Table of contents

We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation $\dot{T}$ of a circle map T we consider (i) the response of the expectation of an observation function and (ii) the response of isolated spectral points of the transfer operator of T. In each case, under mild conditions on the set of feasible perturbations $\dot{T}$ we show there is a unique optimal feasible infinitesimal perturbation perspective, we remark that even a movemen$\dot{T}_\textrm{optimal}$, maximising the increase of the expectation of the given observation function or maximising the increase of the spectral gap of the transfer operator associated to the system. We derive expressions for the unique maximiser $\dot{T}_\textrm{optimal}$ in terms of its Fourier coefficients. We also devise a Fourier-based computational scheme and apply it to illustrate our theory.

We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity:

These systems mainly appear in modeling spatio-temporal patterns during bacterial growth. Central to our study is the diffusion term $g(n)h(b)$, which degenerates at n = 0 and b = 0; and the reaction term $f(n,b)$, which is positive, except for n = 0 or b = 0. Specifically, the existence of traveling wave solutions composed by a couple of strictly monotone functions for every wave speed in a closed half-line is proved, and some threshold speed estimates are given. Moreover, the regularity of the traveling wave solutions is discussed in connection with the wave speed.

In this paper, in both topological and measure-theoretical senses, various types of sequence equicontinuity and bounded sequence complexity via Furstenberg family are introduced, and the interrelationships among which and rigidity are systematically studied, which generalize many known results in the literature. In particular, it is shown that a topological dynamical system (X, T) is uniformly rigid iff it is $\mathcal{F}_{r}$-equicontinuous of type 4 iff it has bounded $\mathcal{F}_{r}$-complexity of type 4; And a measure-preserving system $(X,\mathcal{B}_X, \mu,T)$ is rigid iff it is µ-$\mathcal{F}_{r}$-mean equicontinuous of type 4 iff it has bounded µ-$\mathcal{F}_{r}$-mean complexity of type 4, where $\mathcal{F}_{r}$ is the Furstenberg family consisting of all sequences like $q\mathbb{N}$ ($q\in\mathbb{N}$). Moreover, several examples are also constructed to distinguish the new notions or illustrate the scopes of some conclusions, while concurrently addressing a few questions posed in earlier articles.

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 study the full asymptotic expansion of the monodromy data (i.e. Stokes multipliers) for the first Painlevé transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex WKB method, also known as the method of uniform asymptotics, to approximate the second-order ODEs derived from PI's Lax pair with higher-order accuracy. As an application, we provide a rigorous proof of the full asymptotic expansion of the nonlinear eigenvalues proposed numerically by Bender, Komijani, and Wang. Additionally, we present the full asymptotic expansion for the pole parameters $(p_{n}, H_{n})$ corresponding to the nth pole of the real tritronquée solution of the PI equation as $n \to +\infty$.

Derived sequence is an important theme in combinatorics on words. Let $\mathcal{S}_{n,j}$ be the infinite one-sided sequence over the alphabet $\{a_d,~1\unicode{x2A7D} d\unicode{x2A7D} n\}$ generated by the substitution $\sigma_{n,j}(a_d) = a^{\,j}_1a_{d+1}$ for $1\unicode{x2A7D} d \lt n$ and $\sigma_{n,j}(a_n) = a_1$, which are natural generalizations of the Fibonacci sequence. In this paper, we determine the derived sequences of the (n, j)-bonacci sequences $\mathcal{S}_{n,j}$ for $n\unicode{x2A7E} 2$ and $j\unicode{x2A7E} 1$.

We consider products of matrices of the form $A_n = O_n+\epsilon N_n$ where $(O_n)_{n\unicode{x2A7E} 1}$ is a sequence of d × d orthogonal matrices, N_n has independent standard normal entries and where the $(N_n)_{n\unicode{x2A7E} 1}$ are mutually independent. We study the Lyapunov exponents of the cocycle as a function of ε, giving an exact expression for the jth Lyapunov exponent in terms of the Gram–Schmidt orthogonalization of $I+\epsilon N$. Further, we study the asymptotics of these exponents, showing that $\lambda_j = (d-2j)\epsilon^2/2+O(\epsilon^4|\log\epsilon|^4)$.

