Global Stability of the Fixed Point of a Certain Planar Map

Thursday, June 23, 2016 - 9:00am - 9:50am
Keller 3-180
Robert Sacker (University of Southern California)
Two elementary Ricker maps in the positive half-line are coupled to produce a planar mapping leaving the positive cone invariant. While this mapping appears to provide an exercise fit for a Calculus class, it is one of the most challenging problems ever experienced by the authors, namely to prove global attraction to the interior fixed point, already known to be asymptotically stable. Oddly, the case of very small coupling is the most difficult to analyze. The first attempt to study this problem resulted in a 29 page paper culminating in a theorem containing infinitely many hypotheses. Using a 1975 result that generalizes a problem originally pioneered by Hadamard, we were able to obtain the result for a wide range of parameters and only three hypotheses.
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