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The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

We consider affine iterated function systems in a locally compact non-Archimedean field F . We establish the theory of singular value decomposition in F and compute the box and Hausdorff dimensions of self-affine sets in F n , in generic sense, which is an analogy of Falconer’s result for the real case. In R n , the box and Hausdorff dimensions of self-affine sets can be obtained only when the norms of linear parts of affine transformations are strictly less than 1 2 . However, in a locally compact non-Archimedean field, the same result can be obtained without the restriction of the norms.

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps on arbitrary sets that have no iterative roots of any order. Then, using this result, we prove that when X is [ 0 , 1 ] m , R m or S 1 every non-empty open set of the space of continuous self-maps on X endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order n ⩾ 2 . This, in particular, proves that the complement of , the set of non-iterates, is dense in for these X.

We describe a method, based on "hard" contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian submanifold. We discuss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of the Glauber dynamics for the mean field Ising model., Comment: 40 pages, 5 figures, improved exposition, mild revision of the section on Glauber dynamics

The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space R 3 , as t → ∞ , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is T 2 × R . Our results reveal that the large-time behaviour for T 2 × R is quite different from that for R 3 . We show that any small initial velocity field u 0 ∈ H 2 ( Ω ) leads to a unique global solution u that remains small in H 2 ( Ω ) . More importantly, as t → ∞ , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.

Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the autonomous future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, introduce the concept of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The central idea is to use the autonomous dynamics of the future limit system to analyse R-tipping in the nonautonomous system. We compactify the original nonautonomous system to include the limiting autonomous dynamics. Considering regular R-tipping edge states that are equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation. Second, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.

In this paper, we study the following Hamiltonian elliptic system − ε 2 Δ u + u = H v ( x , u , v ) in R N , − ε 2 Δ v + v = H u ( x , u , v ) in R N , u ( x ) → 0 , v ( x ) → 0 a s | x | → ∞ , where ɛ > 0 is a small parameter, H is a super-quadratic sub-critical Hamiltonian. Our investigation focuses on the cases that H ( x , u , v ) = 1 2 K ( x ) ( u 2 + v 2 ) + F ( u , v ) and H ( x , u , v ) = K ( x ) F ( u , v ) , where K(x) possesses saddle points. By introducing a new penalization associated with the barycenters of functions and applying the Leray–Schauder degree theory, we adopt the local variational arguments in Byeon and Jeanjean (2007 Arch. Ration. Mech. Anal. 185 185–200); Byeon and Tanaka (2013 J. Eur. Math. Soc. 15 1859–99; 2014 Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations (Providence, RI: American Mathematical Society)) and obtain the existence of semi-classical states for the Hamiltonian elliptic system, which concentrate around the saddle points of V as ɛ → 0.

We consider Lagrangians that are defined only on the horizontal distribution of a sub-riemannian manifold. The associated Hamiltonian is neither strictly convex nor coercive. We prove a result on homogenization of the Hamilton–Jacobi equation following the approach in Contreras et al (2015 Calc. Var. Partial Differ. Equ. 52 237–52). To establish the result, we need to extend weak KAM and Aubry–Mather theories to the sub-riemannian setting. We obtain a Tonelli theorem to dispend of the Lagrangian dynamics tools used in standard weak KAM theory. In the way towards our main result we observe that, the long time convergence of the Lax-Oleinik semigroup, and the vanishing discount convergence of the discounted value function hold.

In this paper, we study the role of periodic measures with large support in the sense of density in invariant measures and the exponential growth for systems with periodic shadowing property or periodic approximate product property. These results are more refined versions of the Sigmund’s density result of periodic measures and Bowen’s entropy formula between the growth of periodic measures and entropy. Furthermore, we provide an abstract framework such that the results also hold for general homoclinic classes and shifts with non-uniform structure.

