Stability and convergence in discrete convex monotone dynamical systems

Details Stability and convergence in discrete convex monotone dynamical systems Stability and convergence in discrete convex monotone dynamical systems

2010-03-28

arxiv.org/abs/1003.5346v1

Journal of Fixed Point Theory and Applications Volume 9, Number 2, 295-325, 2011

Mathematics

Dynamical Systems

36 pages, 1 fugure

Scientific paper

10.1007/s11784-011-0052-1

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on $n$ letters, where $n$ is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.

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