Highlights of 2017

Multistability is a ubiquitous feature in systems of geophysical relevance and provides key challenges for our ability to predict a system's response to perturbations. Near critical transitions small causes can lead to large effects and—for all practical purposes—irreversible changes in the properties of the system. As is well known, the Earth climate is multistable: present astronomical and astrophysical conditions support two stable regimes, the warm climate we live in, and a snowball climate characterized by global glaciation. We first provide an overview of methods and ideas relevant for studying the climate response to forcings and focus on the properties of critical transitions in the context of both stochastic and deterministic dynamics, and assess strengths and weaknesses of simplified approaches to the problem. Following an idea developed by Eckhardt and collaborators for the investigation of multistable turbulent fluid dynamical systems, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. The edge state attracts initial conditions belonging to such a boundary and, while being defined by the deterministic dynamics, is the gate facilitating noise-induced transitions between competing attractors. We use a simplified yet Earth-like intermediate complexity climate model constructed by coupling a primitive equations model of the atmosphere with a simple diffusive ocean. We refer to the climatic edge states as Melancholia states and provide an extensive analysis of their features. We study their dynamics, their symmetry properties, and we follow a complex set of bifurcations. We find situations where the Melancholia state has chaotic dynamics. In these cases, we have that the basin boundary between the two basins of attraction is a strange geometric set with a nearly zero codimension, and relate this feature to the time scale separation between instabilities occurring on weather and climatic time scales. We also discover a new stable climatic state that is similar to a Melancholia state and is characterized by non-trivial symmetry properties.

In this paper, we will study the stochastic Swift–Hohenberg equation. The weak martingale solution, stationary martingale solution, invariant measures, mild solution, large deviation principle and random attractors for the stochastic Swift–Hohenberg equation will be considered.

The global attraction is proved for the nonlinear three-dimensional Klein–Gordon equation with a nonlinearity concentrated at one point. Our main result is the convergence of each 'finite energy solution' to the manifold of all solitary waves as $t\to\pm\infty$. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation.

We justify this mechanism by the following strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m, m]$ and satisfies the original equation. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single frequency $\newcommand{\om}{\omega} \omega\in[-m, m]$.

Let X  =  [0,1]. Given a non-uniformly contracting conformal iterated function system (IFS) $\left\{{{w}_{j}}\right\}_{j=1}^{m}$ and a family of positive Dini continuous potential functions $\left\{\,{{p}_{j}}\right\}_{j=1}^{m}$, the triple system $\left(X,\left\{{{w}_{j}}\right\}_{j=1}^{m},\left\{\,{{p}_{j}}\right\}_{j=1}^{m}\right)$, under some conditions, determines uniquely a probability invariant measure, denoted by μ. In this paper, we study the pressure function of the system and multifractal structure of μ. We prove that the pressure function is Gateaux differentiable and the multifractal formalism holds, if the IFS $\left\{{{w}_{j}}\right\}_{j=1}^{m}$ has non-overlapping.

In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.

Real-world networks in technology, engineering and biology often exhibit dynamics that cannot be adequately reproduced using network models given by smooth dynamical systems and a fixed network topology. Asynchronous networks give a theoretical and conceptual framework for the study of network dynamics where nodes can evolve independently of one another, be constrained, stop, and later restart, and where the interaction between different components of the network may depend on time, state, and stochastic effects. This framework is sufficiently general to encompass a wide range of applications ranging from engineering to neuroscience. Typically, dynamics is piecewise smooth and there are relationships with Filippov systems. In this paper, we give examples of asynchronous networks, and describe the basic formalism and structure. In the following companion paper, we make the notion of a functional asynchronous network rigorous, discuss the phenomenon of dynamical locks, and present a foundational result on the spatiotemporal factorization of the dynamics for a large class of functional asynchronous networks.

We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov–Arnol'd–Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.

We investigate the spectral stability of travelling wave solutions in a Keller–Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear, and linear consumption rate. Linearising around the travelling wave solutions, we locate the essential and absolute spectrum of the associated linear operators and find that all travelling wave solutions have parts of the essential spectrum in the right half plane. However, we show that in the case of constant or sublinear consumption there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. For the constant and sublinear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. We observe that this crossing always occurs off of the real axis.

