Random perturbations of systems with periodic impulse effects

Abstract

The principal aim of the present work is to explore limit theorems for small random perturbations of dynamical systems with periodic impulse effects, in the limit of vanishing noise intensity. We start with a system whose time evolution is governed by a nonlinear ordinary differential equation in between impulses, and a nonlinear resetting map at impulses; the latter are assumed to arrive in a time-periodic manner. We next consider small state-dependent Brownian perturbations of this system and explore the zero noise limit on finite, but arbitrary, time horizons. For the resulting stochastic system with impulse effects, we prove convergence to the underlying deterministic impulsive system as the noise goes to zero. More importantly, we prove convergence of the rescaled fluctuation process about the deterministic limit in a strong pathwise sense on finite time intervals to a limiting fluctuation process governed by a linear time-dependent stochastic differential equation in between impulses and a linear time-dependent resetting map at impulses.The results are illustrated numerically for a periodically kicked nonlinear pendulum with state-dependent kick sizes.

Figures (3)

• Figure 1: Sample paths of the components $X_i\triangleq X^\varepsilon_i(t)$, $x_i(t)$, $A^\varepsilon_i\triangleq x_i+\varepsilon Z_i(t)$ and $Error^\varepsilon_i \triangleq X^\varepsilon_i - A^\varepsilon_i$ for $i=1, 2$ and $\varepsilon=2^{-4}$.
• Figure 2: Sample paths of the components $Y_i\triangleq \frac{X^\varepsilon_i(t)- x_i(t)}{\varepsilon}$, $Z_i(t)$ and $Error^\varepsilon_i \triangleq Y^\varepsilon_i - Z_i$ for $i=1, 2$ and $\varepsilon=2^{-5}$ .
• Figure 3: In the figure (A) the components $(e_1, e_2)$ which are the mean of $\sup_{0\le t\le \mathsf{T}}|X^\varepsilon_i(t) - x_i(t)|$ and in the figure (B) the components $(e_1, e_2)$ which are the mean of $\sup_{0\le t\le \mathsf{T}}|X^\varepsilon_i(t) - A^\varepsilon_i(t)|$, over $1000$ sample paths are plotted on a $log_2$--$log_2$ scale. The values of $e_i, i=1, 2$ decrease with increasing $i$, where the values of $\varepsilon=2^{-i}, 1\le i\le 10$.

Theorems & Definitions (26)

• Remark 2.3
• Remark 2.4
• Theorem 2.5: Law of large numbers
• Theorem 2.6: Central limit theorem
• Remark 2.8
• Remark 2.9
• Proposition 2.10: Taylor's formula
• Definition 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['L:LLN-path-diff']}