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This paper studies a type of non-uniformly parabolic problem with nonlocal term u t = u p ( u x x + u − u ¯ ) 0 < t < T m a x , 0 < x < a , u x ( t , 0 ) = u x ( t , a ) = 0 0 < t < T m a x , u ( 0 , x ) = u 0 ( x ) 0 < x < a , where p > 1, a > 0. First the classification of the finite-time blow-up/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. More importantly, we derive that any bounded solution converges to some steady state as t → +∞. The difficulties in proving this convergence result lie in the existence of a continuum of steady states and the lack of the comparison principle due to the introduction of nonlocal term. To conquer these difficulties, we combine the applications of Lojasiewicz–Simon inequality and energy estimates.

We study non-negative travelling wave solutions, u ≡ U(x − ct) with constant wavespeed c > 0, of the cubic nonlocal Fisher-KPP equation in one spatial dimension, namely, ∂ u ∂ t = ∂ 2 u ∂ x 2 + u 2 1 − 1 λ ∫ − ∞ ∞ ϕ y − x λ u y , t d y , for ( x , t ) ∈ R × R + , where u(x, t) is the population density. Here ϕ(y) is a prescribed, piecewise continuous, symmetric, nonnegative and nontrivial, integrable kernel, which is nonincreasing for y > 0, has a finite derivative as y → 0+ and is normalised so that ∫ − ∞ ∞ ϕ ( y ) d y = 1 . The parameter λ is the ratio of the lengthscale of the kernel to the diffusion lengthscale. The quadratic version of the equation, with reaction term u(1 − ϕ*u), has a unique travelling wave solution (up to translation) for all c ⩾ c min = 2. This minimum wavespeed is determined locally in the region where u ≪ 1, (Berestycki et al 2009 Nonlinearity 22 2813–44). For the cubic equation, we find that a minimum wavespeed also exists, but that the numerical value of the minimum wavespeed is determined globally, just as it is for the local version of the equation, (Billingham and Needham 1991 Dynam. Stabil. Syst. 6 33–49). We also consider the asymptotic solution in the limit of a spatially-localised kernel, λ ≪ 1, for which the travelling wave solutions are close to those of the cubic Fisher-KPP equation, u t = u xx + u 2(1 − u). We find that when ϕ = o(y −3) as y → ∞, the minimum wavespeed is 1 2 + O ( λ 4 ) , but that when ϕ = O(y −n ) with 1 < n ⩽ 3, the minimum wavespeed is 1 2 + O ( λ 2 ( n − 1 ) ) . In each case we determine the correction terms. We also compare these asymptotic solutions to numerical solutions and find excellent agreement for some specific choices of kernel.

In this paper, we consider the inviscid Boussinesq system under the presence of magnetic field in porous media. We firstly proved the global well-posedness and large time behaviour of solutions in the whole space R 3 . Precisely, when the H 3 norm of initial data is small, but the higher order derivatives can be arbitrary large, the system is globally well-posed by pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the L p − L 2 (1 ⩽ p ⩽ 2) type of the optimal time decay rates are obtained without any smallness assumption on the L p norm of the initial data. At last, we derive a global weak solution with large initial data and give an explicit decay rate of the solution in R 2 . Our results mathematically explain the stable phenomenon of the system with stratification effects.

In many natural and technological systems the rule that governs the system’s dynamics changes over time. In such switched systems the system’s switching can be a significant source of instability. Here we give a simple sufficient criterium to determine if an i.i.d. stochastically switched system is stable in expectation. This method extends recent results for linear switched systems to nonlinear switched systems. It also extends results known for general switched systems giving improved results for systems with i.i.d. stochastic switching. The paper also considers the effects of time-delays on the stability of switched systems. Such time delays, which are intrinsic to any real-world system, can also have a destabilising effect on the system’s dynamics. Previously, it has been shown that if a dynamical system is intrinsically stable, which is a stronger form of global stability, then it maintains its stability even when time-delays are introduced into the system. Here we extend this notion to stochastically switched systems. We refer to this type of stability as patient stability and give a simple sufficient criterium under which such systems are patiently stable, i.e. cannot be destabilised by time delays. Both criteria introduced in this paper side step the need to use Lyapunov, linear matrix inequalities, and semi-definite programming-type methods. Our examples in this paper demonstrate the simplicity of these criteria.

