### 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.