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Instead of minimizing the loss of the current weights, SAM minimizes the worst-case loss in its neighborhood in the parameter space. In this paper, we investigate the convergence instability of SAM near a saddle point. Using the qualitative theory of dynamical systems, we explain how SAM becomes stuck in the saddle point and theoretically prove that the saddle point can become an attractor under SAM dynamics. Additionally, we show that this convergence instability can also occur in stochastic dynamical systems by establishing the diffusion of SAM. We prove that SAM diffusion is worse than that of vanilla gradient descent in terms of saddle point escape. 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Additionally, we show that this convergence instability can also occur in stochastic dynamical systems by establishing the diffusion of SAM. We prove that SAM diffusion is worse than that of vanilla gradient descent in terms of saddle point escape. 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