Physica D: Nonlinear Phenomena最新文献

We demonstrate the formation of two types of symbiotic nondipolar droplet supersolid in a binary dipolar–nondipolar mixture with an interspecies attraction, where the dipolar (nondipolar) atoms are trapped (untrapped). In the absence of an interspecies attraction, in the first type, a dipolar droplet supersolid exists, whereas in the second type, there are no droplets in the dipolar component. To illustrate, we consider a ¹⁶⁴Dy-⁸⁷Rb mixture, where the untrapped ⁸⁷Rb supersolid sticks to the trapped ¹⁶⁴Dy supersolid due to the interspecies attraction and forms a symbiotic supersolid with overlapping droplets. The first (second) type of symbiotic supersolid emerges for the scattering length $a_1=85a_0$ ($a_1=95a_0$) of ¹⁶⁴Dy atom, while under an appropriate trap a dipolar droplet supersolid exists (does not exist) for no interspecies interaction, where $a_0$ the Bohr radius. This study is based on the numerical solution of an improved binary mean-field model, where we introduce an intraspecies Lee–Huang–Yang interaction in the dipolar component, which stops a dipolar collapse and forms a dipolar supersolid. To observe this symbiotic droplet supersolid, one should prepare the corresponding fully trapped dipolar–nondipolar supersolid and then remove the trap on the nondipolar atoms.

An Excitable Network Attractor (ENA) is a forward-invariant set in phase space that can be used to explain input-driven behaviour of Recurrent Neural Networks (RNNs) trained on tasks involving switching between a discrete set of states. An ENA is composed of two or more attractors and excitable connections that allow transitions from one attractor to another under some input perturbation. The smallest such perturbation that makes a connection between two attractors is called the excitability threshold associated with that connection. The excitability threshold provides a measure of sensitivity of the connection to input perturbations. Errors in performance of such trained RNNs can be related to errors in transitions around the associated ENA. Previous work has demonstrated that ENAs of arbitrary sensitivity and structure can be realised in a RNN by suitable choice of connection weights and nonlinear activation function. In this paper we show that ENAs of arbitrary sensitivity and structure can be realised even using a suitable fixed nonlinear activation function, i.e. by suitable choice of weights only. We show that there is a choice of weights such that the probability of erroneous transitions is very small.

In this work, we define a class of models to understand the impact of population size on opinion formation dynamics, a phenomenon usually related to group conformity. To this end, we introduce a new kinetic model in which the interaction frequency is weighted by the kinetic density. In the quasi-invariant regime, this model reduces to a Kaniadakis–Quarati-type equation with nonlinear drift, originally introduced for the dynamics of bosons in a spatially homogeneous setting. From the obtained PDE for the evolution of the opinion density, we determine the regime of parameters for which a critical mass exists and triggers blow-up of the solution. Therefore, the model is capable of describing strong conformity phenomena in cases where the total density of individuals holding a given opinion exceeds a fixed critical size. In the final part, several numerical experiments demonstrate the features of the introduced class of models and the related consensus effects.

Random collisions of particles occur in various biophysical and physical systems. Inspired by the binding of receptor and ligand on the cell membrane, we devised a method based on stochastic dynamical modeling to quantify the likelihood of two random particles colliding on a circle. We consider the dynamics of a receptor binding to a ligand on the cell membrane, where the receptor and ligand perform different motions and are thus modeled by stochastic differential equations with non-Gaussian noise. We use neural networks based on the Onsager–Machlup function to compute the probability $P_1$ of an unbounded receptor diffusing to the cell membrane. Meanwhile, we compute the probability $P_2$ of the extracellular ligand arriving at the cell membrane by solving the associated nonlocal Fokker–Planck equation. We can then calculate the most probable binding probability by combining $P_1$ and $P_2$. In this way, we conclude with some indication of how the receptors could distribute on the membrane, as well as where the ligand will most probably encounter the receptor, contributing to a better understanding of the cell’s response to external stimuli and communication with other cells.

A minimal model for reservoir computing is studied. We demonstrate that a reservoir computer exists that emulates given coupled maps by constructing a modularised network. We describe a possible mechanism for collapses of the emulation in the reservoir computing by introducing a measure of finite scale deviation. Such transitory behaviour is caused by either (i) an escape from a finite-time stagnation near an unstable chaotic set, or (ii) a critical transition driven by the effective parameter drift. Our approach reveals the essential mechanism for reservoir computing and provides insights into the design of reservoir computer for practical applications.

In this work we consider a new family of algorithms for sequential prediction, Hierarchical Partitioning Forecasters (HPFs). Our goal is to provide appealing theoretical - regret guarantees on a powerful model class - and practical - empirical performance comparable to deep networks - properties at the same time. We built upon three principles: hierarchically partitioning the feature space into sub-spaces, blending forecasters specialized to each sub-space and learning HPFs via local online learning applied to these individual forecasters. Following these principles allows us to obtain regret guarantees, where Constant Partitioning Forecasters (CPFs) serve as competitor. A CPF partitions the feature space into sub-spaces and predicts with a fixed forecaster per sub-space. Fixing a hierarchical partition $H$ and considering any CPF with a partition that can be constructed using elements of $H$ we provide two guarantees: first, a generic one that unveils how local online learning determines regret of learning the entire HPF online; second, a concrete instance that considers HPF with linear forecasters (LHPF) and exp-concave losses where we obtain $O(klogT)$ regret for sequences of length $T$ where $k$ is a measure of complexity for the competing CPF. Finally, we provide experiments that compare LHPF to various baselines, including state of the art deep learning models, in precipitation nowcasting. Our results indicate that LHPF is competitive in various settings.

We numerically analyse solutions of the spherically symmetric gravitational Vlasov–Poisson system close to compactly supported stable steady states. We observe either partially undamped oscillations or macroscopically damped solutions. We investigate for many steady states to which of these behaviours they correspond. A linear relation between the exponents of polytropic steady states and the qualitative behaviour close to them is identified. Undamped oscillations are also observed around not too concentrated King models and around all shells with a sufficiently large vacuum region. We analyse all solutions both at the non-linear and linearised level and find that the qualitative behaviours are identical at both. To relate the observed phenomena to theoretical results, we further include a comprehensive numerical study of the radial particle periods in the equilibria.

We study the nonexistence of multi-dimensional solitary waves for the Euler–Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler–Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves in the $L^1$ space for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler–Poisson flows. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler–Poisson system for ions and electrons.

Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues associated with finite truncations of Koopman operators (e.g., Extended Dynamic Mode Decomposition), such as spurious eigenvalues. Spectral properties computed by ResDMD include spectra, pseudospectra, spectral measures, Koopman mode decompositions, and dictionary verification. In scenarios where the number of snapshots is fewer than the dictionary size, particularly for exact DMD and kernelized Extended DMD, ResDMD has traditionally been applied by dividing snapshot data into a training set and a quadrature set. We demonstrate how to eliminate the need for two datasets through a novel computational approach of solving a dual least-squares problem. We analyze these new residuals for exact DMD and kernelized Extended DMD, demonstrating ResDMD’s versatility and broad applicability across various dynamical systems, including those modeled by high-dimensional and nonlinear observables. The utility of these new residuals is showcased through three diverse examples: the analysis of a cylinder wake, the study of airfoil cascades, and the compression of transient shockwave experimental data. This approach not only simplifies the application of ResDMD but also extends its potential for deeper insights into the dynamics of complex systems.