  This paper investigates the dynamics of cosmological models incorporating
nonlinear electrodynamics (NLED), focusing on their stability and causality. We
explore two specific NLED models: the Power-Law and Rational Lagrangians. We
assess these models' viability in describing the universe's evolution using
dynamical systems theory and Bayesian inference. We present the theoretical
framework of NLED coupled with general relativity, followed by an analysis of
the stability and causality through the squared sound speed of the NLED
Lagrangians. We then conduct a detailed dynamical analysis to identify the
universe's evolution with this matter content. Our results show that the
Power-Law Lagrangian model transitions through various cosmological phases from
a Maxwell radiation-dominated state to a matter-dominated state. For the
Rational Lagrangian model, including the Maxwell term, stable and causal
behavior is observed within specific parameter ranges, with critical points
indicating the evolutionary pathways of the universe. To validate our
theoretical findings, we perform Bayesian parameter estimation using a
comprehensive set of observational data, including cosmic chronometers, Baryon
Acoustic Oscillation (BAO) measurements, and Type Ia Supernovae (SNeIa). The
estimated parameters for both models align with expected values for the current
universe, particularly the matter density $\Omega_m$ and the Hubble parameter
$h$. However, the parameters $\alpha$ and $b$ are not tightly constrained
within the prior ranges. Our model comparison strongly favors the $\Lambda$CDM
model over the NLED models for late-universe observations, as the NLED model
does not exhibit a cosmological constant behavior. Our results highlight the
need for further refinement and exploration of NLED-based cosmological models
to fully integrate them into the standard cosmological framework.
