In this paper we study relations of various types of sensitivity between a t.d.s. (X, T) and t.d.s. (M(X), T-M) induced by (X, T) on the space of probability measures. Among other results, we prove that F -sensitivity of (M(X), TM) implies the same of (X, T) and the converse is also true when F is a filter. We show that (X, T) is multi-sensitive if and only if so is (M(X), TM) and that (X, T) is F -sensitive if and only if (M-n(X), T-M) is F -sensitive (for some/all n is an element of N). We finish the paper providing an example of a minimal syndetically sensitive t.d.s. or a Li-Yorke sensitive t.d.s. such that induced t.d.s. fails to be sensitive.

We consider a nonlinear Schrodinger equation (NLS) posed on a graph (or network) composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of d-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass (L-2-norm). When existing, the minimizer is called ground state and it is the profile of an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists for sufficiently small masses whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state). This is a wide generalization of a result previously obtained for a star-graph with a single vertex. The main part of the proof is devoted to prove the concentration compactness principle for starlike structures; this is non trivial due to the lack of translation invariance of the domain. Then we show that a minimizing, bounded, H-1 sequence for the constrained NLS energy with external linear potentials is in fact convergent if its mass is small enough. Moreover we show that the ground state bifurcates from the vanishing solution at the bottom of the linear spectrum. Examples are provided with a discussion of the hypotheses on the linear part.

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.

We discuss the nonlinear phenomena of irreversible tipping for nonautonomous 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.

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins-Daruud et al in Hawkins-Daruud et al (2011 Int. J. Numer. Math. Biomed. Eng. 28 3-24). The model consists of a Cahn-Hilliard equation for the tumor cell fraction phi coupled to a reaction-diffusion equation for a function sigma representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side of the equation for sigma and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Frechet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

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 u satisfies u.v > 0, where. is the outer normal to the boundary of the domain.

We give a review of results on superpolynomial decay of correlations, and polynomial decay of correlations, for nonuniformly expanding semiflows and nonuniformly hyperbolic flows. A self-contained proof is given for semiflows. Results for flows are stated without proof (the proofs are contained in separate joint work with Balint and Butterley). Applications include intermittent solenoidal flows, suspended Henon attractors, Lorenz attractors and singular hyperbolic attractors, and various Lorentz gas models including the infinite horizon Lorentz gas.

Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed to a system which has strictly less symmetry. We give a geometric approach to study this phenomenon in the setting of Hamiltonian systems. We provide a method for determining the equilibria and relative equilibria that persist after a symmetry breaking perturbation. In particular a lower bound for the number of each is found, in terms of the equivariant Lyusternik-Schnirelmann category of the group orbit.

The Neumann initial-boundary value problem for the chemotaxis system {u(t) = del . (D(u)del u) - .(S(u)del nu), (*) nu(t) = Delta nu - nu + u, is considered in a bounded domain Omega subset of R-n, n >= 1, with smooth boundary. In compliance with refined modeling approaches, the diffusivity function D therein is allowed to decay considerably fast at large densities, where a particular focus will be on the mathematically delicate case when D(s) decays exponentially as s -> infinity. In such situations, namely, straightforward Moser-type recursive arguments for the derivation of L-infinity estimates for u from corresponding L-p bounds seem to fail. Accordingly, results on global existence, and especially on quantitative upper bounds for solutions, so far mainly concentrate on cases when D decays at most algebraically, and hence are unavailable in the present context. This work develops an alternative approach, at its core based on a Mosertype iteration for the quantity e(u), to establish global existence of classical solutions for all reasonably regular initial data, as well as a logarithmic upper estimate for the possible growth of parallel to u(.,t)parallel to(L)infinity((Omega)) as t -> infinity, under the assumptions that with some K1 > 0, K2 > 0, beta(-) > 0 and beta(+) is an element of (-infinity,beta(-)] we have K(1)e(-beta-s) = 0, and that the size of S relative to D can be estimated according to S(s)/D(s) = 0 with some K-3 > 0 and gamma is an element of[beta(broken vertical bar) - beta/2 , beta(broken vertical bar) /2). Making use of the fact that this allows for certain superalgebraic growth of S/D, as a particular consequence of this and known results on nonexistence of global bounded solutions we shall see that in the prototypical case when D(s) = e(-beta s) and S(s) = se(-alpha s) for all s >= 0 and some positive alpha and beta, the assumptions that n >= 2 and that beta>0 and {alpha is an element of(beta/2, beta) if n = 2, alpha is an element of(beta/2, beta) if n >= 3, warrant the existence of classical solutions which are global but unbounded, and for which this infinite-time blow-up is slow in the sense that the corresponding grow-up rate is at most logarithmic. To the best of our knowledge, this inter alia seems to constitute the first quantitative information on a blow-up rate in a parabolic Keller-Segel system of type (star) for widely arbitrary initial data, hence independent of a particular construction of possibly non-generic exploding solutions.

