Nonlinearity

Various definitions of an attractor for a nonlinear dynamical system have been proposed. These use various assumptions on the set of initial conditions that should converge (the basin), and various notions of convergence. A weak assumption on the basin is the measure attractor of Milnor, which requires that the basin has positive measure. A weak assumption of the notion of convergence is the statistical attractor due to Ilyashenko, which requires that limiting to the attractor occurs on a set of future times of full density. We point out that many examples of statistical attractors actually satisfy a stronger definition which we call a bounded-return-time attractor, and we investigate such attractors. We also give an improved definition for the notion of pullback measure attraction. This was originally developed to understand attractors in nonautonomous systems, but we note here that it is helpful for understanding convergence towards statistical attractors in the autonomous setting. We investigate implications between all these different notions of attractors. We also investigate which of these notions are fulfilled by a hyperbolic fixed point with a homoclinic loop.

A notable feature of the elephant trunk is the pronounced wrinkling that enables its great flexibility. Here, we devise a general mathematical model that accounts for the characteristic skin wrinkles formed during morphogenesis in the elephant trunk. Using physically realistic parameters and operating within the theoretical framework of nonlinear morphoelasticity, we elucidate analytically and numerically the effect of skin thickness, relative stiffness, and differential growth on the physiological pattern of transverse wrinkles distributed along the trunk. We conclude that since the skin and muscle components have similar material properties, geometric parameters, such as curvature, play an important role. In particular, our model predicts that, in the proximal region close to the skull, where the curvature is lower, fewer wrinkles form and will form sooner than in the distal narrower region, where more wrinkles develop. Similarly, less wrinkling is found on the ventral side, which is flatter, compared to the dorsal side. In summary, the mechanical compatibility between the skin and the muscle enables them to grow seamlessly, while the wrinkled skin acts as a protective barrier that is both thicker and more flexible than the unwrinkled skin.

We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation.

For countably infinite IFSs on $\mathbb{R}^2$ consisting of affine contractions with diagonal linear parts, we give conditions under which the affinity dimension is an upper bound for the Hausdorff dimension and a lower bound for the lower box-counting dimension. Moreover, we identify a family of countably infinite IFSs for which the Hausdorff and the affinity dimension are equal, and which have full dimension spectrum. The corresponding self-affine sets are related to restricted digit sets for signed Lüroth expansions.

We establish the optimal convergence rate of the hypersonic similarity for two-dimensional steady potential flows with large data past a straight wedge in the $BV\cap L^1$ framework, provided that the total variation of the large data multiplied by $\gamma-1+\frac{a_{\infty}^2}{M_\infty^2}$ is uniformly bounded with respect to the adiabatic exponent γ > 1, the Mach number $M_\infty$ of the incoming steady flow, and the hypersonic similarity parameter $a_\infty$. Our main approach in this paper is first to establish the well-posedness and the Lipschitz continuous map $\mathcal{P}$ that has the properties similar to the Standard Riemann Semigroup of the initial-boundary value problem for the isothermal hypersonic small disturbance equations with large data, and then to compare the Riemann solutions between two systems with boundary locally case by case. Based on them, we derive the global L¹–estimate between the two solutions by employing the Lipschitz continuous map $\mathcal{P}$ and the local L¹–estimates. We further construct an example to show that the convergence rate is optimal.

We consider generic families X_θ of smooth dynamical systems depending on parameters $\theta\in P$ where P is a 2-dimensional simply connected domain and assume that each X_θ only has a finite number of restpoints and periodic orbits. We prove that if over the boundary of P there is a S or Z shaped bifurcation graph containing two opposing fold bifurcation points while over the rest of the boundary there are no other bifurcation points, then, if there is no fold–Hopf bifurcation in P, there is a set of bifurcation curves in P that contain an odd number of cusps. In particular, there is at least one codimension 2 bifurcation point in the interior of P.

In this study, we investigate the behavior of three-dimensional parabolic–parabolic Patlak–Keller–Segel systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.

Treschev made the remarkable discovery that there exists formal power series describing a billiard with locally linearizable dynamics. We show that if the frequency for the linear dynamics is Diophantine, the Treschev example is $(1+ \alpha)$-Gevrey for some α > 0. Our proof is based on an iterative scheme that further clarifies the structure and symmetries underlying the original Treschev construction. Hopefully, our result sheds a light on the more important question of whether this example is convergent.