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank., 57 pages, revisions, adding references

In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem 3.1 inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles (Zhu 2021 Nonlinearity 34 5163–85, theorems 2.12 and 3.9) proved by Zhu under additional assumptions. Then along the line of Misiurewicz (1976 Studia Math. 55 175–200), we introduce a new invariant called relative topological tail entropy and prove a Ledrappier’s type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$., Comment: 23 pages, 7 figures

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved., 30 pages, 4 figures

We consider the 3D cubic nonlinear Schr\"odinger equation (NLS) with a strong toroidal trap. In the first part, we show that as the confinement is strengthened, a large class of global solutions to the time-dependent model can be described by 1D flows solving the 1D periodic NLS (Theorem 1.4). In the second part, we construct a steady state as a constrained energy minimizer, and prove its dimension reduction to the well-known 1D periodic ground state (Theorem 1.6 and 1.7). Then, employing the dimension reduction limit, we establish the local uniqueness and the orbital stability of the 3D ring soliton (Theorem 1.8). These results justify the emergence of stable quasi-1D periodic dynamics for Bose-Einstein condensates on a ring in physics experiments., Comment: 52 pages

We adapt methodology of Tosio Kato to establish necessary and sufficient conditions for the solutions to the Navier–Stokes equations with Dirichlet boundary conditions to converge in a strong sense to a solution to the Euler equations in the presence of a boundary as the viscosity is taken to zero. We extend existing conditions for no-slip boundary conditions to allow for nonhomogeneous Dirichlet boundary conditions and curved boundaries, establishing several new conditions as well. We give a brief comparison of various correctors appearing in the literature used for similar purposes.

The topological and metrical classifications of fractal sets are important topics in analysis. The goal of the present paper is to carry out such studies by using a finite state automaton. Firstly, we introduce Σ-automaton for self-similar sets, and we define topology automaton for fractal gaskets. Next, we show that a fractal gasket is always bi-Hölder equivalent to the pseudo-metric space induced by its topology automaton. Thirdly, we investigate when the pseudo-metric spaces induced by different automata can be bi-Lipschitz equivalent. As an application, we obtain a rather general sufficient condition for two fractal gaskets to be bi-Hölder or bi-Lipschitz equivalent.

We investigate critical points of a Landau–de Gennes (LdG) free energy in three-dimensional (3D) cuboids, that model nematic equilibria. We develop a hybrid saddle dynamics-based algorithm to efficiently compute solution landscapes of these 3D systems. Our main results concern (a) the construction of 3D LdG critical points from a database of two-dimensional (2D) LdG critical points and (b) studies of the effects of cross-section size and cuboid height on solution landscapes. In doing so, we discover multiple-layer 3D LdG critical points constructed by stacking 2D critical points on top of each other, novel pathways between distinct energy minima mediated by 3D LdG critical points and novel metastable escaped solutions, all of which can be tuned for tailor-made static and dynamic properties of confined nematic liquid crystal systems in 3D.

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of axisymmetric patterns. Our analysis covers localised patterns equipped with a wide range of dihedral symmetries, avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gr\"obner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton-Kantorovich procedure to prove the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment., Comment: 60 pages, 15 figures

For a bounded 2D planar domain Ω, we investigate the impact of domain geometry on oscillatory translational instabilities of N-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An N-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in Ω except in N well-separated localized regions of extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in -frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a 2N-vector that characterizes the possible modes (directions) of oscillation. The 2 N × 2 N matrix contains terms associated with a certain Green’s function on Ω, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates r = 1 + σ f ( θ ) with 0 < ε ≪ σ ≪ 1 , θ ∈ [ 0 , 2 π ) , and f ( θ ) 2π-periodic, we show that only the mode-2 coefficients of the Fourier series of f impact the bifurcation threshold at leading order in σ. We further show that when f ( θ ) = cos 2 θ , the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains Ω and N-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.

In this paper, we show that the small-amplitude stationary solutions are time-asymptotically stable to the half-space problem for the one-dimensional radiative and reactive gas (pressure p = R ρ θ + a 3 θ 4 , internal energy e = C v θ + a ρ θ 4 ) provided that the initial perturbation is sufficiently small in some weighted Sobolev spaces. We also obtain the convergence rate of global solutions toward corresponding stationary solutions. The proofs are given by a weighted energy method. A key point is to capture the positivity of the energy dissipation functional and boundary terms with suitable space weight functions and using the precise estimates about stationary solutions.

