Journal of Nonlinear Science最新文献

Predicting the evolution of dynamics from a given trajectory history of an unknown system is an important and challenging problem. This paper presents a model-free method of forecasting unknown chaotic systems through reconstructing vector fields from noisy measured data via an adaptation of optimal control methods. This technique is also applicable to partially observed systems using a Takens delay embedding approach. The algorithms are validated on the Lorenz system and the four-dimensional hyperchaotic Rössler system, and demonstrate successful predictions well beyond the Lyapunov timescale. It is found that for small datasets or datasets with large levels of noise, the prediction accuracy of partially observed systems approaches that of fully observed systems. The presented approach also allows the model-free assessment of local predictability on the attractor by evolving initial condition density through the reconstructed vector fields via estimation of the transfer operator. The method is compared to predictions made by an imperfect model which highlights the utility of model-free approaches when the only available models have significant model error. The capability of this method for reconstruction of continuous and global vector fields may be applied to model validation, forecasting of initial conditions not in the training set, and model-free filtering.

The study of nonlinear Schrödinger-type equations with nonzero boundary conditions introduces challenging problems both for the continuous (partial differential equation) and the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions, which describe solutions decaying on the top of a finite background, we provide a rigorous proof that for the corresponding initial boundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value (A_*) defined by the gain-loss parameters. We argue that this essential property of this infinite lattice cannot be captured by finite lattice approximations of the problem. Commonly, such approximations are provided by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite-dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude (A_*). Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude (A_*) in the long-time evolution of the system.

The occurrence of finite time blow-up phenomenon in the Keller–Segel (KS) model has always been a significant area of interest for mathematicians. Despite extensive research on the blow-up phenomenon in KS models with signal production, Understanding of this phenomenon in models with signal consumption mechanisms has been scarce.This paper marks a preliminary investigation into this unexplored field. In this study, we employ a backward self-similar solution to demonstrate that the finite time blowup indeed occurs within this model. More precisely, in one-dimensional space, finite time blowing up corresponding to the chemotactic collapse phenomenon (the formation of Dirac (delta )-singularity ) happens; in high-dimensional space, the self-similar solution will blow up everywhere. Finally, we also consider the special cases where the diffusion coefficient of bacteria or oxygen is 0. For these cases, chemotactic collapse phenomenon occurs in both one-dimensional and two-dimensional spaces.

We consider piecewise smooth vector fields (Z=(Z_+, Z_-)) defined in ({mathbb {R}}^n) where both vector fields are tangent to the switching manifold (Sigma ) along a submanifold (Msubset Sigma ). We shall see that, under suitable assumptions, Filippov convention gives rise to a unique sliding mode on M, governed by what we call the tangential sliding vector field. Here, we will provide the necessary and sufficient conditions for characterizing such a vector field. Additionally, we prove that the tangential sliding vector field is conjugated to the reduced dynamics of a singular perturbation problem arising from the Sotomayor–Teixeira regularization of Z around M. Finally, we analyze several examples where tangential sliding vector fields can be observed, including a model for intermittent treatment of HIV.

In this paper, we study the existence of spread speeds and periodic traveling waves for a class of space–time periodic and degenerate cooperative systems with nonlocal dispersal and delays. In order to characterize the spread speeds, we first establish the principal eigenvalue theory for linear space–time periodic and degenerate systems with nonlocal dispersal and delays and give some sufficient conditions for the existence of the principal eigenvalue. Then, we prove the existence of a single spreading speed and give the computational formulae of it. Finally, we generalize the monotone iteration scheme combined with the method of sub–super-solutions to prove the existence of the space–time periodic traveling waves.

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities.

Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as ‘fairy circles’. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction–diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction–diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.

We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse interface and is known as the Cahn–Hilliard model. We shall consider a nonlocal version of the Cahn–Hilliard model which replaces the gradient term in the free energy functional into a spatial convolution operator acting on the order parameter and incorporate with it a potential that is assumed to satisfy an arbitrary polynomial growth. The order parameter is influenced by the fluid velocity by means of convection; the temperature affects the interface via a modification of the Landau–Ginzburg free energy. The fluid is governed by the Navier–Stokes equations which is affected by the order parameter and the temperature by virtue of the capillarity between the two fluids. The temperature on the other hand satisfies a parabolic equation that considers latent heat due to phase transition and is influenced by the fluid via convection. The goal of this paper is to prove the global existence of weak solutions and show that, for an appropriate choice of sequence of convolutional kernels, the solutions of the nonlocal system converge to its local version.

In this paper, we are concerned with the minimal regularity of weak solutions implying the law of balance for both energy and helicity in the incompressible Euler equations. In the spirit of recent works due to Berselli (J Differ Equ 368:350–375, 2023) and Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33):1–14, 2024), it is shown that the energy of weak solutions is invariant if the velocity (vin L^{p}(0,T;B^{frac{1}{p}}_{frac{2p}{p-1},c(mathbb {N})} )) with (1<ple 3) and the helicity is conserved if (vin L^{p}(0,T;B^{frac{2}{p}}_{frac{2p}{p-1},c(mathbb {N})} )) with (2<ple 3 ) for both the periodic domain and the whole space, which generalizes the classical work of Cheskidov et al. (Nonlinearity 21:1233–1252, 2008). As an application, we deduce the upper bound of energy dissipation rate of the form (o(mu ^{frac{palpha -1}{palpha -2alpha +1}})) of Leray–Hopf weak solutions in (L^{p}( 0,T;underline{B}^{alpha }_{frac{2p}{p-1},VMO}(mathbb {T}^{d}))) in the Navier–Stokes equations, which extends recent corresponding result obtained by Drivas and Eyink (Nonlinearity 32:4465–4482, 2019).

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.