Poincaré bifurcation of a three-dimensional system

Perturbations of parallel flows on the sphere in R3

“…The general theory of bifurcation method of periodic orbits was established in [3,4]. Further study on Hopf bifurcation and bifurcation near a family of periodic orbits was given in [5,6,[8][9][10] using the method of Poincaré map. It was proved in [7] that a three-dimensional quadratic system having a first integral of the form x 2 + y 2 = h has at most two limit cycles on each cylinder defined by the integral if the cylinder contains no singular points.…”

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

“…By (2.1), similarly to the proof of Lemma 1 in [13], we have Lemma The periodic transformation (1.2) into the following C r system:…”

“…We usually study the bifurcation phenomenon of high-dimensional systems in some cases. In a series of papers, [10][11][12][13], we have concerned with perturbed three-dimensional or four-dimensional systems as follows: ẋ = f (x, y) + εP (x, y, ε), y = g(x, y) + εQ(x, y, ε), (1.1) where x ∈ R 2 , y ∈ R or y ∈ R 2 , 0 < |ε| 1, g(x, 0) = 0. If the unperturbed systems have a k multiple closed orbit, a family of periodic orbits, a center, and a weak focus of order k on the invariant plane y = 0, respectively, we have analyzed the bifurcation of system (1.1) and obtained the sufficient conditions for the existence of periodic solutions of system (1.1) in each of the cases.…”

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

“…In dimension n ≥ 3 the arguments are more delicate, and some recent techniques (see [21]) on Lyapunov-Schmidt reduction have to be used. For instance, in [17] the authors study the same problem using similar techniques however for 3D smooth system.…”