  We consider the energy landscape of a dissipative Klein-Gordon lattice with a
$\phi^4$ on-site potential. Our analysis is based on suitable energy arguments,
combined with a discrete version of the \L{}ojasiewicz inequality, in order to
justify the convergence to a single, nontrivial equilibrium for all initial
configurations of the lattice. Then, global bifurcation theory is explored, to
illustrate that in the discrete regime all linear states lead to nonlinear
generalizations of equilibrium states. Direct numerical simulations reveal the
rich structure of the equilibrium set, consisting of non-trivial topological
(kink-shaped) interpolations between the adjacent minima of the on-site
potential, and the wealth of dynamical convergence possibilities. These
dynamical evolution results also provide insight on the potential stability of
the equilibrium branches, and glimpses of the emerging global bifurcation
structure, elucidating the role of the interplay between discreteness,
nonlinearity and dissipation.
