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<p>The transient probabilistic response analysis of nonlinear stochastic systems subjected to combined Gaussian white noise and non-stationary modulated coloured noise is often formulated in terms of a Fokker–Planck-Kolmogorov (FPK) equation after Markovianisation. However, the state augmentation required in this procedure may lead to a curse of dimensionality and a high computational burden. This difficulty is mainly caused by the finite correlation time of coloured noise and the non-Markovian features induced by non-stationary excitation. To address this issue, this paper proposes a transient analysis framework that combines the extended Novikov-Furutsu (NF) theory with the Laplace path integral (Laplace-PI) method. First, based on the extended NF theory, an equivalent FPK equation is derived for systems driven by Gaussian white noise and deterministically modulated Gaussian coloured noise, without introducing auxiliary state variables. This formulation incorporates the prescribed noise correlation, modulation effects, and equilibrium-dependent dynamical information into a low-dimensional probabilistic description. Subsequently, a Laplace-based path-integral approximation is developed to efficiently evaluate the transient transition probability density. This approximation enables a systematic investigation of how equilibrium structures and dominant transition paths influence the evolution of transient probability density functions (PDFs). Numerical studies on representative strongly nonlinear systems, including Duffing, self-excited, and ship rolling motion oscillators, validate the effectiveness and robustness of the proposed framework under complex dynamical regimes characterized by multistability, limit cycles, and structural transitions in probabilistic states. The results show that the NF-Laplace-PI method provides accurate predictions of stationary probabilistic structures, multimodal distributions, and transient PDF evolution under varying system parameters. Overall, the proposed framework offers an efficient theoretical and computational tool for transient probabilistic analysis under non-stationary stochastic excitation, while substantially reducing the computational burden associated with conventional Markovian dimensional expansion and numerical integration approaches.</p>

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Transient probability density analysis of strongly nonlinear systems subjected to combined Gaussian white noise and non-stationary modulated coloured noise

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Jianmin Xiang,
  • Di Liu

摘要

The transient probabilistic response analysis of nonlinear stochastic systems subjected to combined Gaussian white noise and non-stationary modulated coloured noise is often formulated in terms of a Fokker–Planck-Kolmogorov (FPK) equation after Markovianisation. However, the state augmentation required in this procedure may lead to a curse of dimensionality and a high computational burden. This difficulty is mainly caused by the finite correlation time of coloured noise and the non-Markovian features induced by non-stationary excitation. To address this issue, this paper proposes a transient analysis framework that combines the extended Novikov-Furutsu (NF) theory with the Laplace path integral (Laplace-PI) method. First, based on the extended NF theory, an equivalent FPK equation is derived for systems driven by Gaussian white noise and deterministically modulated Gaussian coloured noise, without introducing auxiliary state variables. This formulation incorporates the prescribed noise correlation, modulation effects, and equilibrium-dependent dynamical information into a low-dimensional probabilistic description. Subsequently, a Laplace-based path-integral approximation is developed to efficiently evaluate the transient transition probability density. This approximation enables a systematic investigation of how equilibrium structures and dominant transition paths influence the evolution of transient probability density functions (PDFs). Numerical studies on representative strongly nonlinear systems, including Duffing, self-excited, and ship rolling motion oscillators, validate the effectiveness and robustness of the proposed framework under complex dynamical regimes characterized by multistability, limit cycles, and structural transitions in probabilistic states. The results show that the NF-Laplace-PI method provides accurate predictions of stationary probabilistic structures, multimodal distributions, and transient PDF evolution under varying system parameters. Overall, the proposed framework offers an efficient theoretical and computational tool for transient probabilistic analysis under non-stationary stochastic excitation, while substantially reducing the computational burden associated with conventional Markovian dimensional expansion and numerical integration approaches.