Bifurcation with memory

A model equation containing a memory integral is posed. The extent of the memory, the relaxation time $\lambda $, controls the bifurcation behavior as the control parameter R is increased. Small (large) $\lambda $ gives steady (periodic) bifurcation. There is a double eigenvalue at $\lambda = \lambda _1 $, separating purely steady $(\lambda \lambda _{1} )$ states with $T \to \infty $ as $\lambda \to \lambda _1^ + $. Analysis leads to the co-existence of stable steady/periodic states and as R is increased, the periodic states give way to the steady states. Numerical solutions show that this behavior persists away from $\lambda = \lambda _1 $.