Spectral Stability of Solutions of Equations Describing the Dynamics of Rods Kathryn Pedings, Dr. Stephane Lafortune College of Charleston

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Spectral Stability of Solutions of Equations
Describing the Dynamics of RodsKathryn Pedings,
Dr. Stephane LafortuneCollege of
Charleston kepeding_at_edisto.cofc.edu
Solutions To find the above solutions, we explore
the solutions of the linear system for ?0 that
come directly from symmetries. For example, the
symmetry ?x gives solution x1.
Abstract This project explores a system of
differential equations that describes the
dynamics of an extensible rod close to its first
writhing instability. We are interested in
traveling wave solutions. We study spectral
stability of these solutions using the Evans
function. Our Evans function computation is based
on symmetry analysis.
Perturbing the Solution We then perturb the
solution and by doing so, get an equation of ?
and a new variable ?. This perturbation will be
used to determine spectral stability.
Rod Equations
Only first order terms of u, w, p, and q are
kept. The question of stability then becomes an
eigenvalue problem If the equation has a L2
solution for Re(?) gt 0, then the solution is
unstable. The above eigenvalue problem can be
rewritten as
These can then be used to reduce the system to a
3 x 3 system which is diagonal in this case
making it easy to solve to attain three more
solutions.

• A(x,t) amplitude- B(x,t) twist density of
  the solution beyond the
  bifurcation threshold- C(x,t) modal tension-
  x space variable - t time
  variable- ? 0 corresponds to an
  inextensible rod 1 corresponds to the
  extensible case

Scaled Equations
Evans Function There are 8 solutions for
A(?,?) The Evans Function is defined as
follows
Remaining Work In system (1), x and A can be
expanded in ?.
Traveling Wave Solutions ? x - vt
We have solved the system for ? 0 completely
and we can use this with variation of parameters
to recursively solve for xi in the system. These
expansions can be used to get an expansion for
the Evans Function.