# Problem with translating "Kozyklus Eigenschaft" to English

I'm struggling with the translation of the book written by Bernd Aulbach about ordinary differential equations. There is one notion that I can't find reference for even on the German webpages about ODEs.

Bernd Aulbach wrote: Gegeben sei eine offene Teilmenge $$D$$ des $$\mathbb R^{1+N}$$, eine stetige und bezüglich $$x$$ Lipschitz-stetige Funktion $$f \colon D \to \mathbb R^N$$ und die somit definierte Differenzialgleichung $$\dot x = f(t,x)$$. Die für alle $$(t, \tau, \xi)$$ aus der Menge $$\Omega$$ definierte Funktion $$\lambda(t, \tau, \xi) = \lambda_{\max}(t, \tau, \xi)$$ nennen wir dann die allgemeine Lösung der Differenzialgleichung. $$\Omega : \{(t, \tau, \xi) \in \mathbb > R^{1+1+N} : (\tau, \xi) \in D, t \in I_{\max}(\tau, \xi)\}$$ Sei $$(\tau, \xi)$$ ein beliebiger Punkt aus $$D$$. Dann gelten für jedes $$\sigma \in I_{\max}(\tau, \xi)$$ die Beziehungen (...) und $$\lambda(t, > \sigma, \lambda(\sigma, \tau, \xi)) = \lambda(t, \tau, \xi)$$ für alle $$t \in I_{\max}(\tau, \xi)$$, die Identität nennen wir die Kozyklus-Eigenschaft der allgemeinen

Here is my (quite accurate, I hope) translation of the second part (which is more important):

Let $$(\tau, \xi)$$ be any point lying in $$D$$. The following equalities are true for all $$\sigma \in I_{\max}(\tau, \xi)$$: (...) and $$\lambda(t, \sigma, \lambda(\sigma, \tau, \xi)) = \lambda(t, \tau, \xi)$$ for all $$t \in I_{\max}(\tau, \xi)$$, this identity is called the cocycle property of general solution.

I can't understand what the "cocycle property" actually is and how to translate it to English.

Internet search shows that $λ(t,τ,ξ)$ is (not so) commonly called the "flow" (Fluss) of the dynamical system.
As some kind of propagator (bijective map) from the fiber over $τ$ to the fiber over $t$, it has a transitivity property in that going first from $τ$ to $σ$ and then from $σ$ to $t$ does not change the result, i.e., $$λ(t,σ,λ(σ,τ,ξ))=λ(t,τ,ξ)$$ Using $λ_{t←τ}(ξ)=λ(t,τ,ξ)$, this composition also reads as $$λ_{t←σ}\circ λ_{σ←τ}= λ_{t←τ}\quad \text{ or }\quad λ_{t←σ}\circ λ_{σ←τ}\circ λ_{τ←t}=id$$ which means that over the cycle $(τ,σ,t)$ (or any larger cycle) the composition of the maps corresponding to the edges of the cycle is closed (in the multiplicative sense).
For autonomous systems, $λ(t,τ,ξ)=ϕ(t-τ,ξ)=ϕ_{t-τ}(ξ)$ only depends on the difference $t-τ$. In this last notation, one is then justified to call $$ϕ_s\circ\phi_t=ϕ_{s+t}$$ the "semi-group property".