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Construction of mean-square Lyapunov-basins for random ordinary differential equations

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  • We propose a straightforward basin search algorithm to determine a suitably large level set of the mean-square Lyapunov-function that corresponds to the linearization about an path-wise equilibrium solution of a random ordinary differential equation (RODE). Noise intensity plays a crucial role for how similar the behavior of solutions of RODEs is compared to the corresponding deterministic system. In this regards, the basin search algorithm also allows to numerically estimate up to which noise intensities linearized mean-square asymptotic stability remains.

    Mathematics Subject Classification: Primary: 34F05, 34D20, 37M05; Secondary: 60H10.

    Citation:

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  • Figure 1.  The path-wise setting and in particular the theory of flows of Random Dynamical Systems connects certain SODEs and RODEs

    Figure 2.  Simulations of the dynamics of our benchmark system (1) for different values of $ c = c_1 = c_2 $. (a) sketches the deterministic ($ c = 0 $) case, and (b) displays some trajectories for $ c = \tfrac{1}{17} $ showing divergent or convergent behaviors. In (c)-(f) results of Monte Carlo runs are displayed for several values of $ c \in \{ \tfrac{1}{17}, \tfrac{1}{8}, \tfrac{1}{2}, 2 \} $. See the text for details

    Figure 3.  (a) Approximation of the region with $ v' < 0 $ (blue) with $ c_1 = c_2 = \tfrac{1}{17} $. An application of our basin search algorithm with some refinement steps is shown in (b)

    Figure 4.  Optimal radii $ r $ of the Lyapunov-basins obtained with our algorithm for increasing noise strengths $ c = c_1 = c_2 $ starting at $ c = 0 $ (deterministic case)

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