In Liu and Zhang (2020 Arch. Ration. Mech. Anal. 235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal. 238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that a 0 ≜ 1 ρ 0 − 1 ∈ B p , 1 3 p ( R 3 ) and the initial velocity u 0 = ( u 0 h , u 0 3 ) ∈ B p , 1 − 1 + 2 p , 1 p ( R 3 ) , with u 0 h belonging to H 1 ( R 3 ) , then the inhomogeneous NS system has a unique global solution provided that ( ∥ a 0 ∥ B p , 1 3 p + ∥ Λ h − 1 ∂ 3 u 0 ∥ B p , 1 − 1 + 2 p , 1 p ) ⋅ f ( u 0 ) being sufficiently small for some bounded function f depending on ∥ u 0 ∥ p , 1 − 1 + 2 p , 1 p and ∥ u 0 h ∥ H 1 . This provide a more general result that of Chemin et al (2014 Commun. Math. Phys. 272 529–66); Chemin and Zhang (2015 Commun. PDE 40 878–96).

In this paper we establish a sharp existence result on weak traveling wave solutions for a general class of multi-species reaction–diffusion systems. Moreover, the minimal speed of the traveling waves is explicitly determined. Such a weak traveling wave solution connects the predator-free equilibrium point E 0 at x = − ∞ but needs not to connect the coexistence equilibrium E ∗ at x = ∞ . We apply this result to three important non-cooperative systems: the classical diffusive SIS system for the spread of infectious disease, a predator–prey system with age structure and a generalised Lotka–Volterra predator–prey system of one predator species feeding on n prey species, and prove with the aid of Lyapunov functions and the LaSalle invariance principle that their weak traveling wave solutions are actually traveling wave solutions that connect E ∗ at x = ∞ . For the SIS system and the generalised Lotka–Volterra predator–prey system, we develop additional techniques to establish the boundedness of their weak traveling wave solutions before applying the LaSalle’s invariance principle.

In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general ϕ 4 n -model for all n ∈ N . This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states − v 2 with v 2 . In the traveling case, we prove the orbital instability in the whole energy space for all n ∈ N , while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all n ∈ N . Furthermore, as a by-product of our analysis, we are able to extend the main result in (de Loreno and Natali 2020 arXiv:2006.01305) (given for a different family of equations) to traveling wave solutions in the whole space, for all n ∈ N .

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

Imitation and aspiration learning rules are frequently observed in humans and animals. The former is an act of copying other’s action, whereas the latter is characterized by the self-evaluation. Here we study the coexistence of these learning mechanisms in structured populations. Both rules have been combined focusing on two different scenarios: (I) adoption of either update rule with a certain probability, and (II) grouping the entire population according to the update rules. We present two pair approximation models, illustrating both scenarios, which yield a nice agreement—under weak selection—with that of agent-based simulations. For weak selection and large population size, we find that the condition for cooperation to dominate defection is similar in both heterogeneous and homogeneous update rules. We examine several variants of the mixed model such as time-evolving aspirations alongside strategies and the coevolution of strategies and update rules. In the former case, our simulation reveals that Prisoner’s dilemma and, in some cases, Stag-hunt experience overall less aspiration levels compared to other games such as Chicken or Trivial. The coevolution of strategies and update rules demonstrates a better cooperation, in contrast to the fixed update rule case, exhibiting the possibility of asymptotic coexistence of both learning mechanisms.

We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of PDEs appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation and the method of separation of variables., 22 pages

We prove homogenization for reaction-advection-diffusion equations with KPP reactions, in the time-periodic spatially stationary ergodic setting, and find an explicit formula for the homogenized dynamic. We also extend this result to models with non-local diffusion and KPP reactions., The content of this paper was originally included in arXiv:2202.07743v1

We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any H s space with s ⩾ 0 . Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus’s aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.