Let $[0;a_1(z),a_2(z),\dots]$ be the Hurwitz continued fraction expansion of a complex irrational number $z\in \mathfrak F: = \{x+iy: x,y\in[-1/2,1/2)\}$, where $a_n(z)$ are Gaussian integers and $|a_n(z)|\unicode{x2A7E} \sqrt 2$. This paper aims to study the growth rate of the product of consecutive partial quotients. Specifically, for any integer $m\unicode{x2A7E} 1$, we provide a criterion for determining the Lebesgue measure of the set

$\left\{z\in\mathfrak F:|a_{n+1}\left(z\right)\cdots a_{n+m}\left(z\right)| \gt \psi\left(n\right)~\textrm{for infinitely many }n\right\},$ where $\psi\colon \mathbb{N}\to\mathbb{R}^+$ is a positive function. Let h be a positive and continuous function on the compact set $\overline{\mathfrak F}$. We further obtain the Hausdorff dimension of the set $\left\{z\in\mathfrak F:|a_{n+1}\left(z\right)\cdots a_{n+m}\left(z\right)| \gt \mathrm e^{h\left(z\right)+\cdots+h\left(T^{n-1}z\right)}~\textrm{for infinitely many }n\right\}$ and show that this set has large intersection property, where $T\colon\mathfrak F\to\mathfrak F$ is the Hurwitz map induced by Hurwitz continued fractions. This extends the main result of Bugeaud et al (2023 arXiv:2306.08254).

We study the following nonlinear Schrödinger system $\begin{cases} -\Delta u + \left(\lambda +P\left(y\right)\right)u = \alpha u^{3+\varepsilon } + \beta u v^{2},\ &\text{in } \ \mathbb{R}^{2},\\ -\Delta v + \left(\lambda +Q\left(y\right)\right)v = \gamma v^{3+\varepsilon } + \beta u^{2}v,\ &\text{in } \ \mathbb{R}^{2},\\ u \gt 0,v \gt 0,\ &\text{in } \ \mathbb{R}^{2},\\ \end{cases}$ with prescribed mass $\int_{\mathbb{R}^2}(u^2+v^2) = 1$, where λ is a Lagrange multiplier, P(y) and Q(y) are real-valued potentials, $\alpha,\beta,\gamma\in \mathbb{R}_{+}$ are constants, and $3+\varepsilon$ is a slightly mass supercritical exponent. In this paper, we show that there exists a segregated vector solution, and it is the number of the critical points of the potentials P(y) and Q(y) that affects the existence and the number of solutions to this problem. We also prove the local uniqueness for the normalized solutions we construct.

Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since 3-periodic trajectories of billiards within ellipses are examples of the Poncelet polygons, our considerations provide a new insight into the relationship between Marden's Theorem and the Poncelet Porism, two gems of exceptional classical beauty. We also show that every parallelogram is a billiard trajectory within a unique ellipse. We prove a similar result for the self-intersecting polygonal lines consisting of two pairs of congruent sides, named 'Darboux butterflies'. In each of three considered cases, we effectively calculate the foci of the boundary ellipses.

