Basin of attraction of triangular maps with applications to di fference equations [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Armengol Gasull


The counterexamples to the so called continuous and discrete Markus-Yamabe conjectures were constructed by using triangular vector fields or maps. In this talk we will recall them as a motivation to study the dynamics of planar triangular maps $$x_{n+1}=f_0(u_n)+f_1(u_n)x_n,\quad u_{n+1}=\phi(u_n).$$ These maps preserve the fibration of the plane given by $\{\phi(u)=c\}$. We assume that there exists an invariant attracting fiber $\{u=u_*\}$ for the dynamical system generated by $\phi$ and that on this fiber the system contains either a global attractor, or it is filled by fixed or $2$-periodic points. Then we study the limit behavior of all the points that lie in the basin of attraction of this invariant fiber. We apply our results to a variety of examples, from particular cases of triangular systems to some planar quasi-homogeneous maps, and some difference equations. Some concrete examples  are the multiplicative and additive difference equations of types $ x_{n+2}=x_ng(x_nx_{n+1})$ and $x_{n+2}=-bx_{n+1}+g(x_{n+1}+bx_{n})$.

The talk is based on the paper:  Cima, Gasull, Mañosa, ``Basin of attraction of triangular maps with applications," that obtained the prize "best paper of J. Difference Equ. Appl. 2014".