4 research outputs found
Interaction of sine-Gordon kinks with defects: The two-bounce resonance
A model of soliton-defect interactions in the sine-Gordon equations is
studied using singular perturbation theory. Melnikov theory is used to derive a
critical velocity for strong interactions, which is shown to be exponentially
small for weak defects. Matched asymptotic expansions for nearly heteroclinic
orbits are constructed for the initial value problem, which are then used to
derive analytical formulas for the locations of the well known two- and
three-bounce resonance windows, as well as several other phenomena seen in
numerical simulations.Comment: 26 pages, 17 figure
Hard loss of stability in Painlev\'e-2 equation
A special asymptotic solution of the Painlev\'e-2 equation with small
parameter is studied. This solution has a critical point corresponding to
a bifurcation phenomenon. When the constructed solution varies slowly
and when the solution oscillates very fast. We investigate the
transitional layer in detail and obtain a smooth asymptotic solution, using a
sequence of scaling and matching procedures
Slow Passage through a Saddle-Node Bifurcation in Discrete Dynamical Systems
We study a discrete non-autonomous system whose autonomous counterpart (with
the frozen bifurcation parameter) admits a saddle-node bifurcation, and in
which the bifurcation parameter slowly changes in time and is characterized by
a sweep rate constant . The discrete system is more appropriate for
modeling realistic systems since only time series data is available. We show
that in contrast to its autonomous counterpart, when the time mesh size is less than the order , there is a bifurcation delay as the
bifurcation time-varying parameter is varied through the bifurcation point, and
the delay is proportional to the two-thirds power of the sweep rate constant
. This bifurcation delay is significant in various realistic systems
since it allows one to take necessary action promptly before a sudden collapse
or shift to different states. On the other hand, when the time mesh size
is larger than the order , the dynamical behavior of
the solution is dramatically changed before the bifurcation point. This
behavior is not observed in the autonomous counterpart. Therefore, the
dynamical behavior of the system strongly depends on the time mesh size.
Finally. due to the very discrete feature of the system, there are no efficient
tools for the analytical study of the system. Our approach is elementary and
analytical