Journal of Nonlinear Science最新文献

This work introduces a parametric model order reduction (PMOR) approach that enhances an existing widely used technique based on proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) for parametrized nonlinear dynamical systems by employing machine learning procedures performed on a Grassmann manifold. In particular, distances between parameters are first computed based on a metric defined on the Grassmann manifold of solution spaces. Then, the distance information is utilized in the K-medoids clustering algorithm to partition parameters into classes with corresponding local solution spaces, which are further used to form a dictionary of local bases. The artificial neural network (ANN) is next used to build a classifier that can automatically identify the most suitable local basis from the dictionary for a given input parameter to construct a parametrized reduced-order model by the POD–DEIM approach. This work numerically demonstrates the significance of using distance on the Grassmann manifold of the solution spaces, instead of directly using the Euclidean distance on the parameter space. To validate the proposed method, numerical studies are performed on a parametrized 1D Burger’s equation and a viscous fingering in a horizontal flow through a 2D porous media domain. The proposed method is shown to have advantage in terms of accuracy when compared to the traditional global basis approach, as well as the local reduced-order basis approach based on the Euclidean metric.

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master–slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.

This paper is dedicated to investigating the chaos of a initial-boundary value (IBV) problem of a multi-dimensional weakly hyperbolic equation subject to two general nonlinear boundary conditions (NBCs). The existence and uniqueness of solution for the IBV problem are established. By employing the snap-back repeller and heteroclinic cycle theories, it has been proven that the IBV problem with a linear and a general NBCs exhibits chaos in the sense of both Devaney and Li–Yorke. Furthermore, these chaotic results are extended to the IBV problem with two general NBCs. Two stability criteria of the IBV problem are established, respectively, for the corresponding two cases of boundary conditions. Finally, numerical simulations are presented to illustrate the theoretical results.

We study the stochastic Camassa–Holm equation with pure jump noise. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. Moreover, we derive the equations for the modulation parameters and show that the remainder converges to the solution of a stochastic linear equation as amplitude of the jump noise tends to zero.

A main goal in the field of statistical shape analysis is to define computable and informative metrics on spaces of immersed manifolds, such as the space of curves in a Euclidean space. The approach taken in the elastic shape analysis framework is to define such a metric by starting with a reparametrization-invariant Riemannian metric on the space of parametrized shapes and inducing a metric on the quotient by the group of diffeomorphisms. This quotient metric is computed, in practice, by finding a registration of two shapes over the diffeomorphism group. For spaces of Euclidean curves, the initial Riemannian metric is frequently chosen from a two-parameter family of Sobolev metrics, called elastic metrics. Elastic metrics are especially convenient because, for several parameter choices, they are known to be locally isometric to Riemannian metrics for which one is able to solve the geodesic boundary problem explicitly—well-known examples of these local isometries include the complex square root transform of Younes, Michor, Mumford and Shah and square root velocity (SRV) transform of Srivastava, Klassen, Joshi and Jermyn. In this paper, we show that the SRV transform extends to elastic metrics for all choices of parameters, for curves in any dimension, thereby fully generalizing the work of many authors over the past two decades. We give a unified treatment of the elastic metrics: we extend results of Trouvé and Younes, Bruveris as well as Lahiri, Robinson and Klassen on the existence of solutions to the registration problem, we develop algorithms for computing distances and geodesics, and we apply these algorithms to metric learning problems, where we learn optimal elastic metric parameters for statistical shape analysis tasks.

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation (u_{t x}=left( 1+nu partial _x^2right) sin u). The key points of the construction are based on the bilinear discrete KP hierarchy and appropriate definition of discrete reciprocal transformations. We derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter limit (brightarrow 0). In particular, one fully discrete gsG equation is reduced to a semi-discrete gsG equation in the case of (nu =-1) (Feng et al. in Numer Algorithms 94:351–370, 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are presented. Dynamics of one- and two-soliton solutions for the discrete gsG equations are analyzed. By introducing a parameter c, we demonstrate that the gsG equation can reduce to the sine-Gordon equation and the short pulse at the levels of continuous, semi-discrete and fully discrete cases. The limiting forms of the N-soliton solutions to the gsG equation in each level also correspond to those of the sine-Gordon equation and the short pulse equation.

We prove that a version of Smagorinsky large eddy model for a 2D fluid in vorticity form is the scaling limit of suitable stochastic models for large scales, where the influence of small turbulent eddies is modeled by a transport-type noise.

We study pairs of symmetrically coupled, identical Lengyel-Epstein oscillators, where the coupling can be through both the fast and slow variables. We find a plethora of strong symmetry breaking rhythms, in which the two oscillators exhibit qualitatively different oscillations, and their amplitudes differ by as much as an order of magnitude. Analysis of the folded singularities in the coupled system shows that a key folded node, located off the symmetry axis, is the primary mechanism responsible for the strong symmetry breaking. Passage through the neighborhood of this folded node can result in splitting between the amplitudes of the oscillators, in which one is constrained to remain of small amplitude, while the other makes a large-amplitude oscillation or a mixed-mode oscillation. The analysis also reveals an organizing center in parameter space, where the system undergoes an asymmetric canard explosion, in which one oscillator exhibits a sequence of limit cycle canards, over an interval of parameter values centered at the explosion point, while the other oscillator executes small amplitude oscillations. Other folded singularities can also impact properties of the strong symmetry breaking rhythms. We contrast these strong symmetry breaking rhythms with asymmetric rhythms that are close to symmetric states, such as in-phase or anti-phase oscillations. In addition to the symmetry breaking rhythms, we also find an explosion of anti-phase limit cycle canards, which mediates the transition from small-amplitude, anti-phase oscillations to large-amplitude, anti-phase oscillations.

Multiple-timescale systems often display intricate dynamics, yet of great mathematical interest and well suited to model real-world phenomena such as bursting oscillations. In the present work, we construct a piecewise-linear version of the Morris–Lecar neuron model, denoted PWL-ML, and we thoroughly analyse its bifurcation structure with respect to three main parameters. Then, focusing on the homoclinic connection present in our PWL-ML, we study the slow passage through this connection when augmenting the original system with a slow dynamics for one of the parameters, thereby establishing a simplified framework for this slow-passage phenomenon. Our results show that our model exhibits equivalent behaviours to its smooth counterpart. In particular, we identify canard solutions that are part of spike-adding transitions. Focusing on the one-spike and on the two-spike scenarios, we prove their existence in a more straightforward manner than in the smooth context. In doing so, we present several techniques that are specific to the piecewise-linear framework and with the potential to offer new tools for proving the existence of dynamical objects in a wider context.

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller–Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction–diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity (d_2=epsilon ^2ll 1) of the chemoattractant concentration field. In the limit (d_2ll 1), steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state N-spike patterns, we analyze the spectral properties associated with both the “large” ({{mathcal {O}}}(1)) and the “small” o(1) eigenvalues associated with the linearization of the Keller–Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that N-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate (d_1) exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant (tau ) is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an N-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of (d_1) where the N-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an ({{mathcal {O}}}(1)) time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis of the KS model with logistic growth in the singular limit (d_2ll 1) is rather closely related to the analysis of spike patterns for the Gierer–Meinhardt RD system.