34,699 research outputs found
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Integral representations and Liouville theorems for solutions of periodic elliptic equations
The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.Comment: 48 page
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation
We show that Pyragas delayed feedback control can stabilize an unstable
periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of
a stable equilibrium in an n-dimensional dynamical system. This extends results
of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback
control can stabilize the UPO associated with a two-dimensional subcritical
Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback
gain matrix for stabilization, as well as knowledge of the period of the
targeted UPO. We apply feedback in the directions tangent to the
two-dimensional center manifold. We parameterize the feedback gain by a modulus
and a phase angle, and give explicit formulae for choosing these two parameters
given the period of the UPO in a neighborhood of the bifurcation point. We
show, first heuristically, and then rigorously by a center manifold reduction
for delay differential equations, that the stabilization mechanism involves a
highly degenerate Hopf bifurcation problem that is induced by the time-delayed
feedback. When the feedback gain modulus reaches a threshold for stabilization,
both of the genericity assumptions associated with a two-dimensional Hopf
bifurcation are violated: the eigenvalues of the linearized problem do not
cross the imaginary axis as the bifurcation parameter is varied, and the real
part of the cubic coefficient of the normal form vanishes. Our analysis of this
degenerate bifurcation problem reveals two qualitatively distinct cases when
unfolded in a two-parameter plane. In each case, Pyragas-type feedback
successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of
the original bifurcation point, provided that the phase angle satisfies a
certain restriction.Comment: 35 pages, 19 figure
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