High Quality Content by WIKIPEDIA articles! In mathematics, a dynamical system is invertible if the forward evolution is one-to-one, not many-to-one; so that for every state there exists a well-defined reverse-time evolution operator. The dynamics are time-reversible if there exists a transformation (an involution) ? which gives a one-to-one mapping between the time-reversed evolution of any one state, and the forward-time evolution of another corresponding state. Any time-independent structures (for example critical points, or attractors) which the dynamics gives rise to must therefore either be self-symmetrical or have symmetrical images under the involution ?.