143 research outputs found
Stable piecewise polynomial vector fields
Consider in R^2 the semi-planes N={y>0} and S={y<0}. In N and S are defined polynomial vector
fields X and Y, respectively, leading to a discontinuous piecewise polynomial
vector field Z=(X,Y). This work pursues the stability and the transition
analysis of solutions of Z between N and S, started by Filippov (1988) and
Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the
regularization method. This method consists in analyzing a one parameter family
of continuous vector fields Z_{\epsilon}$, defined by averaging X and Y. This
family approaches Z when the parameter goes to zero. The results of
Sotomayor-Teixeira and Sotomayor-Machado (2002) providing conditions on (X,Y)
for the regularized vector fields to be structurally stable on planar compact
connected regions are extended to discontinuous piecewise polynomial vector
fields on R^2. Pertinent genericity results for vector fields satisfying the
above stability conditions are also extended to the present case. A procedure
for the study of discontinuous piecewise vector fields at infinity through a
compactification is proposed here
Monodromic nilpotent singular points with odd Andreev number and the center problem
Given a nilpotent singular point of a planar vector field, its monodromy is
associated with its Andreev number . The parity of determines whether
the existence of an inverse integrating factor implies that the singular point
is a nilpotent center. For odd, this is not always true. We give a
characterization for a family of systems having Andreev number such that
the center problem cannot be solved by the inverse integrating factor method.
Moreover, we study general properties of this family, determining necessary
center conditions for every and solving the center problem in the case
Bifurcation of Limit Cycles from a Periodic Annulus Formed by a Center and Two Saddles in Piecewise Linear Differential System with Three Zones
In this paper, we study the number of limit cycles that can bifurcate from a
periodic annulus in discontinuous planar piecewise linear Hamiltonian
differential system with three zones separated by two parallel straight lines,
such that the linear differential systems that define the piecewise one have a
center and two saddles. That is, the linear differential system in the region
between the two parallel lines (i.e. the central subsystem) has a center and
the others subsystems have saddles. We prove that if the central subsystem has
a real or a boundary center, then we have at least six limit cycles bifurcating
from the periodic annulus by linear perturbations, four passing through the
three zones and two passing through the two zones. Now, if the central
subsystem has a virtual center, then we have at least four limit cycles
bifurcating from the periodic annulus by linear perturbations, three passing
through the three zones and one passing through the two zones. For this, we
obtain a normal form for these piecewise Hamiltonian systems and study the
number of zeros of its Melnikov functions defined in two and three zonesComment: arXiv admin note: text overlap with arXiv:2109.1031
Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones
In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential system with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it has at most one limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique limit cycle for all possible combinations of saddles and centers
Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere
Let X be a homogeneous polynomial vector field of degree 2 on S2 having finitely many invariant circles. Then, we prove that each invariant circle is a great circle of S2, and at most there are two invariant circles. We characterize the global phase portrait of these vector fields. Moreover, we show that if X has at least an invariant circle then it does not have limit cycles
Rational first integrals of the Liénard equations : The solution to the Poincaré problem for the Liénard equations
Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations ẍ + f (x)ẋ + x = 0, being f (x) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified
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