Table of contents

We give an explicit family of polynomial maps called centre unstable Hénon-like maps and prove that they exhibit blenders for some parameter values. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.

We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

In this paper, we study the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler–Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli–Kohn–Nirenberg type. We establish the asymptotic behaviour of the branches for large values of the bifurcation parameter. We also perform an expansion in a neighbourhood of the first bifurcation point on the branch of symmetric solutions that characterizes the local behaviour of the non-symmetric branch. These results are compatible with earlier numerical and theoretical observations. Further numerical results allow us to distinguish two global scenarios. This sheds new light on the symmetry breaking phenomenon.

We address a fluid–structure system which consists of the incompressible Navier–Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. Given sufficiently small initial data, we prove the global-in-time existence of solutions by establishing a key energy inequality which in addition provides exponential decay of solutions.

We study synchronization properties of networks of coupled dynamical systems with interaction akin to diffusion. We assume that the isolated node dynamics possesses a forward invariant set on which it has a bounded Jacobian, then we characterize a class of coupling functions that allows for uniformly stable synchronization in connected complex networks—in the sense that there is an open neighbourhood of the initial conditions that is uniformly attracted towards synchronization. Moreover, this stable synchronization persists under perturbations to non-identical node dynamics. We illustrate the theory with numerical examples and conclude with a discussion on embedding these results in a more general framework of spectral dichotomies.

In this paper, we set up a 'dictionary' between discrete Schrödinger operators and holomorphic dynamics on certain affine cubic surfaces, building on previous work by Cantat, Damanik and Gorodetski. To achieve this, we make use of potential theory: a detailed description of the dynamical Green functions is obtained; then basic results concerning the equilibrium measures and the Green functions of compact subsets of $\mathbb{C}$ are used to transfer statements from the dynamical context to the Schrödinger one. This provides a new viewpoint on several recent theorems.

We consider a rather general class of convection–diffusion equations, involving dissipation (of possibly fractional order) which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models, Burgers' equation, the Navier–Stokes equations, the surface quasi-geostrophic equations and the Keller–Segel model for chemotaxis. Here we establish a Petrowsky type parabolic estimate of such equations which entail a precise time decay of higher order Sobolev norms for this class of equations. To this end, we introduce as a main new tool, an 'infinite-order energy functional', ${\mathcal E}(t): = \sum_{n=0}^\infty \alpha_n t^n \|u(\cdot,t)\|_{\dot{{\mathbb H}}^{n\theta+\beta_c}}$ with appropriate Sobolev critical regularity of order β_c. It captures the regularizing effect of all higher order derivatives of u(·, t), by proving—for a careful, problem-dependent choice of weights {α_n}, that ${\mathcal E}(t)$ is non-increasing in time.

In this paper we deal with nonlinear differential systems of the form

$$\begin{equation*}x'(t)=\sum_{i=0}^k\varepsilon^i F_i(t,x)+\varepsilon^{k+1} R(t,x,\varepsilon), \end{equation*}$$

$F_i:\mathbb{R}\times D\rightarrow\mathbb{R}^n$

$R:\mathbb{R}\times D\times(-\varepsilon_0,\varepsilon_0)\rightarrow\mathbb{R}^n$

$\mathbb{R}^n$

$\mathcal{C}^1$

The structure of the periodic orbits analysed in this paper was first reported in the Michelson system by Kent and Elgin (1991 Nonlinearity4 1045–61), who gave it the name noose bifurcation. Recently, it has been found in a piecewise linear system with two linearity zones separated by a plane, which is called the separation plane. In this system, the orbits that take part in the noose bifurcation have two and four points of intersection with the separation plane, and they are arranged in two curves that are connected by a point where the periodic orbit has a crossing tangency with the separation plane. In this work, we analytically prove the local existence of the curve of periodic orbits with four intersections that emerges from the point corresponding to the crossing tangency. Moreover, we add a numerical study of the stability and bifurcations of the periodic orbits involved in the noose curve for the piecewise linear system and check that they exhibit the same configuration as that of the Michelson system.

This paper contains rigorous results on nonequilibrium steady states for a class of particle systems coupled to unequal heat baths. These stochastic models are derived from the mechanical chains studied by Eckmann and Young by randomizing certain quantities while retaining other features of the model. Our results include the existence and uniqueness of nonequilibrium steady states, their relation to Lebesgue measure, tail bounds on total energy and number of particles in the system, and exponential convergence to steady states from suitable initial conditions.