Global bifurcation of periodic solutions to ordinary differential equations (Q1381635)

The author studies bifurcations and multiplicity of solutions to the parametrized ordinary differential equation \[ u''+ \lambda u - g(\lambda, x, u) = 0 \] with periodic boundary conditions \[ u(0) - u(2\pi) = u'(0) - u'(2\pi) = 0, \] where \(g\) is continuously differentiable in the third variable and \(g(\lambda, x, 0) = g_\lambda(\lambda, x, 0) = 0\). The author treats the parameterized boundary value problem as a problem for stationary solutions to the semilinear parabolic problem \[ u_{xx} + \lambda u - g( \lambda, x, u) = u_t u(2\pi, t) - u(O, t) = u_x(2\pi, t) - u_x(O, t) = 0. \] Under certain conditions on \(g\) the author is able to show that there is a continuum of full bounded solutions bifurcating from each point \((k^2, 0)\), \(k\in \mathbb{N}\) which extends globally. The result is based on Rybakowski's extension of the Conley index.