Journal of Nonlinear Science最新文献

Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field (E_0) plays a key role in controlling the shapes and types of waves on the interface.

By introducing bilinear operators of trigonometric type, we propose several novel integrable variants of the famous Toda lattice, two of which can be regarded as integrable discretizations of the Kadomtsev–Petviashvili equation—a universal model describing weakly nonlinear waves in media with dispersion of velocity. We also demonstrate that two one-dimensional reductions of these variants can approximate the nonlinear Schrödinger equation and a generalized nonlinear Schrödinger equation well. It turns out that these equations admit meaningful solutions including solitons, breathers, lumps and rogue waves, which are expressed in terms of explicit and closed forms. In particular, it seems to be the first time that rogue wave solutions have been obtained for Toda-type equations. Furthermore, g-periodic wave solutions are also produced in terms of Riemann theta function. An approximation solution of the three-periodic wave is successfully carried out by using a deep neural network. The introduction of trigonometric-type bilinear operators is also efficient in generating new variants together with rich properties for some other integrable equations.

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.

Characterizing existence or not of periodic orbit is a classical problem, and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system (dot{x}=y,~dot{y}=-g(x)-f(x,y)y) are obtained and by examples shows that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by Andreev (Am Math Soc Transl 8:183–207, 1958) on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters.

We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. We subsequently address the question of discretize-then-optimize versus optimize-then-discrete for both semi-discretization and time integration, by characterizing the commutativity of discretize-then-optimize methods versus optimize-then-discretize methods uniquely in terms of an adjoint-variational quadratic conservation law. For Galerkin semi-discretizations and one-step time integration methods in particular, we explicitly construct these commuting methods by using structure-preserving discretization techniques.

In this paper, we consider the stochastic nutrient–phytoplankton–zooplankton model with nutrient cycle. In order to take stochastic fluctuations into account, we add the stochastic increments to the variations of biomass of nutrition, phytoplankton and zooplankton during time interval (Delta t), thus we obtain the corresponding stochastic model. Subsequently, we explore the existence, uniqueness and stochastically ultimate boundness of global positive solution. By constructing suitable Lyapunov function, we also obtain V-geometric ergodicity of this model. In addition, the sufficient conditions of exponential extinction and persistence in the mean of plankton are established. At last, we present some numerical simulations to validate theoretical results and analyze the impacts of some important parameters.

Based on empirical evidences and previous studies, we introduce and mathematically study a perception-driven model for the dynamics of buyer populations in markets of perishable goods. Buyer behaviours are driven partly by some loyalty to the sellers that they previously purchased at, and partly by the sensitivity to the intrinsic attractiveness of each seller in the market. On the other hand, the sellers update they attractiveness in time according to the difference between the volume of their clientele and the mean volume of buyers in the market, optimising either their profit when this difference is favourable or their competitiveness otherwise. While this negative feedback mechanism is a source of instability that promotes oscillatory behaviour, our analysis identifies the critical features of the dynamics that are responsible for the asymptotic stability of the stationary states, both in their immediate neighbourhood and globally in phase space. Altogether, this study provides mathematical insights into the consequences of introducing feedback into buyer–seller interactions in such markets, with emphasis on identifying conditions for long-term constancy of clientele volumes.

We extend the method of controlled Lagrangians to nonholonomic Euler–Poincaré equations with advected parameters, specifically to those mechanical systems on Lie groups whose symmetry is broken not only by a potential force but also by nonholonomic constraints. We introduce advected-parameter-dependent quasivelocities in order to systematically eliminate the Lagrange multipliers in the nonholonomic Euler–Poincaré equations. The quasivelocities facilitate the method of controlled Lagrangians for these systems, and lead to matching conditions that are similar to those by Bloch, Leonard, and Marsden for the standard holonomic Euler–Poincaré equation. Our motivating example is what we call the pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. We show that the upright spinning of the pendulum skate is stable under certain conditions, whereas the upright sliding equilibrium is always unstable. Using the matching condition, we derive a control law to stabilize the sliding equilibrium.

In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys ((N=2)), a continuum approximation for the (J_1-J_3)-model for discrete spin systems ((N=4)), and models for crystalline surfaces with N different facets (general N). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.