遵守莫训
Theorem: Instability of Perturbed Discrete DynamicsLet the system evolution on manifold \mathcal{M} be governed by the map x_{k+1} = \Phi(x_k) + \Xi_k, where \Xi_k denotes bounded stochastic perturbations. Consider a Lyapunov candidate V(x_k) = x_k^T P x_k with P \succ 0.The orbital difference \Delta V is defined as:\Delta V(x_k) = V(\Phi(x_k) + \Xi_k) - V(x_k)Proposition: If the spectral radius of the linearized operator satisfies \rho(\nabla \Phi) > 1 and the perturbation energy accumulates such that:\Delta V(x_k) \ge \gamma(\|x_k\|) > 0, \quad \forall x_k \in \Omega \setminus \{0\}then the equilibrium is unstable. Consequently, the maximal Lyapunov exponent \lambda_{max} becomes positive definite:\lambda_{max} = \lim_{k \to \infty} \frac{1}{k} \sum_{i=0}^{k-1} \ln \| \nabla \Phi(x_i) \| > 0implying exponential divergence of trajectories and structural failure.
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