Heriot-Watt Mathematics Report Series
HWM98-30, 22 Oct 1998
Global bifurcation in generic systems of nonlinear Sturm-Liouville problems
B P Rynne
Abstract
We consider the system of coupled nonlinear
Sturm-Liouville boundary value problems
\begin{center}$
\begin{array}{c}
L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v), \quad {\rm in}\ (0,1),\\[1 ex]
a_{10} u(0) + b_{10} u'(0) = 0, \quad
a_{11} u(1) + b_{11} u'(1) = 0,
\end{array}
$\end{center}
\begin{center}$
\begin{array}{c}
L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v), \quad {\rm in}\ (0,1),\\[1 ex]
a_{20} v(0) + b_{20} v'(0) = 0, \quad
a_{21} v(1) + b_{21} v'(1) = 0,
\end{array}
$\end{center}
where $\mu$, $\nu$ are real spectral parameters.
It will be shown that if the functions $f$ and $g$
are `generic' then for all integers $m,\,n \ge 0$,
there are smooth 2-dimensional manifolds $\S_m^1$, $\S_n^2$, of
`semi-trivial' solutions of the system which bifurcate from the
eigenvalues $\mu_m$, $\nu_n$, of $L_1$, $L_2$, respectively.
Furthermore, there are smooth curves
$\B_{mn}^1 \subset \S_m^1$, $\B_{mn}^2 \subset \S_n^2$,
along which secondary bifurcations take place,
giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions.
It is shown that there is a single such manifold, $\N_{mn}$,
which `links' the curves $\B_{mn}^1$, $\B_{mn}^2$.
Nodal properties of solutions on $\N_{mn}$ and global properties
of $\N_{mn}$ are also discussed.
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