Snakes: Oriented families of periodic orbits,their sources,sinks, and continuation

Poincaré observed that for a differential equation $\text{x′ = ?(x, α)}$ depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in (x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index φ and a “center” index

= 1) and “sinks” ($\textcolor{red}{}\text{= ?1}$). Generically if the set Q is bounded in (x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter “snake” of orbits. A “snake” is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1.