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In this paper, we study the existence of bifurcation of a van der Pol-Duffing oscillator with quintic terms and its quasi-periodic solutions by means of qualitative and bifurcation theories. Firstly, we analyze the autonomous system and find that it has two kinds of local bifurcations and a global bifurcation: pitchfork bifurcation, Hopf bifurcation, homoclinic bifurcation. It is worth noting that the disappearance of the homoclinic orbit is synchronized with the emergence of a large limit cycle. Then, by discussing the stability of equilibria at infinity and the orientation of the trajectory, the existence and stability of limit circles of the autonomous system are analyzed by combining the Poincar\'{e}-Bendixson theorem and the index theory. The global phase portrait and the numerical simulation of the autonomous system in different parameter values are given. Finally, the existence of periodic and quasi-periodic solutions to periodic forced system is proved by a KAM theorem.

Time series forecasting is a crucial task in various domains. Caused by factors such as trends, seasonality, or irregular fluctuations, time series often exhibits non-stationary. It obstructs stable feature propagation through deep layers, disrupts feature distributions, and complicates learning data distribution changes. As a result, many existing models struggle to capture the underlying patterns, leading to degraded forecasting performance. In this study, we tackle the challenge of non-stationarity in time series forecasting with our proposed framework called U-Mixer. By combining Unet and Mixer, U-Mixer effectively captures local temporal dependencies between different patches and channels separately to avoid the influence of distribution variations among channels, and merge low- and high-levels features to obtain comprehensive data representations. The key contribution is a novel stationarity correction method, explicitly restoring data distribution by constraining the difference in stationarity between the data before and after model processing to restore the non-stationarity information, while ensuring the temporal dependencies are preserved. Through extensive experiments on various real-world time series datasets, U-Mixer demonstrates its effectiveness and robustness, and achieves 14.5\% and 7.7\% improvements over state-of-the-art (SOTA) methods., Comment: Accepted by AAAI2024

Numerous studies have proved their effective strength in detecting Control Area Network (CAN) attacks. In the realm of understanding the human semantic space, transformer-based models have demonstrated remarkable effectiveness. Leveraging pre-trained transformers has become a common strategy in various language-related tasks, enabling these models to grasp human semantics more comprehensively. To delve into the adaptability evaluation on pre-trained models for CAN intrusion detection, we have developed two distinct models: CAN-SecureBERT and CAN-LLAMA2. Notably, our CAN-LLAMA2 model surpasses the state-of-the-art models by achieving an exceptional performance 0.999993 in terms of balanced accuracy, precision detection rate, F1 score, and a remarkably low false alarm rate of 3.10e-6. Impressively, the false alarm rate is 52 times smaller than that of the leading model, MTH-IDS (Multitiered Hybrid Intrusion Detection System). Our study underscores the promise of employing a Large Language Model as the foundational model, while incorporating adapters for other cybersecurity-related tasks and maintaining the model's inherent language-related capabilities., Comment: 13 pages, 13 figures, 6 tables

In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution}, later by D. Li and X. Zhang in \cite{LiZh:NLS, LiZh:NLW} in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in $\dot H^1(\R^N)$ (i.e. the spectral assumption in \cite{MiaoWX:dynamic gHartree}) by the nondegeneracy result of positive bubble solution in $L^{\infty}(\R^N)$ in \cite{LLTX:Nondegeneracy} and the Moser iteration method in \cite{DiMeVald:book}, which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions., Comment: 41 pages. All comments are welcome

Time series forecasting has received wide interest from existing research due to its broad applications and inherent challenging. The research challenge lies in identifying effective patterns in historical series and applying them to future forecasting. Advanced models based on point-wise connected MLP and Transformer architectures have strong fitting power, but their secondary computational complexity limits practicality. Additionally, those structures inherently disrupt the temporal order, reducing the information utilization and making the forecasting process uninterpretable. To solve these problems, this paper proposes a forecasting model, MPR-Net. It first adaptively decomposes multi-scale historical series patterns using convolution operation, then constructs a pattern extension forecasting method based on the prior knowledge of pattern reproduction, and finally reconstructs future patterns into future series using deconvolution operation. By leveraging the temporal dependencies present in the time series, MPR-Net not only achieves linear time complexity, but also makes the forecasting process interpretable. By carrying out sufficient experiments on more than ten real data sets of both short and long term forecasting tasks, MPR-Net achieves the state of the art forecasting performance, as well as good generalization and robustness performance.

In this paper, we establish pointwise boundary ${{C}^{1,\alpha}}$ estimates for viscosity solutions of some degenerate fully nonlinear elliptic equations on ${C^{1,\alpha}}$ domains. Instead of straightening out the boundary, we utilize the perturbation and compactness techniques., Comment: 10 pages

We mainly show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $u(x)$ is a real-valued function, $N\geq 3$, $0<\lambda