Retarded equations on the sphere induced by linear equations

Using a Poincaré compactification, the linear homogeneous system of delay equations {x = Ax(t ? 1) (A is an n × n real matrix) induces a delay system π(A) on the sphere Sⁿ. The points at infinity belong to an invariant submanifold S^{n ? 1} of Sⁿ. For an open and dense set of 2 × 2 matrices A with distinct eigenvalues, the system π(A) has only hyperbolic critical points (including the critical points at infinity). For an open and dense set of 2 × 2matrices $\text{A}$ with complex eigenvalues, the nonwandering set at infinity is the union of an odd number of hyperbolic periodic orbits; if $\text{(}\text{det}\text{A}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{<}\text{3π}\text{2}$, the restriction of $\text{π(}\text{A}\text{)}$ to S¹ is Morse-Smale. For n = 1 there exist periodic orbits of period 4 provided that $\text{?A >}\text{π}\text{2}$ and Hopf bifurcation of a center occurs for ?A near $\text{(}\text{π}\text{2}\text{) + 2kπ, k ? Z}$.