Maps with discontinuities can be shown to have many of the same properties of continuous maps if we include hidden orbits—solutions that include points lying on a discontinuity. We show here how the well known property that ‘period 3 implies chaos’ also applies to maps with discontinuities, but with a twist, namely that if a map has a hidden fixed point and a hidden period 2 orbit, then it has hidden orbits of all periods, or concisely ‘period 1 + 2 implies chaos’. More precisely we show that for a one-dimensional map defined on an interval, if there exist hidden orbits of periods p and q, then there exist hidden orbits of all periods ap + bq, for any a , b ∈ N . To better understand this result we show that, by smoothing or ‘regularising’ across the discontinuity, these hidden orbits become regular unstable orbits, and the familiar results for continuous maps (like ‘period 3 implies chaos’) are restored. We also show examples of hidden orbits in some applications of oscillators, and consider how these results can be used to approximate the behaviour of continuous maps with steep segments.

We give a geometric description of the parabolic bifurcation locus in the space Rat d of all rational functions on P 1 of degree d > 1, generalising the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.

After casting Euler-α equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-α equations in 2D exterior domain provided that the initial data is regular enough.

In this paper we deal with the conditional regularity of the weak solutions of the Navier–Stokes equations on a bounded domain endowed with Navier boundary conditions, Navier-type boundary conditions or Dirichlet boundary conditions. We prove the regularity criteria which are based on the directional derivatives of several fundamental quantities along the streamlines, namely the velocity magnitude, the kinetic energy, the pressure, the velocity field and the Bernoulli pressure. In striking contrast to the known criteria in which the mentioned quantities were differentiated along a fixed vector, our criteria are mostly optimal for the whole range of parameters and have a clear physical meaning.

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (α, β). We show that for any r ⩾ 1, this rotation set only appears for C r diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. More precisely, the dynamics admits bounded deviation along the direction −(α, β) in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small C 0-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism f with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.

In this paper, we study the p-Laplacian equation with a L p -norm constraint: − Δ p u = λ | u | p − 2 u + μ | u | q − 2 u + g ( u ) in R N , ∫ R N | u | p d x = a p , where N ⩾ 2, a > 0, 1 < p < q ⩽ p ¯ ≔ p + p 2 N , μ ∈ R , g ∈ C ( R , R ) and λ ∈ R is a Lagrange multiplier, which appears due to the mass constraint ‖u‖ p = a. We assume that g is odd and L p -supercritical. When q < p ¯ and μ > 0, we use Schwarz rearrangement and Ekeland variational principle to prove the existence of positive radial ground states for suitable μ. When q = p ¯ and μ > 0 or q ⩽ p ¯ and μ ⩽ 0, with an additional condition of g, we obtain a positive radial ground state if μ lies in a suitable range, by the Schwarz rearrangement and minimax theorems. Via a fountain theorem type argument, with suitable μ ∈ R , we obtain infinitely many radial solutions for any N ⩾ 2 and establish the existence of infinitely many nonradial sign-changing solutions for N = 4 or N ⩾ 6. In any case mentioned above, the range of μ depends on the value of a: |μ| can be large if a > 0 is small.

This paper focuses on the so-called Weighted Inertia-Dissipation-Energy (WIDE) variational approach for the approximation of unsteady Leray-Hopf solutions of the incompressible Navier-Stokes system. Initiated in [56], this variational method is here extended to the case of non-Newtonian fluids with power-law index $r\geq11/5$ in three space dimensions and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimization. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition $-\frac12|\boldsymbol{v}|^2\boldsymbol{n}+\mathbb{T}\boldsymbol{n}=0$. This provides the first theoretical explanation for the onset of such boundary conditions., Comment: Preprint, 38 pages

This paper deals with a two-species chemotaxis model with two chemicals in R 2 . In our previous work (2019 Nonlinearity 32 4762–78), the critical mass was obtained that the solutions exist globally if m 1 m 2 − 4π(m 1 + m 2) < 0, and the finite time blow-up of solutions may occur if m 1 m 2 − 4π(m 1 + m 2) > 0, where m 1 and m 2 describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m 1 m 2 − 4π(m 1 + m 2) = 0. To apply the key logarithmic Hardy–Littlewood–Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L 1-norms of the two species in exterior domains R\right\}$?> { x ∈ R 2 : | x | > R } uniformly for t > 0.

