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The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar polynomial vector field with an invariant octothorpe. In this paper we provide sufficient and necessary conditions for structural stability in this model. [ABSTRACT FROM AUTHOR]

These last years an increasing interest appeared in studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. The understanding of the dynamics of these systems has many difficulties. One of them is the study of their limit cycles. In this paper, we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line and formed by combinations of linear centres (consequently isochronous) and cubic isochronous centres with homogeneous nonlinearities. For these classes of planar discontinuous piecewise differential systems we solved the extension of the 16th Hilbert problem, i.e. we provide an upper bound for their maximum number of crossing limit cycles. [ABSTRACT FROM AUTHOR]

The aim of this paper is the study of the center-focus and cyclicity problems inside the class X of 3-dimensional vector fields that admit a first integral that leaves invariant any sphere centered at the origin. We classify the centers of linear, quadratic homogeneous and a family of quadratic vector fields F ⊂ X , restricted to one of these spheres. Moreover, we show the existence of at least 4 limit cycles in family F . [ABSTRACT FROM AUTHOR]

In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations. [ABSTRACT FROM AUTHOR]

Consider an analytical function f : V ⊂ R 2 → R having 0 as its regular value, a switching manifold Σ = f − 1 (0) and a piecewise analytical vector field X = (X + , X −) , i.e. X ± are analytical vector fields defined on Σ ± = { p ∈ V : ± f (p) > 0 }. We characterize when the vector field X has a monodromic singular point in Σ, called Σ-monodromic singular point. Moreover, under certain conditions, we show that a Σ-monodromic singular point of X has a neighborhood free of limit cycles. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide results like Peixoto's Theorem inside a class\mathfrak{X}X of 3-dimensional reversible systems. We deal with reversible systems associated to an involution\varphi_1φ 1 ​ such that\operatorname{Dim}\operatorname{Fix}(\varphi_1)DimFix(φ 1 ​) is equal to one. We further assume the existence of a first integral for any X\in\mathfrak{X}X∈X that leaves invariant any sphere centered at the origin. The main result establishes generic necessary conditions for the structural stability in\mathfrak{X}X. Stability conditions on generic one-parameter families of reversible vector fields on the two-dimensional sphere are also stated. [ABSTRACT FROM AUTHOR]

This paper deals with the Euler equations on the Lie Algebra so(4). These equations are given by a polynomial differential system in R 6 . We prove that this differential system has four 3-dimensional invariant manifolds and we give a complete description of its dynamics on these invariant manifolds. In particular, each of these invariant manifolds are fulfilled by periodic orbits except in a zero Lebesgue measure set. [ABSTRACT FROM AUTHOR]

Our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is x y = 0. We consider the symmetric case. That is, one vector field defined in the odd quadrants and the other in the even ones. We deal with equilibrium points of center-focus type, with matrices in real Jordan form, in each vector field when the infinity is monodromic. In this case, we provide the center classification at infinity, we prove that the maximum order of a weak focus is five. Moreover, we show the existence of systems exhibiting five limit cycles bifurcating from infinity. [ABSTRACT FROM AUTHOR]

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the saddle-fold singularity that is a codimension two singularity for piecewise smooth planar differential systems. The approach consists in apply the Regularization's Method to the unfolding of the saddle-fold singularity in order to obtain a 2-parameter family of smooth vector fields. Our main result says that the regularized family undergoes thought a classical Bogdanov–Takens bifurcation. [ABSTRACT FROM AUTHOR]

A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, ( x ′ , y ′ ) = ( − y + ε f ( x , y , ε ) , x + ε g ( x , y , ε ) ) . In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n , no more than N n − 1 limit cycles appear up to a study of order N . We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Liénard differential systems providing better upper bounds for higher order perturbation in ε , showing also when they are reached. The Poincaré–Pontryagin–Melnikov theory is the main technique used to prove all the results. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed. [ABSTRACT FROM AUTHOR]

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary . We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ { p 0 } ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached. [ABSTRACT FROM AUTHOR]

