| Contents |
| We study dynamics of multi-dimensional maps $F_i$ of the form
$$
F_i: [u_1,u_2,\dots,u_{N-1},u_N]\mapsto [u_1+a_if(x_{N}),u_1,u_2,\dots,u_{N-1}],
$$
where $f$ is a fixed one-dimensional map on $\mathbb R$ and $a_i, i=1,\dots,K$, are real numbers. Such maps result from exact reduction of dynamics in periodic differential delay equations with piece-wise constant argument of the form
$$
x^{\,\prime}(t)=a(t)g(x([t-N])),
$$
where $a(t)$ is a $K$-periodic function, $g$ is a real valued function, and $[\cdot]$ is the integer part function. We address the problems of global stability of a unique fixed point, existence and stability of cycles (periodic points), and other questions of global dynamics in such maps and their compositions. |
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