Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.20517

Transient Stability of Nonlinear Stochastic Dynamics: A Finite-Amplitude Logarithmic Measure Framework

Surya Ratna Prakash D·Soumyendu Raha
View original source ↗Read source document ↗Export metadata ↓

Abstract

Spacecraft guidance, navigation, and control systems operating during mission-critical phases such as atmospheric entry, powered descent, and planetary landing require reliable assessment of finite-time transient behavior under stochastic uncertainty. Existing stochastic stability frameworks primarily characterize asymptotic or infinitesimal behavior and therefore provide limited insight into finite-amplitude transient amplification over operationally relevant time horizons. This paper develops a finite-amplitude logarithmic measure for nonlinear Itô stochastic differential equations, extending classical matrix measures from infinitesimal to finite-amplitude perturbation evolution. The proposed framework establishes finite-time mean and variance bounds for logarithmic perturbation growth, derives Chernoff-type probabilistic bounds on transient amplification, and shows that mean transient stability does not necessarily guarantee finite-time pathwise safety. The theory is further extended to projected stochastic dynamics, leading to a transient-risk index that quantitatively balances deterministic contraction and diffusion-induced variability for transient-risk-aware system design. The proposed framework is validated through Monte Carlo simulations and flight-like lunar descent telemetry. The results demonstrate substantially improved prediction of nonlinear transient growth compared with classical Jacobian-based approaches and successfully distinguish trajectories exhibiting similar nominal behavior but significantly different transient-risk characteristics. The proposed framework provides a unified methodology for finite-time transient stability analysis, probabilistic transient-risk assessment, and transient-risk-aware guidance, navigation, and control of nonlinear stochastic aerospace systems.

Explore related subjects

math.DSmath.OCstat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Surya Ratna Prakash D, Soumyendu Raha. 2026-07-24. Transient Stability of Nonlinear Stochastic Dynamics: A Finite-Amplitude Logarithmic Measure Framework. https://arxiv.org/abs/2604.20517

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