Problem with translating "Kozyklus Eigenschaft" to English

Question

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.

Answer 1

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".

Answer 2

It is a cocycle property. An example is a stochastic differential equation, where the distribution Q is generally defined by noise terms. This generally consists of base flow, the noise, and a cocycle dynamic system on the domain of the physical phase space.

For more info, check out this page:

http://en.wikipedia.org/wiki/Random_dynamical_system