Connection between Lorenz-Fibonacci substitutions and Lorenz-attractor dynamics

Establish the connection between the substitutions \(\zeta_{k,r}\) and \(\phi_{k,r}\), their corresponding dynamical systems, and the dynamics of the Lorenz attractor.

Background

The paper introduces two families of substitutions, the $(k,r)$-Lorenz-Fibonacci substitutions $\zeta_{k,r}$ and the direct Lorenz-Fibonacci substitutions $\phi_{k,r}$. Although these substitutions share the same incidence matrix and are related to integer sequences arising in symbolic models of the Lorenz attractor, their combinatorial structures and associated graphs differ substantially.

The authors identify the relationship between these substitutions, their dynamical systems, and the Lorenz-attractor dynamics as unresolved. This problem seeks a detailed dynamical interpretation of the substitutions and their associated systems within the dynamics of the Lorenz attractor.