131 research outputs found
Symmetric periodic orbits near heteroclinic loops at infinity for a class of polynomial vector fields
For polynomial vector fields in R3, in general, it is very difficult to detect the existence of an
open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial
vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools
for proving this result are, first, the existence in the phase portrait of a symmetry with respect
to a plane and, second, the existence of two symmetric heteroclinic loops
Generation of symmetric periodic orbits by a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2 in R3
In this paper we will find a continuous of periodic orbits passing
near infinity for a class of polynomial vector fields in R3. We consider
polynomial vector fields that are invariant under a symmetry with
respect to a plane and that possess a “generalized heteroclinic loop”
formed by two singular points e+ and e− at infinity and their invariant
manifolds � and . � is an invariant manifold of dimension 1 formed
by an orbit going from e− to e+, � is contained in R3 and is transversal
to . is an invariant manifold of dimension 2 at infinity. In fact, is
the 2–dimensional sphere at infinity in the Poincar´e compactification
minus the singular points e+ and e−. The main tool for proving the
existence of such periodic orbits is the construction of a Poincar´e map
along the generalized heteroclinic loop together with the symmetry
with respect to
Symmetric periodic orbits near a heteroclinic loop in R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds
In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinic
loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere
S2 and a diameter connecting the north with the south pole. The north pole is
an attractor on S2 and a repeller on . The equator of the sphere is a periodic
orbit unstable in the north hemisphere and stable in the south one. The full space
is topologically homeomorphic to the closed ball having as boundary the sphere
S2. We also assume that the flow of X is invariant under a topological straight line
symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient
Poincar´e map, we prove the existence of infinitely many symmetric periodic orbits
of X near L that gives n turns around L in a period. We also exhibit a class of
polynomial vector fields of degree 4 in R3 satisfying this dynamics
The Dirac point electron in zero-gravity Kerr--Newman spacetime
Dirac's wave equation for a point electron in the topologically nontrivial
maximal analytically extended electromagnetic Kerr--Newman spacetime is studied
in a zero-gravity limit; here, "zero-gravity" means , where is
Newton's constant of universal gravitation. The following results are obtained:
the formal Dirac Hamiltonian on the static spacelike slices is essentially
self-adjoint; the spectrum of the self-adjoint extension is symmetric about
zero, featuring a continuum with a gap about zero that, under two smallness
conditions, contains a point spectrum. Some of our results extend to a
generalization of the zero- Kerr--Newman spacetime with different
electric-monopole-to-magnetic-dipole-moment ratio.Comment: 49 pages, 17 figures; referee's comments implemented; the endnotes in
the published version appear as footnotes in this preprin
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Some contributions to the analysis of piecewise linear systems.
This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively.
In the first part, we study several models of memristor oscillators of dimension three and four, providing for the first time rigorous mathematical results regarding the rich dynamics of such memristor oscillators, both in the case of piecewise linear models and polynomial models. Thus, for some families of discontinuous 3D piecewise linear memristor oscillators, we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible so to justify the periodic behavior exhibited by such three dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems.
By using the first-order Melnikov theory, we derive the bifurcation set for a three-parametric family of Bogdanov-Takens systems with symmetry and deformation. As an applications of these results, we study a family of 3D memristor
oscillators where the characteristic function of the memristor is a cubic polynomial. In this family we also show the existence of an infinity number of invariant manifolds. Also, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role of invariant manifolds in these models.
In a similar way than for the 3D case, we study some discontinuous 4D piecewise linear memristor oscillators, and we show that the dynamics in each stratum is topologically equivalent to a continuous 3D piecewise linear dynamical system. Some previous results on bifurcations in such reduced systems, allow us to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.
In the second part of this thesis, we show that the two-dimensional stroboscopic map defined by a second order system with a relay based control and a linear switching surface is topologically equivalent to a canonical form for discontinuous piecewise linear systems.
Studying the main properties of the stroboscopic map defined by such a canonical form, the orbits of period two are completely characterized. At last, we give a conjecture about the occurrence of the big bang bifurcation in the previous map
- …