  A parameter dependent perturbation of the spectrum of the scalar Laplacian is
studied for a class of nonlocal and non-self-adjoint rank one perturbations. A
detailed description of the perturbed spectrum is obtained both for Dirichlet
boundary conditions on a bounded interval as well as for the problem on the
full real line. The perturbation results are applied to the study of a related
parameter dependent nonlinear and nonlocal parabolic equation. The equation
models a feedback system that e.g. can be interpreted as a thermostat device or
in the context of an agent based price formation model for a market. The
existence and the stability of periodic self-oscillations of the related
nonlinear and nonlocal heat equation that arise from a Hopf bifurcation is
proved. The bifurcation and stability results are obtained both for the
nonlinear parabolic equation with Dirichlet boundary conditions and for a
related problem with nonlinear Neumann boundary conditions that model feedback
boundary control. The bifurcation and stability results follow from a Popov
criterion for integral equations after reducing the stability analysis for the
nonlinear parabolic equation to the study of a related nonlinear Volterra
integral equation. While the problem is studied in the scalar case only it can
be extended naturally to arbitrary euclidean dimension and to manifolds.
