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We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in $${\mathbb{R}^{3}}$$ , we show that they can be transformed with a$C$^{$m$}-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

For any dynamical system T : X → X of a compact metric space X with g-almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodiclike recurrences and find that they all carry full topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multifractal analyses for all continuous observable functions. These results are suitable for all β-shifts (β > 1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc. Roughly speaking, we combine many different “eyes” (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

To derive the hidden dynamics from observed data is one of the fundamental but also challenging problems in many different fields. In this study, we propose a new type of interpretable network called the ordinary differential equation network (ODENet), in which the numerical integration of explicit ordinary differential equations (ODEs) are embedded into the machine learning scheme to build a general framework for revealing the hidden dynamics buried in massive time-series data efficiently and reliably. ODENet takes full advantage of both machine learning algorithms and ODE modeling. On one hand, the embedding of ODEs makes the framework more interpretable benefiting from the mature theories of ODEs. On the other hand, the schemes of machine learning enable data handling, paralleling, and optimization to be easily and efficiently implemented. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet exhibits its remarkable capability in handling time-series data even in the presence of large noise. We further apply the ODENet to real actin aggregation data, which shows an impressive performance as well. These results demonstrate the superiority of ODENet in dealing with noisy data, data with either non-equal spacing or large sampling time steps over other traditional machine learning algorithms.