Lyapunov exponent analysis for the coupled Chialvo–Rulkov tri-oscillator network

Develop and carry out a Lyapunov exponent analysis for the six-dimensional discrete-time heterogeneous tri-oscillator chain consisting of two Chialvo map neurons at the edges and one Rulkov map neuron at the center, coupled bidirectionally via static linear couplings (σ12, σ21, σ23, σ32), in order to compute the Lyapunov spectrum of the full coupled network and characterize its stability and chaotic behavior.

Background

The paper introduces a six-dimensional discrete-time heterogeneous neural network composed of two Chialvo neurons (edge nodes) and one Rulkov neuron (center), coupled bidirectionally through linear, static coupling strengths. The authors thoroughly analyze fixed points, Jacobian spectra, noninvertibility, and bifurcations (codimension-1 and -2), and quantify synchrony and complexity via cross-correlation, Kuramoto order parameter, and sample entropy.

Despite these advances, the authors explicitly state that conducting a Lyapunov exponent study for the full coupled network remains unresolved in their work. They suggest that methods developed for temporal networks (e.g., Caligiuri et al., 2023) might inspire an approach, but emphasize that establishing a Lyapunov analysis for this static coupled-map network is still an open question.

References

One challenge the authors have faced is to come up with a Lyapunov exponent study of the network itself, where the nodes are coupled. One approach could be motivated by Caligiuri {\em et al.}. This remains an open question for a static network like our model.

— Dynamical properties of a small heterogeneous chain network of neurons in discrete time  (2405.05675 - Ghosh et al., 2024) in Conclusions and future directions (Section: Conclusions and future directions)

We emphasize that \lambda_1 must be computed numerically for the coupled lattice. There is not currently a way to theoretically predict \lambda_1.

— Spatiotemporal Chaos with Extended Spatial Interactions  (2609.01424 - Raj et al., 1 Sep 2026) in Section 3.2, “Spectra of Lyapunov Exponents,” discussion following Eq. (10)