Exact periods and exact critical values for Hopf bifurcations from multi-peak solutions of the shadow Gierer-Meinhardt model in one spatial dimension

We consider the shadow Gierer-Meinhardt model { ∂ t U = ε 2 U x x − U + U p ξ q for 0 < x < 1 , 0 < t < T , τ ∂ t ξ = ∫ 0 1 ( − ξ + U r ξ s ) d x for 0 < t < T , U x ( 0 , t ) = U x ( 1 , t ) = 0 , where ε > 0 , τ > 0 , p > 1 , q > 0 , r > 0 , s ≥ 0 and q r / ( p − 1 ) ( s + 1 ) > 1 . Wei and Winter showed that if ( p , r ) = ( 3 , 2 ) or ( 2 , 2 ) , then as τ increases, a stable stationary monotone solution is destabilized by Hopf bifurcation, and hence periodic solutions appear. In this paper we consider two cases ( p , r ) = ( 3 , 1 ) and ( 3 , 3 ) . We show that a Hopf bifurcation occurs for n -mode stationary solutions, n ≥ 1 , in a rigorous way, studying eigenvalues in detail. Exact periods and exact critical values of τ can be written by using complete elliptic integrals. A relationship between a period and a shape of a stationary solution is also studied. In particular, a maximum point of the period is studied.