10 research outputs found
Frequency locking by external forcing in systems with rotational symmetry
We study locking of the modulation frequency of a relative periodic orbit in
a general -equivariant system of ordinary differential equations under an
external forcing of modulated wave type. Our main result describes the shape of
the locking region in the three-dimensional space of the forcing parameters:
intensity, wave frequency, and modulation frequency. The difference of the wave
frequencies of the relative periodic orbit and the forcing is assumed to be
large and differences of modulation frequencies to be small. The intensity of
the forcing is small in the generic case and can be large in the degenerate
case, when the first order averaging vanishes. Applications are external
electrical and/or optical forcing of selfpulsating states of lasers.Comment: 5 figure
Frequency locking of modulated waves
We consider the behavior of a modulated wave solution to an
-equivariant autonomous system of differential equations under an
external forcing of modulated wave type. The modulation frequency of the
forcing is assumed to be close to the modulation frequency of the modulated
wave solution, while the wave frequency of the forcing is supposed to be far
from that of the modulated wave solution. We describe the domain in the
three-dimensional control parameter space (of frequencies and amplitude of the
forcing) where stable locking of the modulation frequencies of the forcing and
the modulated wave solution occurs.
Our system is a simplest case scenario for the behavior of self-pulsating
lasers under the influence of external periodically modulated optical signals
Local conservation laws of second-order evolution equations
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78,
531-676], we completely describe local conservation laws of second-order
(1+1)-dimensional evolution equations up to contact equivalence. The possible
dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The
canonical forms of equations with respect to contact equivalence are found for
all nonzero dimensions of spaces of conservation laws.Comment: 11 pages, minor correction