We investigate time-decay rates of strong solutions to the 3D compressible magnetohydrodynamics equations with large initial data. The main novelty of this paper is two-fold: first, we prove the upper optimal decay rates of the higher-order spatial derivatives of the solution, which are the same as those of the heat equation, and faster than the decay rates in the previous related works. Second, if the initial data satisfy some additional low frequency assumption, we also show the lower optimal decay rates of the solution as well as its all-order spatial derivatives. Therefore, our decay rates are optimal in this sense. Our methods mainly involve the Fourier splitting method, low-frequency and high-frequency decomposition and delicate energy estimates.

Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attractor at the boundary between contractivity and expansion of an IFS? The first question is addressed in the paper by providing examples of highly noncontractive IFSs with attractors. The second question leads to the study of two types of transition phenomena associated with an IFS family that depend on a real parameter. These are called lower and upper transition attractors. Their existence and properties are the main topic of this paper. Lower transition attractors are related to the semiattractors, introduced by Lasota and Myjak in 1990s. Upper transition attractors are related to the problem of continuous dependence of an attractor upon the IFS. A main result states that, for a wide class of IFS families, there is a threshold such that the IFSs in the one-parameter family have an attractor for parameters below the threshold and they have no attractor for parameters above the threshold. At the threshold there exists a unique upper transition attractor.

We study quasilinear elliptic equations of the form − div A ( u ) ∇ u + 1 2 A ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3 , A ∈ C 1 ( R , R + ) is a bounded function, V(x) is allowed to be singular at the origin, and h ∈ C ( R , R ) is a general nonlinearity. Such type of equations has been derived as models of several physical phenomena corresponding to various types of A . Despite the lack of a priori compactness condition for the energy functional, we develop a new variational approach to deal with the issues of multiple radial solutions, nonradial solutions and sign-changing solutions.

We study the action of translation on the spaces of uniformly bounded continuous functions on the real line which are uniformly band-limited in a compact interval. We prove that two intervals themselves will decide if two spaces are topologically conjugate, while the length of an interval tells the mean dimension of a space. We also investigate universal real flows. We construct a sequence of compact invariant subsets of a space consisting of uniformly bounded smooth one-Lipschitz functions on the real line, which have mean dimension equal to one, such that all real flows can be equivariantly embedded in the translation on their product space. Moreover, we show that the countable self-product of any among them does not satisfy such a universal property. This, on the one hand, presents a more reasonable choice of a universal real flow with a view towards mean dimension, and on the other hand, clarifies a seemingly plausible impression; meanwhile, it refines the previously known results in this direction. Our proof goes through an approach of harmonic analysis. Furthermore, both the universal space that we provide and an embedding mapping which we build for any real flow are explicit.

For β < 1 3 , we consider C β T 3 × [ 0 , T ] weak solutions of the incompressible Euler equations that do not conserve the kinetic energy. We prove that for such solutions the closed and non-empty set of singular times B satisfies dim H ( B ) ⩾ 2 β 1 − β . This lower bound on the Hausdorff dimension of the singular set in time is intrinsically linked to the Hölder regularity of the kinetic energy and we conjecture it to be sharp. As a first step in this direction, for every β < β ′ < 1 3 we are able to construct, via a convex integration scheme, non-conservative C β T 3 × [ 0 , T ] weak solutions of the incompressible Euler system such that dim H ( B ) ⩽ 1 2 + 1 2 2 β ′ 1 − β ′ . The structure of the wild solutions that we build allows moreover to deduce non-uniqueness of C β T 3 × [ 0 , T ] weak solutions of the Cauchy problem for Euler from every smooth initial datum.

Recently, Kiriki, Nakano and Soma introduced a concept called pointwise emergence as a new quantitative perspective into the study of non-existence of averages for dynamical systems. In the present paper, we consider the set of points with high pointwise emergence for symbolic systems with non-uniform structure and prove that this set carries full topological pressure. For the proof of this result, we show that such systems have ergodic measures of arbitrary intermediate pressures.

This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures.