In this paper, a novel Riemann–Hilbert (RH) approach is reported for a physically significant nonlocal integrable nonlinear Schrödinger equation. In this RH approach, the spectral analysis is performed from the t-part of the Lax pair rather than the x-part to formulate the desired RH problem. As a consequence, the resulting RH problem is determined by the t-part of the Lax pair with the x-part playing an auxiliary role. Compared with the traditional RH method, the novel RH approach in this paper has the merits that (a) the symmetry relations of the scattering data are found to be simple, (b) the general multi-soliton solutions of the equation can be easily obtained in the reflectionless cases. Additionally, to show the remarkable features of the obtained multi-soliton solutions, some special soliton dynamics are theoretically explored and then graphically illustrated by demonstrating their three-dimensional profiles.

In this article, we prove that the set of points whose pointwise emergence have same degree in [ 1 , ∞ ] with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a C 1 + α diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

In this paper, we consider the following nonlinear wave equation 0, \;\; x\in \mathbb{R}^n\\ &u(0 , x ) = u_0(x), \qquad u_t(0 , x) = u_1(x) \qquad x\in \mathbb{R}^n \end{split} \right. \end{equation*}?> u t t − Δ u + a ( t , x ) | u t | m − 1 u t + | u | p − 1 u = 0 , t > 0 , x ∈ R n u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) x ∈ R n and obtain optimal energy decay estimates under suitable assumptions on the space-time nonlinear damping coefficient a, where m satisfies 1 < m ⩽ n + 2 n − 2 . We establish a new Liapunov type function to obtain energy decay estimates in the case of nonlinear damping instead of the Kormonik lemma.

This paper is devoted to studying a system consisting of a reaction–diffusion equation with multi-valued right-hand side and an ordinary differential equation in absence of dissipation term, which is defined on the whole space R N . The system is driven by time-dependent forces and coloured noise with nonlinear diffusion. We first establish the global existence of strong/mild solutions for initial-value problems. The measurability of solution map with respect to sample points and initial values is then obtained via the upper semicontinuity, which indicates that these solutions define a (non-autonomous) multi-valued random dynamical system. Finally, we prove the existence of pullback attractor for the dynamical system.

Letr⩾2be an integer. The generalized Brezin–Gross–Witten (BGW) tau-function for the Gelfand–Dickey hierarchy of(r−1)dependent variables (aka ther-reduced Kadomtsev–Petviashvili hierarchy) is defined as a particular tau-function that depends on(r−1)constant parametersd1,…,dr−1. In this paper we show that this tau-function satisfies a family of linear equations, called theW-constraints of the second kind. The operators giving rise to the linear equations also depend on(r−1)constant parameters. We show that there is a one-to-one correspondence between the two sets of parameters.

As a model to provide a hands-on, elementary understanding of "vortex dynamics", we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalized Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents., Comment: 14 pages, 3 figures

We consider two approaches to study non-reversible Markov processes, namely the hypocoercivity theory and general equations for non-equilibrium reversible–irreversible coupling; the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker–Planck equation. We do this both for linear Markov processes and for a class of nonlinear Markov process as well. We then characterise the structure of the large deviation functional of generalised-reversible processes; this is a class of non-reversible processes of large relevance in applications. Finally, we show how our results apply to two classes of Markov processes, namely non-reversible diffusion processes and a class of piecewise deterministic Markov processes (PDMPs), which have recently attracted the attention of the statistical sampling community. In particular, for the PDMPs we consider we prove entropy decay.

In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems., 30 pages; the main theorem of the previous version is slightly weakened, several new theorems added

We investigate Newton’s method as a root finder for complex polynomials of arbitrary degrees. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical mathematics, numerics, computer graphics and physics, known methods may have excellent theoretical complexity but cannot be used in practice, or are practically efficient but lack a successful theory behind them. We provide precise and strong upper bounds for the theoretical complexity of Newton’s method and show that it is near-optimal with respect to the known set of starting points that find all roots. This theoretical result is complemented by a recent implementation of Newton’s method that finds all roots of various polynomials of degree more than a billion, significantly faster than our upper bounds on the complexity indicate, and often much faster than established fast root finders. Newton’s method thus stands out as a method that has strong merits both from the theoretical and from the practical point of view. Our study is based on the known explicit set of universal starting points, for each degreed, that are guaranteed to find all roots of polynomials of degreed(appropriately normalized). We show that this set containsdpoints that converge very quickly to thedroots: the expected total number of Newton iterations required to find alldroots with precisionɛisO(d3 log3d+d log | log ε|), which can be further improved toO(d2 log4d+d log | log ε|). The key argument shows that many root finding orbits are ‘R-central’ in the sense that they stay forever in a disk of radiusR, and each iteration ‘uses up’ an explicit amountAn,k(ℓ)of area within this disk.

