Most Probable KAM Tori in Stochastic Hamiltonian Systems

Abstract

This paper conducts an in-depth investigation into the impact of stochastic perturbations-particularly multiplicative noise-on the integrable structures of Hamiltonian systems, with a central focus on developing a KAM theory for stochastic Hamiltonian dynamics. We begin by deriving the Onsager-Machlup functional for Hamiltonian systems driven by multiplicative noise and identifying the most probable transition paths of system trajectories. This analysis reveals the fundamental differences in how additive versus multiplicative noise influences the integrability of Hamiltonian systems. Building upon this, we establish a large deviation principle for the system and derive a rate function that quantitatively characterizes trajectory deviations, especially in the regime of rare events. The main contribution of this work lies in demonstrating that, under the small noise limit, the quasi-periodic invariant tori of the unperturbed system persist in a probabilistic sense, indicating the stability of KAM structures under stochastic perturbations. Furthermore, we show that the exponential rate of deviation from the invariant tori exactly matches the large deviation rate function, thus providing a quantitative characterization of the structural persistence and fluctuation geometry of quasi-periodic motions in stochastic Hamiltonian systems. These results extend the classical KAM framework to a stochastic setting and offer new insights into the behavior of complex dynamical systems under the influence of noise.

Figures (4)

• Figure 1: Comparison of phase space trajectories between the deterministic (bold solid line) and stochastically perturbed (thin dashed lines) harmonic oscillators. The system is modeled with the following parameters: mass $m = 1$, spring constant $k = 1$, time step $\Delta t = 0.0001$, and total simulation time $T = 300$. The deterministic trajectory follows a stable circular orbit in phase space, indicating energy conservation and periodic motion. In contrast, the stochastic system introduces white noise perturbations using the Euler-Maruyama method, leading to diffusive and irregular trajectories that deviate from the original orbit over time.
• Figure 2: The distribution curve of the Hamiltonian $H(q, p) = \frac{p^2}{2m} + \frac{1}{2}kq^2$. The system is modeled with the following parameters: mass $m = 1$, spring constant $k = 1$, time step $\Delta t = 0.001$, total simulation time $T = 300$, and 5000 simulation runs. The maximum value of the Hamiltonian $H$ is observed at 1249.6399, with a corresponding density of 0.0006. This result illustrates the distribution of the Hamiltonian in the presence of stochastic perturbations, showing that the system's energy tends to concentrate around specific values in most cases.
• Figure 3: The first row of three plots represents the phase space trajectories of the solutions to the stochastic nearly integrable Hamiltonian system \ref{['30']}, the nearly integrable Hamiltonian system \ref{['31']}, and the corresponding integrable Hamiltonian system, respectively, under a perturbation strength $\gamma = 0.001$. The second row of plots shows the corresponding projections of the trajectories from the first row onto the X-Y plane.
• Figure 4: The comparison figure of Fig \ref{['F.3']} when $\gamma = 0.01$ and $\gamma = 0.1$, respectively.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1
• Remark 2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7