Arnoldi Jean-François, Haegeman Bart
Center for Biodiversity Theory and Modelling , Station d'Écologie Théorique et Expérimentale , 2 route du CNRS, 09200 Moulis, France.
Proc Math Phys Eng Sci. 2016 Sep;472(2193):20150874. doi: 10.1098/rspa.2015.0874.
We exhibit a fundamental relationship between measures of dynamical and structural stability of linear dynamical systems-e.g. linearized models in the vicinity of equilibria. We show that dynamical stability, quantified via the response to external perturbations (i.e. perturbation of dynamical variables), coincides with the minimal internal perturbation (i.e. perturbations of interactions between variables) able to render the system unstable. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a linear system's response to white-noise perturbations directly reflects the intensity of internal white-noise disturbance that it can accommodate before becoming stochastically unstable.
我们展示了线性动力系统的动力学稳定性和结构稳定性度量之间的基本关系，例如平衡附近的线性化模型。我们表明，通过对外部扰动（即动态变量的扰动）的响应来量化的动力学稳定性，与能够使系统不稳定的最小内部扰动（即变量之间相互作用的扰动）相一致。首先，通过重新表述控制理论的一个结果，我们解释了谐波外部扰动反映了平衡点处雅可比矩阵相对于其系数的恒定变化的谱灵敏度。然而，为了使这种等价关系成立，必须允许雅可比系数的虚数变化。因此，对于实际动力系统，与动力学稳定性的联系就丧失了。我们表明，通过将随机噪声视为外部和内部扰动，可以避免这个问题，从而恢复动力学稳定性和结构稳定性之间的基本联系。更确切地说，我们证明了线性系统对白噪声扰动的响应直接反映了它在变得随机不稳定之前能够承受的内部白噪声干扰的强度。
