Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

Agraïments: The first author is supported by NSFC grant #11471228. The third author is supported by NSFC grants #11231001, #11221101.It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear center at the origin of \R^2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \R^2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function

The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension

In this paper, we establish an exponential periodic turnpike property for linear quadratic optimal control problems governed by periodic systems in infinite dimension. We show that the optimal trajectory converges exponentially to a periodic orbit when the time horizon tends to infinity. Similar results are obtained for the optimal control and adjoint state. Our proof is based on the large time behavior of solutions of operator differential Riccati equations with periodic coefficients

Cyclicity of (1,3)-switching FF type equilibria

Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of (1, 2)-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of (1, 3)-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3

Industrial Anomaly Detection with Domain Shift: A Real-world Dataset and Masked Multi-scale Reconstruction

Industrial anomaly detection (IAD) is crucial for automating industrial quality inspection. The diversity of the datasets is the foundation for developing comprehensive IAD algorithms. Existing IAD datasets focus on the diversity of data categories, overlooking the diversity of domains within the same data category. In this paper, to bridge this gap, we propose the Aero-engine Blade Anomaly Detection (AeBAD) dataset, consisting of two sub-datasets: the single-blade dataset and the video anomaly detection dataset of blades. Compared to existing datasets, AeBAD has the following two characteristics: 1.) The target samples are not aligned and at different scales. 2.) There is a domain shift between the distribution of normal samples in the test set and the training set, where the domain shifts are mainly caused by the changes in illumination and view. Based on this dataset, we observe that current state-of-the-art (SOTA) IAD methods exhibit limitations when the domain of normal samples in the test set undergoes a shift. To address this issue, we propose a novel method called masked multi-scale reconstruction (MMR), which enhances the model's capacity to deduce causality among patches in normal samples by a masked reconstruction task. MMR achieves superior performance compared to SOTA methods on the AeBAD dataset. Furthermore, MMR achieves competitive performance with SOTA methods to detect the anomalies of different types on the MVTec AD dataset. Code and dataset are available at https://github.com/zhangzilongc/MMR.Comment: submit to Computers in Industr

Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space

In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded domain to an unbounded one. The proof is based on the locally parabolic-type frequency function method. An observability estimate from measurable sets in time for the same equation is also derived.Comment: 26 page

CA2: Class-Agnostic Adaptive Feature Adaptation for One-class Classification

One-class classification (OCC), i.e., identifying whether an example belongs to the same distribution as the training data, is essential for deploying machine learning models in the real world. Adapting the pre-trained features on the target dataset has proven to be a promising paradigm for improving OCC performance. Existing methods are constrained by assumptions about the number of classes. This contradicts the real scenario where the number of classes is unknown. In this work, we propose a simple class-agnostic adaptive feature adaptation method (CA2). We generalize the center-based method to unknown classes and optimize this objective based on the prior existing in the pre-trained network, i.e., pre-trained features that belong to the same class are adjacent. CA2 is validated to consistently improve OCC performance across a spectrum of training data classes, spanning from 1 to 1024, outperforming current state-of-the-art methods. Code is available at https://github.com/zhangzilongc/CA2.Comment: Submit to AAAI 202

Restricted independence in displacement function for better estimation of cyclicity

Agraïments: The author is partially supported by NSFC grant #11471228 (X. Chen), NSFC grant # 11501083 (Z. Wang), and NSFC grants # 11231001 and # 11221101 (W. Zhang).Since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center-focus equilibrium, in this paper we consider a restricted independence to a parametric curve, which gives a method not only to increase the lower bound for the cyclicity of the center-focus equilibrium but also to be available when those focal values are not independent. We apply the method to a nondegenerate cubic center-focus variety and prove that the cyclicity reaches its an upper bound

The multivariable finite elements based on B-spline wavelet on the interval for 1D structural mechanics

Wavelet finite elements with two kinds of variables for 1D structural mechanics are constructed based on B-spline wavelet on the interval (BSWI) and the generalized variational principle. In contrast to the traditional method, the BSWI element with two kinds of variables (TBSWI) can improve the solution accuracy of the generalized stress apparently, because generalized displacement and stress are interpolated separately. Another superiority of the elements constructed is the interpolating function BSWI, which has very good approximation property, further guarantees solution accuracy. Euler beam, Timoshenko beam and Elastic foundation beam are studied providing several numerical examples to verify the efficiency