This paper considers the periodic solutions of a two-dimensional nonlinear wave equation with x-dependent coefficients v ( x ) y t t − ( v ( x ) y x 1 ) x 1 − ( v ( x ) y x 2 ) x 2 + β y + g ( y ) = f ( x , t ) , subject to the Sturm–Liouville boundary conditions a 11 y ( 0 , x 2 , t ) − b 11 y x 1 ( 0 , x 2 , t ) = 0 , a 12 y ( π , x 2 , t ) + b 12 y x 1 ( π , x 2 , t ) = 0 , a 21 y ( x 1 , 0 , t ) − b 21 y x 2 ( x 1 , 0 , t ) = 0 , a 22 y ( x 1 , π , t ) + b 22 y x 2 ( x 1 , π , t ) = 0 , where x = (x 1, x 2) ∈ (0, π) × (0, π), t ∈ R and a i j 2 + b i j 2 ≠ 0 for i, j = 1, 2. Such a model arises from the forced vibrations of a nonhomogeneous membrane and the propagation of waves in nonhomogeneous media. By using the duality principles of Brézis (1983 Bull. Am. Math. Soc. 8 409–26), we first set up the theoretical framework and give an abstract theorem for the existence of periodic solution. Then, by analysing the spectral asymptotic behaviours of the weighted Sturm–Liouville problem and classifying the boundary conditions, we obtain the fundamental properties of the variable coefficients wave operator under appropriate assumptions for different class of boundary conditions. Finally, based on these properties and with the help of the abstract theorem, we establish some results on the existence and regularity of periodic solutions. To the best of our knowledge, the results are entirely new, and this is the first time to obtain such results for the two-dimensional nonlinear wave equation with x-dependent coefficients.

In this paper, we investigate the following Schrödinger equation 0.1 − Δ u − μ | x | 2 u = g ( u ) + | u | 2 * − 2 u i n R N \ { 0 } , where N ⩾ 3, μ < ( N − 2 ) 2 4 , 2 * ≔ 2 N N − 2 is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any μ ∈ 0 , ( N − 2 ) 2 4 , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 − Δ u = g ( u ) + | u | 2 * − 2 u i n R N . when μ < 0, we prove that the mountain pass level of problem (0.1) in H 1 ( R N ) cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in H 1 ( R N ) . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.

In the previous paper by Alt et al (1984 Indiana Univ. Math. J. 33 367–91), the existence of the symmetric plane flows of two fluids with two free boundaries has been established by variational approach. However, the uniqueness of the two-phase fluids with two free boundaries is an unsolved problem. In the present paper, we will deal with the uniqueness of the two-phase fluids. In addition, if we assume that the two nozzles both converge, we shall show the monotonicity of the two free boundaries of the incompressible two-phase fluids issuing from these two nozzles.

In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier–Stokes equations in Ω ⊂ R 2 . The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data ( ρ 0 , u 0 ) ∈ L ∞ × H 0 s with s > 0, which constitutes a positive answer to the question raised by Danchin and Mucha (2019 Commun. Pure Appl. Math. 72 1351–85), in which the initial velocity u 0 ∈ H 0 1 (see also Li (2017 J. Differ. Equ. 263 6512–36).

We study the analyticity and Gevrey regularity for the solutions to the compressible Navier–Stokes–Korteweg system. It was observed by Charve et al (2021 Indiana Univ. Math. J. 70 1903–44) that this system has the full parabolicity property due to the coupling of the equations given by the Korteweg capillarity term, which allows them to prove the analyticity of the solutions in the whole space R d with radius bounded from below by C t . In this paper, we improve this lower bound, motivated by the works of Chemin et al (2020 Math. Res. Lett. 27 1631–43) concerning the classical Navier–Stokes system. We also study the compressible Navier–Stokes–Korteweg system with periodic boundary condition and we show that the radius of analyticity for the mild solution is bounded from below by Ct.

We prove the reducibility of 1D quantum harmonic oscillators in R perturbed by a quasi-periodic in time potential V(x, ωt) under the following conditions, namely there is a C > 0 such that | V ( x , θ ) | ⩽ C , | x ∂ x V ( x , θ ) | ⩽ C , ∀ ( x , θ ) ∈ R × T σ n . The corresponding perturbation matrix ( P i j ( θ ) ) is proved to satisfy ( 1 + | i − j | ) | P i j ( θ ) | ⩽ C and i j | P i + 1 j + 1 ( θ ) − P i j ( θ ) | ⩽ C for any θ ∈ T σ n and i, j ⩾ 1. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element P i j ( θ ) as well as the discrete difference matrix element P i + 1 j + 1 ( θ ) − P i j ( θ ) . For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a B\"{a}cklund transformation, (e) use the B\"{a}cklund transformation to obtain an infinite number of conservation laws., Comment: 27 pages. This is a standalone paper based on Section 6 of arXiv:2005.10781v1