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our main goal is to study stability problems inside the class of 3 − dimensional refractive piecewise smooth vector fields. Our concern is to study refractive vector fields that admit a first integral that leaves invariant any sphere centered at the origin. Global stability conditions on generic one-parameter families of refractive piecewise smooth vector fields on a two-dimensional sphere are presented and used to prove our main result, which establishes necessary conditions for the structural stability inside that class. • Qualitative and geometric aspects of non smooth dynamical systems theory. • Stability problems inside the class of 3-dimensional refractive vector fields. • Necessary conditions for structural stability of refractive vector fields. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate Σ-center). We prove that any nondegenerate Σ-center is Σ-equivalent to a particular normal form . Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from that have k hyperbolic limit cycles bifurcating from the nondegenerate Σ-center of (the same holds for ). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from . As a consequence we prove that has infinite codimension. [Copyright &y& Elsevier]

Abstract: The structural stability of vector fields with impasse regular curves on is studied and a version of Peixotoʼs Theorem is established. Moreover a global analysis of normal forms of the constrained systems in the Poincaré ball (i.e. in the compactification of with the sphere of the infinity) is made. [Copyright &y& Elsevier]

Abstract: In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set. [Copyright &y& Elsevier]

This paper presents results concerning bifurcations of 2D piecewise-smooth vector fields. In particular, the generic unfoldings of codimension-three fold-saddle singularities of Filippov systems, where a boundary-saddle and a fold coincide, are considered and the bifurcation diagrams exhibited. [ABSTRACT FROM AUTHOR]

We prove that the static, spherically symmetric Einstein-Yang-Mills equations do not have periodic solutions when r > 0. [ABSTRACT FROM AUTHOR]

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of nonsmooth vector fields are studied, and the bifurcation diagrams are exhibited. Our main results describe a particular unfolding of the so-called fold-cusp singularity by means of the variation of 3 parameters. [ABSTRACT FROM AUTHOR]

Abstract: This article extends results contained in Buzzi et al. (2006) , Llibre et al. (2007, 2008) concerning the dynamics of non-smooth systems. In those papers a piecewise discontinuous vector field Z on is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider a polynomial function defined on the open subset . The set divides into subdomains , with border . These subdomains provide a Whitney stratification on . We consider smooth vector fields and we get a discontinuous vector field with discontinuities in . Our approach combines several techniques such as ε-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an ε-regularization of Z (see Sotomayor and Teixeira, 1996 ; Llibre and Teixeira, 1997 ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 ), in systems with hysteresis (Seidman, 2006 ) and in mechanical systems with impacts (di Bernardo et al., 2008 ). [Copyright &y& Elsevier]

Abstract: For a class of reversible quadratic vector fields on we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart . More specifically, we prove that for all , there exists such that the reversible quadratic polynomial differential system in , with , , , , , and , for has at least n periodic orbits near the heteroclinic loop. [Copyright &y& Elsevier]

Abstract: In this paper we deal with discontinuous vector fields on and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work. [Copyright &y& Elsevier]

In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typically follow one of three scenarios: splitting (from two pairs of imaginary eigenvalues to a quadruplet on the complex plane), passing (on the imaginary axis), or crossing (a quadruplet crossing the imaginary axis). We give a complete description of the behaviour ofreversibleperiodic orbits in the vicinity of such a bifurcation point. For non-reversible periodic solutions, in the case of Hopf bifurcation with crossing eigenvalues, we obtain a generalization of the equivariant Hopf Theorem. [ABSTRACT FROM AUTHOR]

We apply the averaging method in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set Σ = { x y = 0 }. They also motivate to consider a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions on the four parameters of a Hamiltonian system, related with the Friedmann–Robertson–Walker Hamiltonian system in a rotating reference frame, which guarantee the existence of 12 continuous families of periodic orbits, parameterized by the values of the Hamiltonian, which born at the equilibrium point localized at the origin of coordinates. The main tool for finding analytically these families of periodic orbits is the averaging theory for computing periodic orbits adapted to the Hamiltonian systems. The technique here used can be applied to arbitrary Hamiltonian systems. • Provide sufficient conditions for the existence of periodic orbits. • To study analytically the periodic orbits of a system. • Using the averaging theory to find periodic orbits. • Periodic orbits of the Friedmann–Robertson–Walker system. [ABSTRACT FROM AUTHOR]

Abstract: The authors M. Bellamy and R.E. Mickens in the article “Hopf bifurcation analysis of the Lev Ginzburg equation” published in Journal of Sound and Vibration 308 (2007) 337–342, claimed that this differential equation in the plane can exhibit a limit cycle. Here we prove that the Lev Ginzburg differential equation has no limit cycles. [Copyright &y& Elsevier]

In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]