Invariant Parallels, Invariant Meridians and Limit Cycles of Polynomial Vector Fields on Some 2-Dimensional Algebraic Tori in \mathbb R ^3

We consider the polynomial vector fields of arbitrary degree in $\mathbb R ^3$ R 3 having the 2-dimensional algebraic torus $$\begin{aligned} \mathbb T ^2(l,m,n)=\{(x,y,z)\in \mathbb R ^3 : (x^{2l}+y^{2m}-r^2)^2+z^{2n}-1=0\}, \end{aligned}$$ T 2 ( l , m , n ) = { ( x , y , z ) ∈ R 3 : ( x 2 l + y 2 m - r 2 ) 2 + z 2 n - 1 = 0 } , where $l,m$ l , m , and $n$ n positive integers, and $r\in (1,\infty )$ r ∈ ( 1 , ∞ ) , invariant by their flow. We study the possible configurations of invariant meridians and parallels that these vector fields can exhibit on $\mathbb T ^2(l,m,n)$ T 2 ( l , m , n ) . Furthermore, we analyze when these invariant meridians or parallels are limit cycles.     

Integrability Conditions for Complex Homogeneous Kukles Systems

In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ? = x + P_n(x, y), ? = ?y, where P_n(x, y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2,?. . .?, 7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components.     

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

The creation or destruction of a crossing limit cycle when a sliding segment changes its stability, is known as pseudo-Hopf bifurcation. In this paper, under generic conditions, we find an unfolding for such bifurcation, and we prove the existence and uniqueness of a crossing limit cycle for this family.

     

Limit Cycles Bifurcating from the Period Annulus of Quasi-Homogeneous Centers

We provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous center, which can be obtained using the Abelian integral method of first order. We show that these bounds are the best possible using the Abelian integral method of first order. We note that these centers are in general non-Hamiltonian. As a consequence of our study we provide the biggest known number of limit cycles surrounding a unique singular point in terms of the degree n of the system for arbitrary large n.      

A posteriori finite element bounds for sensitivity derivatives of partial-differential-equation outputs

We present a Neumann-subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, “constant-free” upper and lower bounds for sensitivity derivatives of functionals of the solutions of partial differential equations. The design motivation for sensitivity derivative error control is discussed; the a posteriori finite element procedure is described; the asymptotic bounding properties and computational complexity of the method are summarized; and illustrative numerical results are presented.     

Singular Instantons and Extra Dimensions

Hawking and Turok (HT) have recently proposedthat an open universe can be created from nothing. Theinstanton describing this process is singular, andtherefore its validity has been subject to question. In particular, Vilenkin has shown that aninstanton with the same singular structure as Hawkingand Turok's would lead to the unsuppressed decay of flatspace. However, Vilenkin's solution can be seen as the dimensional reduction of a five-dimensionalnonsingular instanton. In this context the unsuppressedinstability of flat space can be traded formetastability with a low decay rate, provided that the size of the extra dimension is large comparedwith the Planck scale. Implications for the HT model arediscussed.