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 $\mathbb{R}^n$ ($n>2$).
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
The classical approaches to show the existence of periodic orbits in high dimensions is the Andronov-Hopf bifurcation and the torus principle. Their limitations are obvious. The first one is local, and conditions of the second one are hard to check in practice.
In the paper [R.A. Smith. Orbital stability for ordinary differential equations. Journal of Differential Equations, 69(2) (1987): 265-287] 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 inertial manifold, which attracts all trajectories by trajectories lying on the manifold (the so-called exponential tracking property). A generalization of the P-B theorem is possible in the case $j=2$ and it is almost obvious if the inertial manifolds view is used (note that R.A. Smith did not know he was dealing with inertial manifolds, but he implicitly used their properties). This squeezing condition can be effectively verified for a large class of vector fields $f(x)=Ax + BF(Cx)$, where $A,B,C$ are linear and $F$ is nonlinear, with the use of the so-called Frequency Theorem (in his works Smith did not use that theorem, but rather a "pocket" version of it, and this connection is treated in the mentioned paper of I.M. Burkin). In his subsequent works, R.A.Smith extended his approach to certain delay and parabolic equations via the method a priori integral estimates.
In my recent work I joined all of the main results of R.A. Smith for autonomous (the P-B theory) and periodic (convergence theorems) ODEs, delay equations and parabolic equations, treating them in the context of inertial manifolds for abstract cocycles in Banach spaces. In applications, frequency conditions arise from applications of infinite-dimensional versions of the Frequency Theorem.
The papers of R.A. Smith also 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 (in general) under the abstract monotonicity.
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?
