studies of certain dynamical systems were given to illustrate the effectiveness of obtained results.

Keywords:

• Set difference equations
• stability in terms of two measures
• discrete dynamical systems
• mixed volume

AMS Subject Classifications:

• 93D30
• 52A39

Acknowledgements

The author is grateful to Professor Gnana Bhaskar T. for discussion of the obtained results.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

Publication is based on the research provided by the grant support of the State Fund For Fundamental Research [project F–61].