I have a quite general and naive question. Is the existence of a steady state in a nonlinear system well-defined? In linear systems, e.g.
$$\frac{\mathrm{d}f}{\mathrm{d}t} = Af,$$
the steady-state solution can, in principle, be obtained by inverting the generator $\Phi(t,t_0)$ of the dynamics $f(t)=\Phi(t,t_0)f(t_0)$ by
$$f_\mathrm{ss} = \Phi^{-1}(t,t_0)Af(t_0).$$
However, for a nonlinear system, e.g.
$$\frac{\mathrm{d}f(t)}{\mathrm{d}t} = A(g(t))f(t)$$ $$\frac{\mathrm{d}g(t)}{\mathrm{d}t} = B(f(t))g(t),$$
it seems like the inversion of the generators of the dynamics of $f,g$, $\Phi_f(t,t_0)$ and $\Phi_g(t,t_0)$, requires knowing all possible states of $f,g$ at all times $t$. In principle, this is not too different from the linear case, since, also there all possible bases of $f$ need to be propagated in order to obtain $\Phi(t,t_0)$. However, thinking about attractors as specific examples of nonlinear systems, it is clear that the steady state of a nonlinear system does not necessarily exist. In particular, I am interested in the conditions for the steady states existence.
Could you please provide references, if they exist, with general statements on steady states for nonlinear systems? I am not at all familiar with literature in the context of control theory/engineering, where such problems seem to be most prevalent, as my background is in condensed matter physics.