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A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space transiently visiting the neighbourhoods of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303, 1948).…
In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge…
Learning generalizable self-supervised graph representations for downstream tasks is challenging. To this end, Contrastive Learning (CL) has emerged as a leading approach. The embeddings of CL are arranged on a hypersphere where similarity…
We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We…
High-fidelity modeling of turbulent flows is one of the major challenges in computational physics, with diverse applications in engineering, earth sciences and astrophysics, among many others. The rising popularity of high-fidelity…
Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical…
It is given notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic…
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian…
This paper presents a rigorous study of advanced functional spaces, with a focus on Sobolev and Besov spaces, to investigate key aspects of fluid dynamics, including the regularity of solutions to the Navier-Stokes equations, hypercomplex…
Linear perturbations of solutions of dynamical systems exhibit different asymptotic growth rates, which are naturally characterized by so-called covariant Lyapunov vectors (CLVs). Due to an increased interest of CLVs in applications,…
We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an…
We derive a hyperbolic system of equations approximating the two-layer dispersive shallow water model for shear flows recently proposed by Gavrilyuk, Liapidevskii \& Chesnokov (J. Fluid Mech., vol. 808, 2016, pp. 441--468). The use of this…
We consider the relationship between Eulerian modal decompositions and Lagrangian coherent structures (LCSs). The model sensitivity framework developed by Kasz\'as and Haller (2020) is used to express data-driven modal representations of…
In this work we perform a detailed study of the scaling properties of Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices: the Fermi-Pasta-Ulam and $\Phi^4$ models. In this case, characteristic (also called…
Vlasov kinetic theory is the dynamics of a bunch of particles flowing according to symplectic Hamiltonian dynamics. More recently, this geometry has been extended to contact Hamiltonian dynamics. In this paper, we introduce geometric…
Motivated by recent success in the dynamical systems approach to transitional flow, we study the efficiency and effectiveness of extracting simple invariant sets (recurrent flows) directly from chaotic/turbulent flows and the potential of…
In this paper, we study transversely holomorphic partially hyperbolic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove in the seven-dimensional case that under the assumption that the subcenter…
The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only…
This paper is concerned with boundary stabilization of two-dimensional hyperbolic systems of partial differential equations. By adapting the Lyapunov function previously proposed by the second author for linearized hyperbolic systems with…
Three-dimensional two-layer incompressible Euler fluids are studied from a Hamiltonian perspective. A natural Hamiltonian structure for the effective 2D model described by the interface-value of the field variables is obtained by means of a…