Basins of Attraction of Equilibrium Points of Monotone Difference Equations
DOI:
https://doi.org/10.5644/SJM.05.2.06Keywords:
Basin, difference equation, attractivity, invariant manifolds, periodic solutionsAbstract
We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quad
n=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.
2000 Mathematics Subject Classification. 39A10, 39A11
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