We study the evolution of angular variable (phase) for general (not necessarily Hamiltonian) perturbations of Hamiltonian systems with one degree of freedom near separatrices of the unperturbed system. To this end, we use the averaged system of order 2. We obtain estimates for the accuracy of the order 2 averaged system near separatrices and use these estimates to prove a formula for the phase change when solutions of the perturbed system approach separatrices of the unperturbed system (such formula is known when the perturbation is Hamiltonian). As an application of this formula, we show that two natural definitions of probability of capture into different domains after separatrix crossing proposed by Arnold and Anosov lead to the same formula for this probability.

Starting from a nonlinear 2D/1D fluid–structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, in the vanishing limit of the relative fluid thickness, we rigorously derive a sixth-order thin-film equation describing the dynamics of vertical displacements of the structure. The procedure is essentially based on quantitative energy estimates in terms of the relative fluid thickness and a uniform no-contact result between the structure and the solid substrate. The sixth-order thin-film equation is justified in the sense of strong convergence of rescaled structure displacements to the unique positive classical solution of the thin-film equation. Moreover, the limiting fluid velocity and pressure can be expressed solely in terms of the solution to the thin-film equation.

We are concerned with the existence of multibump solutions to the nonlinear Schrödinger equation − Δ u + λ a ( x ) u + μ u = | u | 2 σ u in R N with an L 2-constraint ‖ u ‖ L 2 ( R N ) 2 = ρ in the L 2-subcritical case σ ∈ (0, 2/N) and the L 2-supercritical case σ ∈ (2/N, 2*/N), where the usual critical Sobolev exponent is 2* = +∞ if N = 1, 2 and 2* = 2N/(N − 2) if N ⩾ 3. Here μ ∈ R will arise as a Lagrange multiplier, and 0 ⩽ a ∈ L loc ∞ ( R N ) has a bottom int a −1(0) composed of ℓ 0 (ℓ 0 ⩾ 1) connected components { Ω i } i = 1 ℓ 0 , where int a −1(0) is the interior of the zero set a − 1 ( 0 ) = { x ∈ R N | a ( x ) = 0 } of a. When ρ is fixed either large in the L 2-subcritical case or small in the L 2-supercritical case, we construct a ℓ-bump (1 ⩽ ℓ ⩽ ℓ 0) positive normalized solution which is localised at ℓ prescribed components { Ω i } i = 1 ℓ for large λ. The asymptotic profile of the solution is also analysed through taking the limit as λ → +∞, and subsequently as ρ → +∞ in the L 2-subcritical case or ρ → 0+ in the L 2-supercritical case. In particular, we find ℓ-bump normalized solutions to the related Dirichlet problem 0\quad \text{for}\ i=1,\dots ,\ell .\hfill \end{aligned}\right.\end{equation*}?> − Δ v + μ v = | v | 2 σ v , v ∈ H 0 1 ( ∪ i = 1 ℓ Ω i ) , ∑ i = 1 ℓ ∫ Ω i v 2 = ρ , v | Ω i > 0 for i = 1 , … , ℓ .

We consider absorbing chemical reactions in a fluid flow modelled by the coupled advection–reaction–diffusion equations. In these systems, the interplay between chemical diffusion and fluid transportation causes the enhanced dissipation phenomenon. We show that the enhanced dissipation time scale, together with the reaction coupling strength, determines the characteristic time scale of the reaction.

As a toy model for understanding the soliton resolution phenomenon we consider a characteristic initial boundary value problem for the 4$d$ equivariant Yang-Mills equation outside a ball. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. In particular, within this approach it is evident that the endstate of evolution must be non-radiative (meaning vanishing flux of energy at future null infinity). In our toy model such non-radiative configurations are given by a static solution (called the half-kink) plus an alternating chain of $N$ decoupled kinks and antikinks. We show numerically that the configurations $N=0$ (static half-kink) and $N=1$ (superposition of the static half-kink and the antikink which recedes to infinity) appear as generic attractors and we determine a codimension-one borderline between their basins of attraction. The rates of convergence to these attractors are analyzed in detail., 15 pages, 7 figures

Considering age of infected cells and intracellular delay in both virus-to-cell and cell-to-cell transmissions, this paper develops an age-structured HIV model with delay to investigate the global threshold dynamics, which show the uninfected and infected steady states of the model. Using the Lyapunov function and LaSalle's invariance principle, we show that the global threshold dynamics of the model can be determined by utilizing basic reproduction number. Moreover, given that sudden environmental changes can lead to uncertainty in parameters of the model, a stochastic age-structured HIV model with Markovian switching is developed to study the finite-time contraction stability, which characterizes transmission properties of virus over a finite time. The sufficient conditions of the finite-time contraction stability are obtained by employing the Lyapunov function and stochastic comparison theorem. Numerical examples are presented to illustrate the theoretical results, and numerical results show that different noise intensity and delay affect stability of the HIV models.

