In this work, we mainly characterize stochastic bifurcations and tipping phenomena of insect outbreak dynamical systems driven by α-stable Lévy processes. In one-dimensional insect outbreak model, we find the fixed points representing refuge and outbreak from the bifurcation curves, and carry out a sensitivity analysis with respect to small changes in the model parameters, the stability index and the noise intensity. We perform the numerical simulations of dynamical trajectories using Monte Carlo methods, which contribute to looking at stochastic hysteresis phenomenon. As for two-dimensional insect outbreak system, we identify the global stability properties of fixed points and express the probability density function by the stationary solution of the nonlocal Fokker-Planck equation. Through numerical modelling, the phase portrait makes it possible to detect critical transitions among the stable states. It is then worth describing stochastic bifurcation associated with tipping points induced by noise through a careful examination of the dynamical paths of the insect outbreak system with external forcing. The results give new insight into the sensitivity of ecosystems to realistic environmental changes represented by stochastic perturbations.

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) Lévy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with α -stable Lévy noise from sample path data based on the properties of α -stable distributions. More specifically, we first estimate the Lévy jump measure and noise intensity via computing mean and variance of the amplitude of the increment of the sample paths. Then we approximate the drift coefficient by combining nonlocal Kramers–Moyal formulas with normalizing flows. Numerical experiments on one- and two-dimensional prototypical examples including simulated and real world measurement data illustrate the accuracy and effectiveness of our method. This approach will become an effective scientific tool in discovering stochastic governing laws of complex phenomena and understanding dynamical behaviors under non-Gaussian fluctuations.

Stochastic vegetation–water dynamical systems are fundamental to understanding ecological stability, biodiversity conservation, water resource sustainability, and climate change adaptation. In this study, we introduce an innovative machine learning framework for analyzing rare events in stochastic vegetation–water systems driven by multiplicative Gaussian noise. By integrating the Freidlin–Wentzell large deviation theory with deep learning techniques, we establish rigorous asymptotic formulations for both the quasipotential and the mean first exit time. Leveraging vector field decomposition principles, we develop a novel neural network architecture capable of accurately computing the most probable transition paths and mean first exit times across diverse boundary conditions, including both non-characteristic and characteristic scenarios. Our findings demonstrate that the proposed method significantly enhances the predictive capabilities for early detection of vegetation degradation, thereby offering robust theoretical foundations and advanced computational tools for ecological management and conservation strategies. Furthermore, this approach establishes a scalable framework for investigating more complex, high-dimensional stochastic dynamical systems, opening new avenues for research in ecological modeling and environmental forecasting.

In the present paper, the noise-induced escape from stable invariant tori with the frame of general Langevin dynamics is investigated under weak random perturbations. Based on the large deviation theory, the quasi-potential, a quantity exponentially dominating the mean first escape time and the stationary or quasi-stationary probability density, is explored both analytically and numerically. The results show that whether the stochastic trajectories are ergodic on a torus has a crucial influence on the behavior of the quasi-potential. Specifically, there are two sources of ergodicity. One is the ergodicity of the deterministic flow, and the other requires the nondegeneracy of the noise in the tangential directions of a torus. It is found that if the ergodicity holds, the quasi-potential will be independent of the initial position on a torus, but not when the ergodicity is broken. In particular, it indicates that, for nonlinear systems driven by combined Gaussian white noise and multiple harmonic excitations, the mean first exit time varies discontinuously with respect to the frequency vector of these harmonic excitations as the noise intensity approaches zero. Adding noise to the phase of these harmonic excitations will eliminate the dependence on the initial position and thus, make the systems more robust. It reminds us that great care must be taken when dealing with noise-induced problems involving systems that possess unstable geometric structures.

The anti-missile interception technique of unpowered phase is of much importance in the military field, which depends on the prediction of the missile trajectory and the establishment of the missile model. With rapid development of data science field and large amounts of available data observed, there are more and more powerful data-driven methods proposed recently in discovering governing equations of complex systems. In this work, we introduce an anti-missile interception technique via a data-driven method based on Koopman operator theory. More specifically, we describe the dynamical model of the missile established by classical mechanics to generate the trajectorial data. Then we perform the data-driven method based on Koopman operator to identify the governing equations for the position and velocity of the missile. Numerical experiments show that the trajectories of the learned model agree well with the ones of the true model. The effectiveness and accuracy of this technique suggest that it will be realized in practical applications of anti-missile interception.

The existing data-driven identification methods for hybrid dynamical systems such as sparse optimization are usually limited to parameter identification for coefficients of pre-defined candidate functions or composition of prescribed function forms, which depend on the prior knowledge of the dynamical models. In this work, we propose a novel data-driven framework to discover the hybrid dynamical systems from time series data, without any prior knowledge required of the systems. More specifically, we devise a dual-loop algorithm to peel off the data subject to each subsystem of the hybrid dynamical system. Then, we approximate the subsystems by iteratively training several residual networks and estimate the transition rules by training a fully connected neural network. Several prototypical examples are presented to demonstrate the effectiveness and accuracy of our method for hybrid models with various dimensions and structures. This method appears to be an effective tool for learning the evolutionary governing laws of hybrid dynamical systems from available data sets with wide applications.