Bifurcation with Memory

A model equation containing a memory integral is posed. The extent of the memory, the relaxation time λ, controls the bifurcation behavior as the control parameter R is increased. Small (large) λ gives steady (periodic) bifurcation. There is a double eigenvalue at λ = λ ₁, separating purely steady (λ < λ ₁ ) from combined steady/T-periodic (λ > λ ₁ ) states with T → ∞ as λ → λ 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 λ = λ ₁.