We prove the hair-trigger effect for a class of nonlocal nonlinear evolution equations on R-d which have only two constant stationary solutions, 0 and theta > 0. The effect consists in that the solution with an initial condition non identical to zero converges (when time goes to infinity) to. locally uniformly in R-d. We also find sufficient conditions for existence, uniqueness and comparison principle in the considered equations.

Following the study of complex elliptic-function-type asymptotic behaviours of the Painleve equations by Boutroux, Joshi and Kruskal for P-I and P-II, we provide new results for elliptic-function-type behaviours admitted by P-III, P-IV, and P-V, in the limit as the independent variable z approaches infinity. We show how the Hamiltonian E-J of each equation P-J, J = I, . . . , V, varies across a local period parallelogram of the leading-order behaviour, by applying the method of averaging in the complex z-plane. Surprisingly, our results show that all the equations P-I-P-V share the same modulation of E to the first two orders.

We consider an initial-boundary value problem for the one-dimensional equations of compressible isentropic magnetohydrodynamic (MHD) flows. The non-resistive limit of the global solutions with large data is justified. As a by-product, the global well-posedness of the compressible non-resistive MHD equations is established. Moreover, the thickness of the magnetic boundarylayer of the value O(nu(alpha)) with 0 0 is the resistivity coefficient. The proofs of these results are based on a full use of the so-called 'effective viscous flux', the material derivative and the structure of the equations.

In this paper, we are concerned with the Schrodinger-Poisson system {-Delta u + u + empty set u = vertical bar u vertical bar(p-2)u in R-d, Delta empty set - u(2) in R-d. (0.1) Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case d >= 3. In contrast, much less information is available in the planar case d = 2 which is the focus of the present paper. It has been observed by Cingolani S and Weth T (2016 On the planar Schrdinger-Poisson system Ann. Inst. Henri Poincare 33 169-97) that the variational structure of (0.1) differs substantially in the case d = 2 and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrodinger-Poisson system Ann. Inst. Henri Poincare 33 169-97) is restricted to the case p >= 4 which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case 2 < p < 4 with a different variational approach.

We study the nodal intersections number of random Gaussian Loral Laplace eigenfunctions ('arithmetic random waves') against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for 'generic' curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar 'static' curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For 'generic' curves we prove the central limit theorem (at least, for 'most' of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.

In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Planck equations in terms of large-deviation rate functionals.

This paper initiates the study of semitoric integrable systems with two degrees of freedom and with proper momentum-energy map, but with possibly nonproper S-1-momentum map. This class of systems includes many standard examples, such as the spherical pendulum. To each such system we associate a subset of R-2, invariant under a natural notion of isomorphism and encoding the integral affine structure of the singular Lagrangian fibration, in the spirit of Delzant polygons for toric systems.

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.

We prove that the Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

We consider feed-forward networks, that is, networks where cells can be divided into layers, such that every edge targeting a layer, excluding the first one, starts in the prior layer. A feed-forward system is a dynamical system that respects the structure of a feed-forward network. The synchrony subspaces for a network, are the subspaces defined by equalities of some cell coordinates, that are flow-invariant by all the network systems. The restriction of a network system to each of its synchrony subspaces is a system associated with a smaller network, which may be, or not, a feed-forward network. The original network is then said to be a lift of the smaller network. We show that a feed-forward lift of a feed-forward network is given by the composition of two types of lifts: lifts that create new layers and lifts inside a layer. Furthermore, we address the lifting bifurcation problem on feed-forward systems. More precisely, the comparison of the possible codimension-one local steady-state bifurcations of a feed-forward system and those of the corresponding lifts is considered. We show that for most of the feed-forward lifts, the increase of the center subspace is a sufficient condition for the existence of additional bifurcating branches of solutions, which are not lifted from the restricted system. However, when the bifurcation condition is associated with the internal dynamics and the lift occurs inside an intermediate layer, we prove that the existence of a bifurcation branch not lifted from the restricted system does depend generically on the chosen feed-forward system.

We study invasion fronts in the FitzHugh-Nagumo equation in the oscillatory regime using singular perturbation techniques. Phenomenologically, localized perturbations of the unstable steady-state grow and spread, creating temporal oscillations whose phase is modulated spatially. The phase modulation appears to be selected by an invasion front that describes the behavior in the leading edge of the spreading process. We construct these invasion fronts for large regions of parameter space using singular perturbation techniques. Key ingredients are the construction of periodic orbits, their unstable manifolds, and the analysis of pushed and pulled fronts in the fast system. Our results predict the wavenumbers and frequencies of oscillations in the wake of the front through a phase locking mechanism. We also identify a parameter regime where nonlinear phase locked fronts are inaccessible in the singularly perturbed geometry of the traveling-wave equation. Direct simulations confirm our predictions and point to interesting phase slip dynamics.