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An outstanding problem in the study of networks of heterogeneous dynamical units concerns the development of rigorous methods to probe the stability of synchronous states when the differences between the units are not small. Here, we address this problem by presenting a generalization of the master stability formalism that can be applied to heterogeneous oscillators with large mismatches. Our approach is based on the simultaneous block diagonalization of the matrix terms in the variational equation, and it leads to dimension reduction that simplifies the original equation significantly. This new formalism allows the systematic investigation of scenarios in which the oscillators need to be nonidentical in order to reach an identical state, where all oscillators are completely synchronized. In the case of networks of identically coupled oscillators, this corresponds to breaking the symmetry of the system as a means to preserve the symmetry of the dynamical state— a recently discovered effect termed asymmetry-induced synchronization (AISync). Our framework enables us to identify communication delay as a new and potentially common mechanism giving rise to AISync, which we demonstrate using networks of delay-coupled Stuart–Landau oscillators. The results also have potential implications for control, as they reveal oscillator heterogeneity as an attribute that may be manipulated to enhance the stability of synchronous states.

Let N be a compact hyperbolic manifold, $M\subset N$ an embedded totally geodesic submanifold, and let $\newcommand{\DN}{\Delta_N} \newcommand{\h}{\hbar} -\h^2\DN$ be the semiclassical Laplace–Beltrami operator.

For any $\newcommand{\vareps}{\varepsilon} \vareps>0$ we explicitly construct families of quasimodes of energy width at most $\newcommand{\vareps}{\varepsilon} \newcommand{\h}{\hbar} \vareps\frac{\h}{\vert \log\h\vert }$ which exhibit a 'strong scar' on M in that their microlocal lifts converge weakly to a probability measure which places positive weight on $S^*M$$\newcommand{\h}{\hbar} \newcommand{\embed}{\hookrightarrow} \newcommand{\e}{{\rm e}} (\embed S^*N)$. An immediate corollary is that any invariant measure on $S^*N$ occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width $\newcommand{\vareps}{\varepsilon} \newcommand{\h}{\hbar} \vareps \frac{\h}{\vert \log\h\vert }$.

The existence of weak solutions to the steady compressible, multicomponent, chemically reacting gas is established without any restriction on the size of the data in the bounded 3D domain with slip boundary conditions. We extend the result in Zatorska (2011 Nonlinearity24 3267–78) for the adiabatic ratio $\gamma>\frac{7}{3}$ to $\gamma>2$. Our proof relies upon the weighted estimates of pressure and kinetic energy.

Let $(X, T)$ be a dynamical system, where X is a compact metric space and $\newcommand{\ra}{\rightarrow} T : X\ra X$ a continuous onto map. For weak Gibbs measures we prove large deviations estimates.

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston–Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.

The starting point of our analysis is a class of one-dimensional interacting particle systems with two species. The particles are confined to an interval and exert a nonlocal, repulsive force on each other, resulting in a nontrivial equilibrium configuration. This class of particle systems covers the setting of pile-ups of dislocation walls, which is an idealised setup for studying the microscopic origin of several dislocation density models in the literature. Such density models are used to construct constitutive relations in plasticity models.

Our aim is to pass to the many-particle limit. The main challenge is the combination of the nonlocal nature of the interactions, the singularity of the interaction potential between particles of the same type, the non-convexity of the the interaction potential between particles of the opposite type, and the interplay between the length-scale of the domain with the length-scale $\newcommand{\e}{{\rm e}} \ell_n$ of the decay of the interaction potential. Our main results are the Γ-convergence of the energy of the particle positions, the evolutionary convergence of the related gradient flows for $\newcommand{\e}{{\rm e}} \ell_n$ sufficiently large, and the non-convergence of the gradient flows for $\newcommand{\e}{{\rm e}} \ell_n$ sufficiently small.

The purpose of this article is to analyze the connection between Eynard–Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left({\rm e}^{\hbar\frac{\rm d}{{\rm d}x}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hbar$-differential systems to this setting. We apply our results to a specific $\hbar$-difference system associated to the quantum curve of the Gromov–Witten invariants of $\mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard–Orantin differentials computed from the topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$. Finally, identifying the large x expansion of the correlation functions, proves a recent conjecture made by Dubrovin and Yang regarding a new generating series for Gromov–Witten invariants of $\mathbb{P}^1$.

We study several fractal properties of the Weierstrass-type function where $\tau :[0, 1)\to[0, 1)$ is a cookie cutter map with possibly fractal repeller, and λ and g are functions with proper regularity. In the first part, we determine the box dimension of the graph of W and Hausdorff dimension of its randomised version. In the second part, the Hausdorff spectrum of the local Hölder exponent is characterised in terms of thermodynamic formalism. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.

In this paper we study C¹-structurally stable diffeomorphisms, that is, C¹ Axiom A diffeomorphisms with the strong transversality condition. In contrast to the case of dynamics restricted to a hyperbolic basic piece, structurally stable diffeomorphisms are in general not expansive and the conjugacies between C¹-close structurally stable diffeomorphisms may be non-unique, even if there are assumed C⁰-close to the identity. Here we give a necessary and sufficient condition for a structurally stable diffeomorphism to admit a dense subset of points with expansiveness and sensitivity to initial conditions. Morever, we prove that the set of conjugacies between elements in the same conjugacy class is homeomorphic to the C⁰-centralizer of the dynamics. Finally, we use this fact to deduce that any two C¹-close structurally stable diffeomorphisms are conjugated by a unique conjugacy C⁰-close to the identity if and only if these are Anosov.

Consider the following Schrödinger system: where either $\newcommand{\Om}{\Omega} \Om=\mathbb{R}^n (n=2, 3, 4)$ or $\newcommand{\Om}{\Omega} \newcommand{\su}{\subset} \Om\subset\mathbb{R}^4$ is a smooth bounded domain. Note that the cubic nonlinearities and the coupling terms are of critical growth whenever dimension $n=4$. We give a characterization of the least energy solutions when $\newcommand{\Om}{\Omega} \Om=\mathbb{R}^n (n=2, 3)$ or $\newcommand{\Om}{\Omega} \Om$ is a smooth bounded domain of $\newcommand{\R}{{\mathbb R}} \R^4$, if the coupling matrix ${\mathcal{B}}:=(\beta_{ij})$ is positively or negatively definite with $\beta_{ij}\geqslant0, \beta_{kk}>0, \forall k; i\neq j;$ and ${\mathcal{B}}^{-1}=(\beta_{ij}){\hspace{0pt}}^{-1}=(a_{ij})$ exists and satisfies $\newcommand{\su}{\subset} \sum_{j}a_{ij}>0$ for $i=1, 2, ..., N$. Furthermore, when $\newcommand{\Om}{\Omega} \newcommand{\R}{{\mathbb R}} \Om=\R^4$ and $\newcommand{\la}{\lambda} \la_j=0, j=1, 2, ..., N$, we obtain a nonexistence theorem about the least energy solutions provided attraction and repulsion coexist, i.e. some of $\newcommand{\bb}{\beta} \bb_{ij}, i\neq j$ are positive but some others are negative.