Linearization of Isochronous Centers

Periodic orbits from second order perturbation via rational trigonometric integrals

“…The first steps follow as in the proof of Proposition 13 for the family S 3 . System (34) transforms, via the rational change of coordinates (29), to a linear center with the rational perturbation (30). Moving to polar coordinates we can proceed analogously and obtain…”

“…Quadratic isochronous centers are classified in [29] in four families called S 1 , S 2 , S 3 and S 4 . A unified proof of the isochronicity property, as well as their linearization, can be found in [30]. The expressions of the vector fields S i that we use in this paper come from [4].…”

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

“…More precisely, we will apply Theorem 1, which gives a method to determine bifurcation of periodic solutions from a submanifold of isochronous periodic solutions. We note that the degenerate center of system (11) it is not isochronous because it cannot be linearized see [12] but, after a change of variables to polar coordinates (r, θ), it becomes isochronous with respect to the new time θ and Theorem 1 can be applied.…”

An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities

“…By comparing with (17) we get that β = 1 2m . Consequently the from Theorem 14 case (b) we get the first integral (22).…”

“…Now we show that differential system (28) is integrable. Indeed from equations (28) and (17) it follows that β = λ 2 . Hence, from Theorem 14 after some computations we get the proof of statements (a), (b), (c) and (d) of Theorem 28.…”