In this work, we study global dynamics in a no-flux initial-boundary value problem (IBVP) for the following fully parabolic minimal Keller–Segel system with matrix-valued sensitivity and indirect signal production: $\begin{cases} u_t = \Delta u-\chi\nabla\cdot\left(uS\nabla v\right) &\text{in } \Omega\times\left(0,\infty\right), \\ v_t = \Delta v-v+w &\text{in } \Omega\times\left(0,\infty\right), \\ w_t = \Delta w-w+u &\text{in } \Omega\times\left(0,\infty\right), \end{cases} \nonumber$ in a bounded and smooth domain $\Omega\subset\mathbb{R}^n(n\unicode{x2A7E} 1)$ with chemosensitivity parameter χ > 0, nonnegative initial datum $(u_0,v_0,w_0)$ and no-flux boundary conditions. Here, S is a given n × n matrix representing rotational effect. First, when $n\in\{1,2,3\}$, without further conditions on the underlying domains, initial data and sensitivity matrices, we establish global boundedness of classical solutions to the corresponding IBVP. Next, for S commuting with rotation or $S = aI+A$ with $A^T = -A$ and $a\in\mathbb{R}$, we show that the above IBVP preserves the radial symmetry of initial data. Then, when $\Omega = B_R\subset \mathbb{R}^4$ and $S = \cos\theta\cdot I+A$ with $A^T = -A$ and $\theta\in (-\frac{\pi}{2}, \frac{\pi}{2})$, we show global boundedness for $\|u_0\|_{L^1(\Omega)} \lt \frac{(8\pi)^2}{\chi\cos\theta}$ and the occurrence of blow-up for $\|u_0\|_{L^1(\Omega)} \gt \frac{(8\pi)^2}{\chi\cos\theta}$. Finally, under certain smallness of $(u_0, \nabla v_0, w_0-\bar w_0)$ in optimal Lebesgue spaces for $n\unicode{x2A7E} 1$ (smallness of $\chi\bar{u}_0\cos\theta$ in radial setting for $n = 1,2$), we demonstrate boundedness and exponential convergence of the globally bounded solution $(u,v,w)$ to its average $(\bar{u}_0,\bar{u}_0,\bar{u}_0)$. These findings in particular identify that anti-symmetry of the sensitivity matrix A plays no effect in radial framework. This work provides quantitative and qualitative effects of the sensitivity matrix S on global boundedness, preservation of radial symmetry, blow-up and exponential convergence of bounded solutions in the minimal indirect Keller–Segel model. The exponential convergence is new and fills up a gap even for the (scalar) indirect Keller–Segel model, namely, S being the identity I. Therefore, this work significantly generalises the previous knowledge on the (scalar) indirect Keller–Segel model and the (direct) Keller–Segel model with rotational sensitivity to the indirect Keller–Segel model with rotational effect.

In this paper, the m-component modified KP hierarchy is introduced as an identity expressed in terms of free fermions and tau functions, where tau functions are defined as group $GL_{\infty}$ orbitals. Subsequently, the Hirota bilinear identity, which as a system of partial differential equations, is derived using simple lattice vertex algebra. Finally, we present the bilinear identity in the form of m × m matrix functions. Based on this, we obtain the wave operator, Sato equation, Lax equations, and other pseudo-differential operators in matrix form.

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $\partial_t u - \mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right) +|\nabla u|^{q} = 0 \qquad {\mathrm{in}} \ \left(0,\infty\right)\times\mathbb{R}^N$ are studied when $N\unicode{x2A7E}1$ and the exponents (p, q) satisfy $p_c = \frac{2N}{N+1} \lt p \lt 2, \qquad p-1 \lt q \lt \frac{p}{2}.$Existence and uniqueness of such a solution are established in dimension N = 1. In dimension $N\unicode{x2A7E}2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behaviour as $|x|\to\infty$ is provided.

We study the propagation dynamics of a Lotka–Volterra competition system in which one growth rate behaves like a monotonically decreasing wave profile that shifts with a given speed and is also periodic in the first spatial variable, while the other growth rate behaves similarly, except that its profile is monotonically increasing with respect to the shifting variable. Furthermore, both growth functions are assumed to be sign-changed, which implies that the environments in which the species live switch spatially from 'good' regions (suitable for survival) to 'bad' regions (not suitable for survival) and vice versa. We reveal that the model admits a forced pulsating wave only when the forced speed lies within a finite interval $(c_*, c^*)$ that contains zero. Biologically, this corresponds to the formation of a shifting cline. Moreover, we find that $c_* \lt 0$ and $c^* \gt 0$ can be calculated in terms of the Fisher-KPP speeds related to the linearized equations of each species. By applying a sliding technique, we show that the forced pulsating wave is unique. We also prove that the forced pulsating wave is Lyapunov-stable. Finally, the spreading dynamics of spatial gap formation in the two species are also investigated when the forced wave speed is either less than $c_*$ or greater that $c^*$. We employ a novel approach to demonstrate how the species invade in response to a shifting environment.

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type A_n. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the general solution at zero and infinity. To reach this goal we are using a fusion of the Iwasawa factorization in the loop group theory and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou which is applicable to 2D Toda in view of its Lax integrability. A principal technical challenge is the extension of the nonlinear steepest descent analysis to Riemann-Hilbert problems of matrix rank greater than 2. In this paper, we meet this challenge for the case n = 2 (the rank 3 case) and it already captures the principal features of the general n case.

