The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in ...calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center.
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré–Liapunov method to detect centers ...with purely imaginary eigenvalues can be used to detect nilpotent centers.
The canard explosion is a significant phenomenon in singularly perturbed system which has attracted lots of attentions in the literature. Such a periodic behavior often appears near a Hopf ...bifurcation and variety of methods have been developed for studying it. In the present work, we introduce a degenerate canard explosion of which the canard cycle does not arise from a Hopf bifurcation (a linear center perturbation) but from a nonlinear nilpotent center perturbation. Moreover, we demonstrate an algorithm to find the asymptotic expansions for this type of canard explosion, whereas some classical iterative methods fail to do so. Specifically, our approach provides the exact expressions of the first three terms of the critical value as well as the explicit analytical approximation of the slow manifold in the blow-up coordinates (but not in the original ones) up to the second-order. In fact, the presence of the error function in the involved expressions prevents obtaining best approximations. As far as we know, it is possibly the first time that a high-order analytical approximation of the critical value of the parameter is obtained for this degenerate canard explosion. Numerical results are also given for illustration and they are compared with the analytical predictions.
•A degenerate canard explosion in a generalization of van der Pol system is studied.•The periodic orbit emerges from the perturbation of a nonlinear nilpotent center.•An algorithm that improves some classical iterative methods is designed.•In the computations, the error function prevents obtaining better approximations.•The exact expressions of the first three terms of the critical value are found.
This work is focused in the center problem for nilpotent singularities of differential systems in the plane. Although there are involved methods to approach the center problem in this work, we ...present a new stability criteria based on the existence of the complex separatrices passing through the singular point, and independent of the normal form. The algebraic method needs only the computation of such complex separatrices up to certain order, which can be done with a computer algebra package.
We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.
Following the work done in 8 we provide the bifurcation diagrams for the global phase portraits in the Poincaré disk of all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar ...polynomial vector fields.
To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and ...second for studying the bifurcation of limit cycles from it or from its period annulus. We provide necessary conditions for detecting nilpotent centers based on recent developments. Moreover we survey the last results on this problem and illustrate our approach by means of examples.
In this work we present techniques for bounding the cyclicity of a wide class of monodromic nilpotent singularities of symmetric polynomial planar vector fields. The starting point is identifying a ...broad family of nilpotent symmetric fields for which existence of a center is equivalent to existence of a local analytic first integral, which, unlike the degenerate case, is not true in general for nilpotent singularities. We are able to relate so-called “focus quantities” to the “Poincaré–Lyapunov quantities” arising from the Poincaré first return map. When we apply the method to concrete examples, we show in some cases that the upper bound is sharp. Our approach is based on computational algebra methods for determining a minimal basis (constructed by focus quantities instead of by Poincaré–Lyapunov quantities because of the easier computability of the former) of the associated polynomial Bautin ideal in the parameter space of the family. The case in which the Bautin ideal is not radical is also treated.
We provide the normal forms of all $\mathbb{Z}_2$-symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin. Furthermore, we complete the classification of ...the global phase portraits in the Poincare disk of the above systems initiated by Dias, Llibre and Valls 9.
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