In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the formfor |ε| = 0 sufficiently small. Here g i : D → R n , for i = 0, 1, . . . , k, are smooth functions being D ⊂ R n an open bounded set. Then we use this result to compute the bifurcation functions which allow to study the periodic solutions of the following T -periodic smooth differential systemIt is assumed that the unperturbed differential system has a sub-manifold of periodic solutions Z, dim(Z) ≤ n. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.
In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3D vector fields.2010 Mathematics Subject Classification. Primary: 34C23, 34C29, 34C45.
Recently sixteen 3-dimensional differential systems exhibiting chaotic motion and having no equilibria have been studied, and it has been graphically observed that these systems have a period-doubling cascade of periodic orbits providing a route to chaos. Here using new results on the averaging theory we prove that these systems exhibit, for some values of their parameters different to the ones having chaotic motion, either a zero-Hopf or a Hopf bifurcation, and graphically we observed that the periodic orbit starting in those bifurcations is at the beginning of the mentioned period-doubling cascade.
The Rössler system is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of Rössler systems exhibiting a zero-Hopf equilibrium. For Rössler systems near to one of these families, we provide generic conditions ensuring the existence of a torus bifurcation. In this case, the torus surrounds a periodic solution that bifurcates from the zero-Hopf equilibrium. For Rössler systems near to the other family, we provide generic conditions for the existence of a periodic solution bifurcating from the zero-Hopf equilibrium. This improves currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed.
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