We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.

In this paper we examine the discrete Shnirelman's inequality (Shnirelman 1985 Math. USSR Sb.56 82–109), which relates the L²-distance of two discrete configurations of a fluid to the $L^1_tL^2_x$-norm of the vector field connecting them. Our proof is inspired by (Shnirelman 1985 Math. USSR Sb.56 82–109), where it was obtained $\alpha = \frac{1}{64}$ in dimension ν = 2, while here we get $\alpha\unicode{x2A7E}\frac{2}{7}$. Moreover, this bound can be taken $\alpha\unicode{x2A7E}\frac{1}{\nu+1}$ for any dimension $\nu\unicode{x2A7E} 3$. This is the best bound achieved in a purely discrete context, and connections to the continuous Shnirelman's inequality are discussed. Our method uses an alternative approach based on volume estimates of permutations, which count the number of maximum cubes that are moved by a permutation P.

Semitoric integrable systems were symplectically classified by Pelayo and V Ngọc in 2009–2011 in terms of five invariants. Four of these invariants were already well-understood prior to the classification, but the fifth invariant, the so-called twisting index invariant, came as a surprise. Intuitively, the twisting index encodes how the structure in a neighborhood of a focus–focus fiber compares to the large-scale structure of the semitoric system and it was originally defined by comparing certain momentum maps. In the first half of the present paper, we produce several new formulations of the twisting index which give rise to dynamical, geometric, and topological interpretations. More specifically, we describe it in terms of differences of action variables, Taylor series, and homology cycles. In the second half of the paper, we compute the twisting index invariant of a specific family of systems with two focus–focus singular points (the so-called generalized coupled angular momenta), which is the first time that the twisting index has been computed for a system with more than one focus–focus point. Moreover, we also compute the terms of the Taylor series invariant up to second order. Since the other invariants of this family were already computed, this becomes the third family of semitoric systems for which all invariants are known, after the coupled spin oscillators and the coupled angular momenta.

We investigate dimension theoretic properties of concentric topological spheres, which are fractal sets emerging both in pure and applied mathematics. We calculate the box dimension and Assouad spectrum of such collections, and use them to prove that fractal spheres cannot be shrunk through consecutive disjoint similar copies into a point at a polynomial rate. We also apply these dimension estimates to quasiconformally classify certain spiral shells, a generalization of planar spirals in higher dimensions. This classification also provides a bi-Hölder map between shells and constitutes an addition to a general programme of research proposed by Fraser (2021 Nonlinearity34 3251–70).

We prove that if f is a $C^{1+}$ partially hyperbolic diffeomorphism satisfying certain conditions then there is a C¹-open neighborhood $\mathcal{A}$ of f so that every $g\in \mathcal{A}\cap \operatorname{Diff}^{1+}(M)$ has a unique equilibrium state.

In this manuscript, we present a method to prove constructively the existence and spectral stability of solitary waves in both the Whitham and the capillary–gravity Whitham equations. By employing Fourier series analysis and computer-aided techniques, we successfully approximate the Fourier multiplier operator in this equation, allowing the construction of an approximate inverse for the linearization around an approximate solution u₀. Then, using a Newton–Kantorovich approach, we provide a sufficient condition under which the existence of a unique solitary wave $\tilde{u}$ in a ball centered at u₀ is obtained. The verification of such a condition is established combining analytic techniques and rigorous numerical computations. Moreover, we derive a methodology to control the spectrum of the linearization around $\tilde{u}$, enabling the study of spectral stability of the solution. As an illustration, we provide a (constructive) computer-assisted proof (CAP) of existence of stable solitary waves in both the case with capillary effects (T > 0) and without capillary effects (T = 0). Moreover, we provide an existence proof for a branch of solitary waves in the case T = 0 via a rigorous continuation in the wave velocity. The methodology presented in this paper can be generalized and provides a new approach for addressing the existence and spectral stability of solitary waves in nonlocal nonlinear equations. All CAPs, including the requisite codes, are accessible on GitHub at Cadiot (2023 https://github.com/matthieucadiot/WhithamSoliton.jl).

