Final answer:
For a two-dimensional system with a stable limit cycle, the largest Lyapunov exponent is non-positive, reflecting the stable nature of the trajectories converging towards the limit cycle.
Step-by-step explanation:
To find the largest Lyapunov exponent for a two-dimensional system with a stable limit cycle, one has to consider the behavior of infinitesimally close trajectories over time. A positive Lyapunov exponent indicates chaos, while a zero or negative exponent suggests stability.
For a stable limit cycle, trajectories converge towards the cycle, implying that the largest Lyapunov exponent should be non-positive to indicate stability. For two-dimensional systems, this typically means one Lyapunov exponent is zero (indicating neutral stability along the limit cycle itself) and the other is negative (indicating attraction towards the cycle).
We can reason that in a system with a stable limit cycle, small perturbations in initial conditions will lead to trajectories that eventually return to the limit cycle. Hence, the stability of the system is maintained. The largest Lyapunov exponent in this case reflects the rate at which trajectories either converge to the limit cycle (when negative) or possibly diverge (which would not be the case for a stable limit cycle).