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In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl.499 (2017)], the authors proved that when parameter a ∈ (− 1 , 1) \ { 0 , − 2 / 3 } is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when a = − 2 / 3 , the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we consider a planar dynamical system with a piecewise linear function containing two dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system, with five singular points, can have at most four limit cycles, three of which surround the foci one by one and the fourth limit cycle surrounds all of the singular points of this system. [ABSTRACT FROM AUTHOR]

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented. [ABSTRACT FROM AUTHOR]

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass $\overline{QW2}$ which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class $\overline{QW2}$ also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for $\overline{QW2}$, 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class $\overline{QW2}$ reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation. [ABSTRACT FROM AUTHOR]

This paper investigates limit cycling behavior of observer-based controlled mechanical systems with friction compensation. The limit cycling is induced by the interaction between friction and friction compensation, which is based on the estimated velocity. The limit cycling phenomenon, which is experimentally observed in a rotating arm manipulator, is analyzed through computational bifurcation analysis. The computed bifurcation diagram confirms that the limit cycles can be eliminated by enlarging observer gains and controller gains at the cost of a steady state error. The numerical results match well with laboratory experiments. [ABSTRACT FROM AUTHOR]