Lyapunov or Markov

Lyapunov analysis studies stability through suitable scalar functions; Markov modeling represents next-state probabilities using a sufficient current state. They are complementary mathematical ideas, not a division between convergent and permanently wandering systems.

Lyapunov analysis and Markov modeling answer different questions and are not mutually exclusive system types. Lyapunov methods seek a function that certifies a property of specified dynamics. A Markov model assumes the chosen current state contains the information needed for the next-state probability rule. Markov chains can settle into absorbing states, approach stationary distributions, cycle, or exhibit other behavior.

Choose the mathematical question before the tool. Convergence of a probability distribution is different from each realized trajectory stopping at one state. A state description may also include accumulated information, so a process can be Markov in that richer state while earlier events still affect its current condition. Neither label alone determines the importance of timing or the appropriate intervention.

When to use it

When choosing how to analyze a specified dynamic or stochastic model, especially when claims about convergence, memory, and initial conditions are being confused.

How it can help

Identify the question and state representation first. Use appropriate convergence or stability conditions, and distinguish individual paths from distributions before drawing a practical implication.