What is the current status on methods to find limit cycles? What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$ 
Where $\vec{x}\in \mathbb{R}^n$.
I am familiar with Poincare-Bendixson Theorem or Dulac's Criterion, but would like to know what is the current status on limit cycles in systems in $\mathbb{R}^n$ ($n>2$).
 A: This contains the question of the existence of periodic orbits (ot periodic solutions) in dynamical systems, a very wide question indeed. In dimensions >=3 it is much more complicated than in dimension 2; Poincare-Bendixon and Dulac are very specific to dimension 2; in higher dimensions, there are no general criteria; in general even a bounded orbit can accumulate on a complicated minimal set, not on a periodic orbit nor cycle. A huge amount of works have been made in many particular frames, since Poincare.... Do you know more about your vector field f?
A: There are many principles to show the existence of periodic orbits in high- and infinite-dimensional systems, in particular, there are generalizations of the Poincaré-Bendixson theorem. I mention here several papers, which in itself contains a number of references on the topic


*

*Ciesielski K. The Poincaré-Bendixson theorem: from Poincaré to the XXIst century. Cent. Eur. J. Math., 10(6), 2110-2128 (2012).
The paper contains a historical background on the development of the P-B theorem in two dimensions from the classical smooth version upto semi-flows on the plane. Moreover, it contains many references on high-dimensional analogs (but do not give any analysis of them).

*Li B. Periodic orbits of autonomous ordinary differential equations: theory and applications. Nonlinear Anal.-Theor., 5(9), 931-958 (1981). A review of several works on the topic is given.

*Burkin I. M. Method of "transition into space of derivatives": 40 years of evolution. Differential Equations, 51(13), 1717-1751 (2015). Another review, which includes author's original approach and especially treats the works of R. A. Smith, who, in my opinion, did a great contribution to the development of P-B theory in high dimensions (below I will explain why).


The classical approaches to show the existence of periodic orbits in high dimensions is the Andronov-Hopf bifurcation and the torus principle. Their limitations is obvious. The first one is local, and the conditions of the second one are hard to check in practice.
In the paper [R. A. Smith. Existence of periodic orbits of autonomous ordinary differential equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 85(1-2) (1980): 153-172] it was used a certain monotonicity condition w. r. t. a symmetric matrix $P$ (this condition is very similar to the squeezing property used in the theory of inertial manifolds) to obtain a generalization of the P-B theorem. The condition is (here $\nu > 0$, $\delta>0$)
$$(P(x-y),f(x)-f(y) + \nu (x-y)) \leq -\delta |x-y|^{2}.$$
It turns out that if the matrix $P$ has $j$ negative and $n-j$ positive eigenvalues then there exists a $j$-dimensional invariant manifold, which attracts all compact solutions. The generalization of the P-B theorem is possible in the case $j=2$. This squeezing condition can be effectively verified for a large class of vector fields $f(x)=Ax + B\varphi(c^{*}x)$ (it may also contain several nonlinearities $\varphi$) with the use of the so-called frequency-domain theorem (in his works Smith did not use the frequency theorem and this connection is treated in the mentioned paper of Burkin). Non-global (and the most interesting) applications of this approach are also possible via constructing invariant sets (the squeezing condition is often cannot be satisfied globally). The papers of Smith motivated the study of systems, which are monotone w. r. t. high-rank cones, where only abstract monotonicty w. r. t. the pseudo-order given by such a cone is considered (L. A. Sanchez, Cones of rank 2 and the Poincare-Bendixson property for a new class of monotone systems. J. Differ. Equations, 246(5), 1978-1990 (2009); Feng L., Wang Yi and Wu J. Semiflows "Monotone with Respect to High-Rank Cones" on a Banach Space. SIAM J. Math. Anal., 49(1), 142-161 (2017)). In this direction a generalization of the P-B theorem is also possible. But it seems some topological consequences such as the existence of inertial manifolds or existence of stable periodic orbits are unreachable under the abstract monotonicity. In his subsequent works Smith directly extended his approach (without any monotonicity and using many a priori estimates) to delay and parabolic equations. In my two recent works (first and second) I joined all the main results of R. A. Smith for autonomous (the P-B theory) and periodic (Convergence theory) ODEs, delay equations and parabolic equations from around 10 papers, treating them in the abstract topological context and using infinite-dimensional versions of the frequency theorem.
