On the structure of the phase portrait of an endomorphism of the plane at the moment of bifurcation of its diagonal attractor. (Q1889528)

The author studies the two-parameter family of piecewise linear endomorphisms on the plane \[ \Phi\colon (x,y)\mapsto \Bigl(1-a(1-\zeta)| x| -a\zeta | y| , 1-a(1-\zeta)| y| -a\zeta| x| \Bigr). \] It is shown that for \(a\in (1,2)\) and \(\zeta< (a-1)/(2a)\), the segment \(\Lambda=\left[\Phi^2(0,0),\Phi(0,0)\right]\) is an asymptotically stable set. For \(a \in (1,2)\) and \(\zeta = (a-1)/(2a)\), the existence of invariant sets is discussed.