The stroboscopic (time-$P$) map of a $P$-periodic first-order system

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, let $P>0$, and let $F:\mathbb{R}\times U\to {\mathbb{R}}^n$ be a map that is continuous and locally Lipschitz in the state variable, in the sense of the hypotheses of Every Picard–Lindelöf initial value problem has one maximal solution on an open interval, and that is $P$-periodic:

$F(t+P,z)=F(t,z)(t\in \mathbb{R},z\in U)\mathrm{.}$

Consider the first-order system $x^{\mathrm{\prime }}(t)=F(t,x(t))$ of First-order systems, initial value problems, and solutions on intervals on the open domain $\mathbb{R}\times U$. For $z\in U$ let

$x_z:J_z⟶U$

be the unique maximal solution of the initial value problem $x^{\mathrm{\prime }}(t)=F(t,x(t))$, $x(0)=z$; here $J_z$ is the open maximal interval of existence containing $0$, and by Every Picard–Lindelöf initial value problem has one maximal solution on an open interval every solution of this initial value problem is a restriction of $x_z$ (Extensions, maximal solutions, maximal intervals, and global solutions, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

The domain of the stroboscopic map is

$D_P:=\{z\in U:[0,P]\subseteq J_z\},$

the set of initial states whose solution exists over one full period. The stroboscopic map (or time-$P$ map) of the system is the function

${\theta }_P:D_P⟶U,{\theta }_P(z):=x_z(P)\mathrm{.}$

This is a well-defined function: for $z\in D_P$ the interval $J_z$ contains $P$, so $x_z(P)$ is a point of $U$, and by the uniqueness clause of the maximal-solution theorem the value $x_z(P)$ does not depend on any choice, since $x_z$ is the only maximal solution through $(0,z)$.

• The definition depends on the chosen initial time $0$. Replacing the initial time by $t_0$ and defining ${\theta }_P^{t_0}(z):=x_{t_0,z}(t_0+P)$ on the corresponding domain gives the same family of maps up to the time-translation flow of the $P$-periodic system; only the initial time $0$ is used on this page.

• The domain $D_P$ is the set where the solution exists up to time $P$; it need not be invariant under ${\theta }_P$ or equal all of $U$. Openness of this domain is not proved here. Formal iterates ${\theta }_P^k$ are defined only on the subset of points for which each successive value remains in $D_P$; the precise statement is Iterates of the stroboscopic map, and equilibria as fixed points.

• If $F$ does not depend on $t$ (the autonomous case), then ${\theta }_P$ is the time-$P$ map of the autonomous system and every equilibrium $z_{\ast }$ of the autonomous right-hand side satisfies ${\theta }_P(z_{\ast })=z_{\ast }$. A fixed point $z\in D_P$ yields a $P$-periodic solution: repeat $x_z{\mathrm{\mid }}_{[0,P]}$ on every interval $[kP,(k+1)P]$, $k\in \mathbb{Z}$. Values at each joining point equal $z$, and the one-sided derivatives agree because $F(P,z)=F(0,z)$. Periodicity of $F$ makes each segment solve the same equation; hence the repeated curve is a solution on $\mathbb{R}$, and uniqueness identifies it with $x_z$. Conversely a $P$-periodic solution through $z$ gives ${\theta }_P(z)=z$. Constant solutions are included, so a fixed point alone does not imply a nonconstant periodic orbit.

• The map ${\theta }_P$ can be studied as a discrete-time map. Applying the hyperbolic-fixed-point definition of Hyperbolic equilibria, hyperbolic fixed points, periodic orbits, limit cycles and homoclinic trajectories requires a differentiable self-map of an open neighbourhood of the fixed point; such a restriction is not supplied here. No relation to a continuous-time stability criterion is asserted here.