We develop a tool in order to analyse the dynamics of differentiable flows with singularities. It provides an abstract model for the local dynamics that can be used in order to control the size of invariant manifolds. This work is the first part of the results announced in Crovisier and Yang (2017 arXiv:1702.05994).

Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an interesting way: for the Ising model on a free group, the free-boundary state typically fails to be equilibrium as long as it is not the only Gibbs state. For every temperature between the uniqueness and reconstruction thresholds a typical sofic approximation gives this state finite but non-maximal pressure, and for every lower temperature the pressure is non-maximal over every sofic approximation. We also show that, for more general interactions on sofic groups, the local on average limit of Gibbs states over a sofic approximation Σ, if it exists, is a mixture of Σ-equilibrium states. We use this to show that the plus- and minus-boundary-condition Ising states are Σ-equilibrium if Σ is any sofic approximation to a free group. Combined with a result of Dembo and Montanari, this implies that these states have the same entropy over every sofic approximation.

We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions. We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension. We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when n = 1 the Assouad dimension is the greatest, and this latter statement is false for n > 1. This answers a question of Rutar.

In this paper, we consider the large time behavior of planar shock wave for 3D compressible isentropic Navier–Stokes equations (CNS) in half space. Providing the strength of the shock wave and initial perturbations are small, we proved the planar shock wave for 3D CNS is nonlinearly stable in half space with Navier boundary condition. The main difficulty comes from the compressibility of shock wave, which leads to lower order terms with bad sign, see the third line in (4.36). We apply a decomposition of the solution into zero and non-zero modes: we take the anti-derivative for the zero mode and obtain the space-time estimates for the energy of perturbation itself. Then combining the fact that the Poincaré inequality is available for the non-zero mode, we have successfully controlled the lower order terms with bad sign in (4.36). To overcome the difficulty that comes from the boundary, we introduce the two crucial estimates on boundary lemma (4.2) and fully utilize the property of Navier boundary conditions, which means that the normal velocity is zero on the boundary and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary. Finally, the nonlinear stability is proved by the weighted energy method.

In this paper, we prove the global wellposedness of the Gross–Pitaevskii equation with white noise potential in dimension 2, i.e. a cubic nonlinear Schrödinger equation with harmonic confining potential and spatial white noise multiplicative term. This problem is ill-defined and a Wick renormalisation is needed in order to give a meaning to solutions. In order to do this, we introduce a change of variables which transforms the original equation into one with less irregular terms. We construct a solution as a limit of solutions of the same equation but with a regularised noise. This convergence is shown by interpolating between a diverging bound in a high regularity Hermite–Sobolev space and a Cauchy estimate in $\mathbb{L}^2({\mathbb{R}}^2)$.

We consider the Oldroyd-B model for a two-dimensional dilute corotational polymer fluid with centre-of-mass diffusion that is interacting with a one-dimensional viscoelastic shell. We show that any family of strong solutions of the system described above that is parametrized by the centre-of-mass diffusion coefficient converges, as the coefficient goes to zero, to a weak solution of a corotational polymer fluid-structure interaction system without centre-of-mass diffusion but with essentially bounded polymer number density and extra stress tensor. As a consequence, we also obtain a weak-strong uniqueness result that says that the weak solution of the latter is unique in the class of the strong solution of the former as the centre-of-mass diffusion vanishes.

Lyme borreliosis, tick-borne encephalitis, and human granulocytic anaplasmosis are remarkable examples of tick-borne diseases. Motivated by their notable epidemiological role, this paper explores the interplay among seasonality, diapause, and diffusion on the population dynamics of ticks. To reach this goal, we analyze a metapopulation model in a fragmented ecosystem of m-patches, with ticks as a motivating example. Specifically, we offer a new methodology to derive criteria of global attraction in nonautonomous metapopulation models with delay (without assuming monotonicity requirements). A strength of our approach is that the results apply to metapopulations with any number of patches and topology. From a practical point of view, our theoretical analysis allows us to corroborate some previous experimental work. Another important conclusion of this paper is that the presence of sinks can benefit the tick population, simplifying the dynamical behavior of the whole metapopulation and increasing the total population size. To assess the real repercussions of the results, we analyze our model with parameters derived from real observations.

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.