Nonlinear dynamical systems with control parameters may not be well modeled by shallow neural networks. In this paper, the stable fixed-point solutions, periodic and chaotic solutions of the parameter-dependent Lorenz system are learned simultaneously via a very deep neural network. The proposed deep learning model consists of a large number of identical linear layers, which provide excellent nonlinear mapping capability. Residual connections are applied to ease the flow of information and a large training dataset is further utilized. Extensive numerical results show that the chaotic solutions can be accurately forecasted for several Lyapunov times and long-term predictions are achieved for periodic solutions. Additionally, the dynamical characteristics such as bifurcation diagrams and largest Lyapunov exponents can be well recovered from the learned solutions. Finally, the principal factors contributing to the high prediction accuracy are discussed.

Reconstructing attractors of airfoil systems from observations facilitates understanding of aeroelasticity, especially the onset of flutter. However, it is generally difficult due to observation noise and the nonlinear nature of the underlying dynamics. In this study, a hybrid strategy is proposed which incorporates data preprocessing and next generation reservoir computing (NG-RC) for reconstructing attractors of an airfoil system. This approach first estimates the system states from noisy observations via a state estimation method and then trains the NG-RC model to predict the responses of the airfoil system. The NG-RC employs nonlinear functions of past states to approximate the dynamics, requiring less training data and fewer hyperparameters than the conventional reservoir computing. To reduce the model complexity, both [ ] and smoothed [ ] norm penalties are introduced to promote the sparsity of trainable weights, where the optimal weights are determined by simple iterative optimization. Simulation results show that the proposed method can predict various vibration patterns and reconstruct the attractors of the airfoil system from limited, noisy observations. The smoothed [ ] norm penalty can lead to sparser weights and, in some cases, enhance performance. The findings support applications of the present method like flutter boundary prediction and flight accident analysis.

The influences of external Lévy noise in CO oxidation on the Ir(111) surfaces are numerically examined. Depending on the control parameters, such as the temperature $T\hspace{0.17em}$ and the CO fraction $Y\hspace{0.17em}$ in the total reactant input gas flux, the reaction exhibits one or two stable stationary states in the absence of noise. Lévy noise is introduced in the reaction by randomly varying the quality of the influx mixture. We then observe a large variety of different types of behaviors depending on the initial CO fraction $Y$ , in particular, the phenomenon of noise-induced shifts of the steady state in the monostable regime is regarded. We also show how the stationary probability density changes in terms of stochastic bifurcation when Lévy noise is applied in the bistable regime. The results provide a strong evidence for noise-induced transitions. We concentrate on the effects of the stability index $\alpha ,$ the skewness parameter $\beta ,\hspace{0.17em}$ and the noise intensity $D\hspace{0.17em}$ that emphasize the differences with Gaussian fluctuations.