Secondary Bifurcation and Multiple Eigenvalues

Secondary bifurcation of general nonlinear operator equations is studied in the neighborhood of multiple eigenvalues of the linearized problem. Necessary conditions for the existence of secondary bifurcation are derived and the location of points of secondary bifurcation are found. The geometrical significance of these conditions is discussed and conditions on primary bifurcation at a multiple eigenvalue are found which guarantee the existence of points of secondary bifurcation.