In this paper we study those cubic systems which are invariant under a rotation of 2π/4 radians. They are written as ż = εz + p z2z̄ − z̄3, where z is complex, the time is real, and ε = ε1+ iε2, p = p1+ ip2 are complex parameters. When they have some critical points at infinity, i.e. |p2| ≤ 1, it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On ...

When the saddle-center bifurcation occurs in an analytic family of area-preserving maps, rst a parabolic xed point appears at the origin and then this point bi-furcates, creating an elliptic xed point and a hyperbolic one. Separatrices of the hyperbolic xed point form a small loop around the elliptic point. In general the sep-aratrices intersect transversely and the splitting is exponentially s...

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a boun...

Article history: Received 8 August 2008 Available online 18 April 2009

We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: d-dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension d ≥ 2, w...

In order to effectively realize and control the critical coupling, a graphene-based hyperbolic metamaterial has been proposed to replace the absorbing thin film in the critically coupled resonance structure. Our calculations demonstrate that the critical coupling effect (near-perfect light absorption) can be achieved at the near-infrared wavelength by using this layered structure, while the cri...

An isolated critical point is a peculiar thermodynamic that occurs in the phase diagram of hyperbolic black holes Kth-order Lovelock gravity higher dimensions (with K odd) for special tuned coupling constants. It corresponds to "merger" two swallowtails and characterized by non-standard exponents. Upon employing recent proposal assigning topological charge points, we argue offers an interpretat...