Global branches of periodic solutions for forced delay differential equations on compact manifolds (Q868834)

This paper deals with the global bifurcation of \(T\)-periodic solutions for the following delay differential equation \[ x'(t)=\lambda f(t,x(t),x(t-1)), \eqno{(1)} \] where \(\lambda\geq 0\) is a parameter and \(f\in C(\mathbb R\times M\times M,\mathbb R^k)\) satisfying \[ f(t+T,p,q)=f(t,p,q)\in T_p(M)~~\forall (t,p,q)\in\mathbb R\times M\times M. \] Here, \(M\subset\mathbb R^k\) is a smooth manifold with (possibly empty) boundary and \(T_p(M)\subset\mathbb R^k\) denotes the tangent space of \(M\) at \(p\). A pair \((\lambda, x)\) of (1) is said to be \(T\)-periodic if \(\lambda\geq 0\) and \(x:\mathbb R\to M\) is a \(T\)-periodic solution of (1) corresponding to \(\lambda\). A \(T\)-periodic pair \((\lambda,x)\) is called trivial if \(\lambda=0\). This nice paper proves that if \(M\) is compact with nonzero Euler-Poincaré characteristic, \(T\geq 1\) and \(f\) satisfies a natural inward condition along the boundary of \(M\) (when nonempty), then there exists an unbounded (with respect to \(\lambda\)) connected branch of nontrivial \(T\)-periodic pairs whose closure intersects the set of the trivial \(T\)-periodic pairs in a nonempty set of bifurcation points. The unusual notion used in this paper is called atypical bifurcation or co-bifurcation. The approach is based on the fixed point index theory for maps on ANRs.