Related papers: Nonlinear normal modes and spectral submanifolds: …
Recently it was demonstrated that the concept of a spectral singularity (SS) can be generalized to waves propagating in nonlinear media, like matter waves or electromagnetic waves in Kerr media. The corresponding solutions represent…
The identification of slow invariant manifolds (SIMs) is an essential part in model-order reduction for reactive systems. The mathematical definition of the SIM by Fenichel can be considered unsatisfactory, because it is only applicable to…
The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural…
Recurrent neural networks have gained widespread use in modeling sequential data. Learning long-term dependencies using these models remains difficult though, due to exploding or vanishing gradients. In this paper, we draw connections…
Solving nonlinear systems is an important problem. Numerical continuation methods efficiently solve certain nonlinear systems. The Asymptotic Numerical Method (ANM) is a powerful continuation method that usually converges faster than…
In this paper we present our detailed investigations on the nature of the phase transition in the scalar noise model (SNM) of collective motion. Our results confirm the original findings of Vicsek et al. [Phys. Rev. Lett. 75 (1995) 1226]…
We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one dimensional quantum walks are universal, as are CMV…
Nonlinear systems exist widely in nature, however, how to construct systems with accurate expected non-linearity artificially is still a problem, which greatly limits their experimental study and engineering application. In this paper, we…
We present the theory of non-stationary normal forms for uniformly contracting smooth extensions with sufficiently narrow Mather spectrum. We give coherent proofs of existence, (non)uniqueness, and a description of the centralizer results.…
Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating…
We introduce a manifold analysis technique for neural network representations. Normalized Space Alignment (NSA) compares pairwise distances between two point clouds derived from the same source and having the same size, while potentially…
We explore the O(N)-invariant Non-Linear Sigma Model (NLSM) in a different perturbative regime from the usual relativistic-free-field one, by using non-canonical basic commutation relations adapted to the underlying O(N) symmetry of the…
Resonances, also known as quasi normal modes (QNM) in the non-Hermitian case, play an ubiquitous role in all domains of physics ruled by wave phenomena, notably in continuum mechanics, acoustics, electrodynamics, and quantum theory. In this…
We obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy, possibly with different growth rates in the uniform and nonuniform parts. We…
We demonstrate the application of the efficient semi-inverse asymptotic method to resonant interaction of the nonlinear normal modes belonging to different branches of the CNT vibration spectrum. Under condition of the 1:1 resonance of the…
We unveil instances where nonautonomous linear systems manifest distinct nonuniform $\mu$-dichotomy spectra despite admitting nonuniform $(\mu, \varepsilon)$-kinematic similarity. Exploring the theoretical foundations of this lack of…
A Nehari manifold optimization method (NMOM) is introduced for finding 1-saddles, i.e., saddle points with the Morse index equal to one, of a generic nonlinear functional in Hilbert spaces. Actually, it is based on the variational…
Quasi-normal modes (QNMs) are ubiquitous throughout photonics and are utilized in a wide variety of applications, but determining these modes remains a formidable task in general. Here we show that by exploiting the structure of Maxwell's…
Normal mode analysis (NMA) provides a mathematical framework for exploring the intrinsic global dynamics of molecules through the definition of an energy function, where normal modes correspond to the eigenvectors of the Hessian matrix…
An analysis of discrete systems is important for understanding of various physical processes, such as excitations in crystal lattices and molecular chains, the light propagation in waveguide arrays, and the dynamics of Bose-condensate…