Differential equation models are crucial to scientific processes across many disciplines, and the values of model parameters are important for analyzing the behaviour of solutions. Identifying these values is known as a parameter estimation, a type of inverse problem, which has applications in areas that include industry, finance and biomedicine. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. Checking the global identifiability of model parameters is a useful tool when exploring the well-posedness of a given model. This problem has been intensively studied for ordinary differential equation models, where theory, several efficient algorithms and software packages have been developed. A comprehensive theory for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

This paper considers a susceptible-infected-susceptible epidemic reaction-diffusion model with no-flux boundary conditions and varying total population. The interaction of the susceptible and infected people is described by the nonlinear transmission mechanism of the form $S^qI^p$, where $0 \lt p\unicode{x2A7D} 1$ and q > 0. In Peng et al (SIAM J. Math. Anal. (arXiv:2411.00582)), we have studied a model with a constant total population. In the current paper, we extend our analysis to a model with a varying total population, incorporating birth and death rates. We investigate the asymptotic profiles of the endemic equilibrium when the dispersal rates of susceptible and/or infected individuals are small. Our work is motivated by disease control strategies that limit population movement. To illustrate the main findings, we conduct numerical simulations and provide a discussion of the theoretical results from the view of disease control. We will also compare the results for the models with constant or varying total population.

We present a framework for constructing a first-order hyperbolic system whose solution approximates that of a desired higher-order evolution equation. Constructions of this kind have received increasing interest in recent years, and are potentially useful as analytical or computational tools for understanding the corresponding higher-order equation. We perform a systematic analysis of a family of linear model equations and show that for each member of this family there is a stable hyperbolic approximation whose solution converges to that of the model equation in a certain limit. We then show through several examples that this approach can be applied successfully to a very wide range of nonlinear partial differential equations of practical interest.

We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles is asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies.

This paper investigates uniqueness results for perturbed periodic Schrödinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\Delta + V + v$, where Δ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-\Delta u + V u + v u = \lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of λ.

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If κ denotes the strength of the diffusion, then we show that with probability at least $1 - o(\kappa^N)$ enhanced dissipation occurs on time scales of order $\vert \ln \kappa \rvert$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert'94.

A quasilinear elliptic equation with irregular double obstacles involving almost variable exponent growth is widely extended to a degenerate/singular one by dealing with matrix weights connected to the gradient component in the nonlinearity. An optimal Calderón–Zygmund type estimate for the gradient of a weak solution is obtained in the setting of a weighted Musielak–Orlicz space by finding a minimal regularity assumption on the matrix weight which turns out a smallness in $\log$-BMO. To this end, we compare a solution to the underlying solution with constant-matrix equations. In the process, we make a series of comparison estimates starting from a degenerate/singular double obstacle problem to the associated obstacle problem and then to the limiting problem having a desired Lipschitz regularity. Our regularity result provides a newly achieved theory in the realm of highly nonlinear elliptic equations with kinds of nonstandard growth conditions as we deal with such degenerate/singular nonlinearities in terms of $\log$-BMO instead of BMO.

Pattern forming systems allow for a wealth of states, where wavelengths and orientation of patterns varies and defects disrupt patches of monocrystalline regions. Growth of patterns has long been recognized as a strong selection mechanism. We present here recent and new results on the selection of patterns in situations where the pattern-forming region expands in time. The wealth of phenomena is roughly organised in bifurcation diagrams that depict wavenumbers of selected crystalline states as functions of growth rates. We show how a broad set of mathematical and numerical tools can help shed light into the complexity of this selection process.

In this survey we recall basic notions of disintegration of measures and entropy along unstable laminations. We review some roles of unstable entropy in smooth ergodic theory including the so-called invariance principle, Margulis construction of measures of maximal entropy, physical measures and rigidity. We also give some new examples and pose some open problems.

This article is firstly a historic review of the theory of Riemann–Hilbert problems with particular emphasis placed on their original appearance in the context of Hilbert's 21st problem and Plemelj's work associated with it. The secondary purpose of this note is to invite a new generation of mathematicians to the fascinating world of Riemann–Hilbert techniques and their modern appearances in nonlinear mathematical physics. We set out to achieve this goal with six examples, including a new proof of the integro-differential Painlevé-II formula of Amir et al (2011 Commun. Pure Appl. Math.64 466–537) that enters in the description of the Kardar–Parisi–Zhang crossover distribution. Parts of this text are based on the author's Szegő prize lecture at the 15th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA) in Hagenberg, Austria.

For a wide range of values of the intensity of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in the past our planet flipped between these two states. The main physical mechanism responsible for such an instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attraction. In the weak-noise limit, large deviation laws define the invariant measure, the statistics of escape times, and typical escape paths called instantons. By constructing the instantons empirically, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first-order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we put forward a new method for constructing Melancholia states from direct numerical simulations, which provides a possible alternative with respect to the edge-tracking algorithm.

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the 'turbulent' and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe–Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d  =  1; among them are the KdV, BBM, Camassa–Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

We present a framework for constructing a first-order hyperbolic system whose solution approximates that of a desired higher-order evolution equation. Constructions of this kind have received increasing interest in recent years, and are potentially useful as analytical or computational tools for understanding the corresponding higher-order equation. We perform a systematic analysis of a family of linear model equations and show that for each member of this family there is a stable hyperbolic approximation whose solution converges to that of the model equation in a certain limit. We then show through several examples that this approach can be applied successfully to a very wide range of nonlinear partial differential equations of practical interest.