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

A well-known regularity criterion in He and Xin (2005 J. Differ. Equ. 213 235–54) shows that if the velocity gradient tensor belongs to L q ( 0 , T ; L p ( R 3 ) ) with 2 / q + 3 / p = 2 , 3 / 2 < p ⩽ ∞ and 1 ⩽ q < ∞ , then the corresponding weak solution to the 3D magnetohydrodynamic (MHD) equations is smooth on ( 0 , T ] . In this paper, we prove that the role of the velocity gradient tensor can be replaced by the combination of the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor or by the combination of the diagonal part of the magnetic gradient tensor and the non-diagonal part of the velocity gradient tensor. The main interest among others is that the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Thus, our theorems may provide us with a new viewpoint to understand the potential singularity of the 3D MHD flow.

In this paper we are concerned with convergence properties of pullback attractors with respect to order of the fractional oscillon equations, that is, we study the fast growing dissipative semilinear oscillon equations as a limiting problem of semilinear equations with the main part being the fractional powers of the oscillon operators. We show that the family of pullback attractors associated with this approximations problems behave upper semicontinuously and we also show a result of continuity of the pullback attractors with respect to order of the fractional oscillon equations in each point of a dense residual subset of the interval [ 0 , 1 ] .

This paper investigates the global structures of periodic orbits that appear in Rayleigh–Bénard convection, which is modelled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period m and the winding number n become odd when an m-periodic orbit is symmetric with respect to the horizontal and vertical centre lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude ɛ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.

In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng- Huang's variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin-Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure., Comment: 25 pages. arXiv admin note: text overlap with arXiv:1012.1103, arXiv:1111.7121 by other authors

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N \gamma_n(q_1 p_1 + q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $\gamma_n$ are arbitrary coefficients, and $N\in \mathbb N$. For $N=2$, being both $\gamma_1,\gamma_2$ different from zero, this reduces to the classical Zernike system. We prove that $\mathcal H_N$ always provides a superintegrable system (for any value of $\gamma_n$ and $N$) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, $\mathcal H_N $ is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1:1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a $(2N-1)$th-order polynomial algebra. As a byproduct, the Hamiltonian $\mathcal H_N $ is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that $\mathcal H_N$ (and so the Zernike system as well) is endowed with a Poisson $\mathfrak{sl}(2,\mathbb R)$-coalgebra symmetry which would allow for further possible generalizations that are also discussed., Comment: 27 pages, 4 figures. Minor corrections

We consider in dimensions d = 1 , 2 , 3 the ɛ-dependent stochastic Cahn–Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order , γ > 0. The initial condition is non-layered and independent from ɛ. Under general assumptions on the noise diffusion σ, we prove moment estimates in H 1 (and in L ∞ when d = 1). Higher H 2 regularity p-moment estimates are derived when σ is bounded, yielding as well space Hölder and L ∞ bounds for d = 2 , 3 , and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter ɛ. As in the deterministic case, in H 1, H 2, the bounds admit a negative polynomial order in ɛ. Finally, assuming layered initial data of initial energy uniformly bounded in ɛ, as proposed by Chen (1996 J. Differ. Geom. 44 262–311), we use our H 1 2d-moment estimate and prove the stochastic solution’s convergence to ± 1 as ε → 0 a.s. when the noise diffusion has a linear growth.

In this work, we consider the initial boundary value problem for the modified version of the Cahn–Hilliard equation with the inertial term in two dimensional smooth bounded domain. Under the optimal regularity condition on the quartic nonlinearity, by using the splitting method, we prove the existence of the absorbing set for the weak solutions. Then, applying the energy method, we show that the semigroup generated by the weak solutions possesses a global attractor of the optimal regularity, and thereby we give a positive answer to the question raised in (Grasselli et al 2009 On the 2D Cahn–Hilliard equation with inertial term Commun. PDE 34 137–70).