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Is there a method for stability analysis of nonlinear dynamical systems $\dot x= f(x)$, except Lyapunov theory?

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You can always:

  • Linearize and determine the eigenvalues of the $A$ matrix, which gives local guarantees.
  • Determine the fixed points of $f(x)$ and analyze whether these are stable by analyzing the Jacobians of the fixed points (also local guarantees)

Moreover, you have methods that explore more global guarantees such as contraction (see Lohmiller and Slotine), convergence (Demidovich conditions) and incremental stability/dissipativity (Angeli 2002), but these methods are Lyapunov variants.

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