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If κ denotes the strength of the diffusion, then we show that with probability at least $1 - o(\kappa^N)$ enhanced dissipation occurs on time scales of order $\vert \ln \kappa \rvert$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert'94.

It is a well known fact that the geometry of a superconducting sample influences the distribution of the surface superconductivity for strong applied magnetic fields. For instance, the presence of corners induces geometric terms described through effective models in sector-like regions. We study the connection between two effective models for the offset of superconductivity and for surface superconductivity introduced in Bonnaillie-Noël and Fournais (2007 Rev. Math. Phys.19 607–37) and Correggi and Giacomelli (2021 Calc. Var. PDE60 236), respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength, and, as a byproduct, we deduce the existence of a minimizer at the threshold for both effective problems. Furthermore, as a consequence, we disprove a conjecture stated in Correggi and Giacomelli (2021 Calc. Var. PDE60 236) concerning the dependence of the corner energy on the angle close to the threshold.

In this article, we address the decay of correlations for dynamical systems that admit an induced weak Gibbs Markov (GM) map (not necessarily full branch). Our approach generalizes Young's coupling arguments to estimate the decay of correlations for the tower map of the induced weak GM map in terms of the tail of the return time function. For that we initially discuss how to ensure the mixing property of the tower map. Additionally, we yield results concerning the central limit theorem and large deviations.

We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation.

We develop a system of non-linear stochastic evolution equations that describes the continuous measurements of quantum systems with mixed initial state. We address quantum systems with unbounded Hamiltonians and unbounded interaction operators. Using arguments of the theory of quantum measurements we derive a system of stochastic interacting wave functions (SIWFs for short) that models the continuous monitoring of quantum systems. We prove the existence and uniqueness of the solution to this system under conditions general enough for the applications. We obtain that the mixed state generated by the SIWF at any time does not depend on the initial state, and satisfies the diffusive stochastic quantum master equation, which is also known as Belavkin equation. We present two physical examples. In one, the SIWF becomes a system of non-linear stochastic partial differential equations. In the other, we deal with a model of a circuit quantum electrodynamics.

Various definitions of an attractor for a nonlinear dynamical system have been proposed. These use various assumptions on the set of initial conditions that should converge (the basin), and various notions of convergence. A weak assumption on the basin is the measure attractor of Milnor, which requires that the basin has positive measure. A weak assumption of the notion of convergence is the statistical attractor due to Ilyashenko, which requires that limiting to the attractor occurs on a set of future times of full density. We point out that many examples of statistical attractors actually satisfy a stronger definition which we call a bounded-return-time attractor, and we investigate such attractors. We also give an improved definition for the notion of pullback measure attraction. This was originally developed to understand attractors in nonautonomous systems, but we note here that it is helpful for understanding convergence towards statistical attractors in the autonomous setting. We investigate implications between all these different notions of attractors. We also investigate which of these notions are fulfilled by a hyperbolic fixed point with a homoclinic loop.

For countably infinite IFSs on $\mathbb{R}^2$ consisting of affine contractions with diagonal linear parts, we give conditions under which the affinity dimension is an upper bound for the Hausdorff dimension and a lower bound for the lower box-counting dimension. Moreover, we identify a family of countably infinite IFSs for which the Hausdorff and the affinity dimension are equal, and which have full dimension spectrum. The corresponding self-affine sets are related to restricted digit sets for signed Lüroth expansions.

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend the notion of metric mean dimension to discontinuous maps in terms of suitable subshifts. As an application, we show that the metric mean dimension of the Gauss map and that of induced maps of the Manneville–Pomeau family is equal to the box dimension of the corresponding set of discontinuity points, which also coincides with a critical parameter of the pressure operator associated to the geometric potential.

In this article, we present a comprehensive framework for constructing smooth, localized solutions in systems of semi-linear partial differential equations, with a particular emphasis to the Gray–Scott model. Specifically, we construct a natural Hilbert space $\mathcal{H}$ for the study of systems of autonomous semi-linear PDEs, on which products and differential operators are well-defined. Then, given an approximate solution u₀, we derive a Newton–Kantorovich approach based on the construction of an approximate inverse of the linearization around u₀. In particular, we derive a condition under which we prove the existence of a unique solution in a neighborhood of u₀. Such a condition can be verified thanks to the explicit computation of different upper bounds, for which analytical details are presented. Furthermore, we provide an extra condition under which localized patterns are proven to be the limit of an unbounded branch of (spatially) periodic solutions as the period tends to infinity. We then demonstrate our approach by proving (constructively) the existence of four different localized patterns in the 2D Gray–Scott model. In addition, these solutions are proven to satisfy the D₄-symmetry. That is, the symmetry of the square. The algorithmic details to perform the computer-assisted proofs are available on GitHub (2024 LocalizedPatternsGS.jl https://github.com/dominicblanco/LocalizedPatternsGS.jl).