The authors study the behavior of solutions of the equation \(u_t=u_x+bu,\;x \in(0,1],\;t\in{\mathbb{R}}_{+}\) which satisfies a nonlocal boundary condition \(u|_{x=1}=h(u)\) \(u_{t}|_{x=0}\) where \(h:{\mathbb{R}}\to {\mathbb{R}}\) is a given \(C^2\)-function with negative derivative and such that \((w-\overline w)h''(w)>0\) for all \(w\in{\mathbb{R}}\) and some \(\overline w\in{\mathbb{R}}\).
Let \(u_\varphi\) stands for a solution which satisfies the condition \(u|_{t=0}=\varphi (x)\), \(\varphi \in C^1([0,1];{\mathbb{R}})\). Then the behavior of \(u_\varphi\) is determined by the one-dimensional mapping \[ f_{b,\varphi}:w\mapsto \lambda(f(w)+\varphi(1)-f(\varphi(0))) \] where \(f(w)= \int_{0}^{w}h(u) du\), \(\lambda =e^b\), and by properties of the shift operator \[ S^t[\varphi](x)= e^{-bx}(f_{b\varphi}^{[x+t]}\circ \widetilde\varphi)(\{x+t\}) \] where \([\cdot]\) \((\{\cdot\})\) stands for the entire (fractional) part of a number, \(\widetilde\varphi(\tau)=\lambda ^\tau\varphi(\tau)\). The authors describe a set \(\Omega(h)\) of pairs \((b,\varphi)\) corresponding to bounded solutions and study the structure of \(\omega\)-limit sets of orbits for infinite-dimensional dynamical systems generated by \(S^t,\;t\geq 0\), on the set \(\Phi_b=\{\varphi:(b,\varphi)\in\Omega(h)\}\). Two kinds of metrics on \(\Phi_b\) are introduced. The first one allows to complete \(\Phi _b\) by functions which are semicontinuous from above, and the second one – by random functions. For both cases the authors announce results concerning the asymptotic periodicity and boundedness of solutions, their dependence on initial values, fractal dimension of graphs of functions from \(\omega\)-limit sets, and the topological entropy of the corresponding dynamical system.