In this paper, we propose a susceptible-infective-recovered (SIR) epidemic model to describe the geographic spread of an infectious disease in two groups/sub-populations living in a spatially continuous habitat. It is assumed that the susceptibility of individuals for infection and the infectivity of individuals are distinct between these two groups/sub-populations. It is also assumed that the infectious disease has a fixed latent period and the latent individuals may diffuse. We investigate the traveling wave solutions and obtain complete information about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number ${{R}_{0}}\left(S_{1}^{0},S_{2}^{0}\right)>1$ at the disease free equilibrium $\left(S_{1}^{0},S_{2}^{0},0,0\right)$, there exists a critical number c^*  >  0 such that for each c  >  c^*, the system admits a nontrivial traveling wave solution with wave speed c, and for c  <  c^*, the system admits no nontrivial traveling wave solution. When ${{R}_{0}}\left(S_{1}^{0},S_{2}^{0}\right)\leqslant 1$, we show that there exists no nontrivial traveling wave solution. In addition, for the case ${{R}_{0}}\left(S_{1}^{0},S_{2}^{0}\right)>1$ and c  >  c^*, we also find that the final sizes of susceptible individuals, denoted by $\left({{S}_{1,0}},{{S}_{2,0}}\right)$, satisfies ${{R}_{0}}\left({{S}_{1,0}},{{S}_{2,0}}\right)<1$, which means that there is no outbreak of this the infectious disease anymore. At last, we analyze and simulate the continuous dependence of the minimal speed c^* on the parameters.

We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth 'parameter shift' from one asymptotic value to another. We express tipping in terms of properties of local pullback attractors and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we introduce the notion of forward basin stability and use this to give a number of sufficient conditions for the presence or absence of rate-induced tipping. We apply our results to give criteria for irreversible rate-induced tipping in a conceptual climate model.

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members ${{ \Omega }_{m}}(t)$ ($1\leqslant m<\infty$) are made up from the respective sum of the L^2m-norms of vorticity and the density gradient. Each ${{ \Omega }_{m}}(t)$ has a lower bound in terms of the inverse Rossby number, Ro⁻¹, that turns out to be crucial to the argument. For convenience, the ${{ \Omega }_{m}}$ are also scaled into a new set of variables D_m(t). By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the D_m(t) in terms of Ro⁻¹ and the Reynolds number Re. These upper bounds vary across bands in the $\left\{{{D}_{1}},\,{{D}_{m}}\right\}$ phase plane. The boundaries of these bands depend subtly upon Ro⁻¹, Re, and the inverse Froude number Fr⁻¹. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of ${{ \Omega }_{1}}$ deviates from Re^3/4 as a function of $R{{o}^{-1}},\,Re$ and Fr⁻¹.

We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng. 30 726–54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn–Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity $\mathbf{u}$ satisfies $\mathbf{u}\centerdot \nu >0$, where ν is the outer normal to the boundary of the domain.

In this paper, we study a class of Schrödinger equations

$$\begin{eqnarray}\begin{array}{*{35}{l}} -\bigtriangleup u=k(u), & x\in {{\mathbb{R}}^{N}}, \end{array}\end{eqnarray}$$

$N\geqslant 3$

In this paper, we are interested in standing waves of the nonlinear Schrödinger equation coupled with the Maxwell equation. Firstly we describe conditions for the existence of minimizers with prescribed charge in terms of a coupling constant e. Secondly we study the existence of ground states for the stationary problem, the uniqueness of ground states for small e and a link between minimizers and ground states. Finally we study the orbital stability of standing waves for the quadratic nonlinearity.

We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with two and three degrees of freedom. The first example verifies the existence of tiny elliptic islands inside large chaotic domains for a quartic potential. In the 3-body problem we prove the KAM stability of the well-known figure eight orbit and two selected orbits of the so called family of rotating eights. Some additional theoretical and numerical information is also given for the dynamics of both examples.