Ruelle gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle's formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space-split sensitivity, or S3, algorithm. In S3, Ruelle's formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle's formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle's formula., Comment: v2: added case of higher dimensional unstable manifolds v3: fixed some mistakes, improvements in the exposition v4: Electronic copy of final peer-reviewed manuscript accepted for publication

We prove a duality between the asymmetric simple exclusion process (ASEP) with non-conservative open boundary conditions and an asymmetric exclusion process with particle-dependent hopping rates and conservative reflecting boundaries. This is a reverse duality in the sense that the duality function relates the measures of the dual processes rather than expectations. Specifically, for a certain parameter manifold of the boundary parameters of the open ASEP this duality expresses the time evolution of a family of shock product measures with $N$ microscopic shocks in terms of the time evolution of $N$ particles in the dual process. The reverse duality also elucidates some so far poorly understood properties of the stationary matrix product measures of the open ASEP given by finite-dimensional matrices., 40 pages

The Farley-Buneman and Gradient Drift instabilities have been investigated using a fluid model, in a partially ionized dusty electrojet region in which dust and neutral particles constitute a uniform static background. The effects of dissociative electron-ion recombination and dust charge fluctuation on the instabilities also have been taken into account. The electron-ion dynamics are considered to derive the perturbed densities which further lead to the generalized dispersion relation. The dispersion relation describes the propagation of electrojet instabilities having frequency within dust ion acoustic range in a magnetized partially ionized dusty plasma. The dispersion relation is separately solved numerically and analytically for the two values of anisotropy parameters which correspond to the two different altitudes in the electrojet region. It is found that Gradient drift instability is unstable at a much longer wavelength as compared to Farley-Buneman instability both with or without dust. At lower altitudes(90 km) the increase of negative charge on dust decreases the threshold electron drift velocity for Farley-Buneman instability while it shows the opposite behavior at higher altitudes(100 km). A much lower electron drift velocity is required to excite the Gradient drift instability than the Farley-Buneman instability at both altitudes. The dissociative electron-ion recombination damps both modes much more than the dust charge fluctuation. A significant changes in threshold drift velocity is observed for the Farley-Buneman mode as compared to the Gradient Drift mode due to the two main damping mechanisms. The present analysis is applicable in the lower ionospheric electrojet region where meteoric ablation processes are dominant.

A general entanglement-based witness of non-classicality has recently been proposed, which can be applied to testing quantum effects in gravity. This witness is based on generating entanglement between two quantum probes via a mediator. In this paper we provide a "temporal" variant of this witness, using a single quantum probe to assess the non-classicality of the mediator. Within the formalism of quantum theory, we show that if a system $M$ is capable of inducing a coherent dynamical evolution of a quantum system $Q$, in the presence of a conservation law, then $M$ must be non-classical. This argument supports witnesses of non-classicality relying on a single quantum probe, which can be applied to a number of open issues, notably in quantum gravity or quantum biology., 8 pages, 1 figure. J. Phys. A: Math. Theor. (2023)

Accurate near-surface characterization and velocity model building are important for a number of geotechnical and geophysical applications. 3D full waveform inversion can be used to generate a detailed velocity model, but must be provided with a good initial velocity model. A method for producing such an initial velocity model is explored in this paper. Assuming that the near-subsurface can be locally approximated by an effective flat-layered medium, a specialized form of acoustic full waveform inversion is proposed. The inversion is based on regularized least-squares in the Laplace-Fourier domain as a measure to mitigate the cycle-skipping problem. Forward modeling is carried out by solving a number of independent finite-difference problems in parallel, for a set of horizontal wavenumbers. The set is determined adaptively, using a pair of algorithms discussed in detail. A numerical implementation of the inverse Hankel transform, used to synthesize data in the frequency-offset domain, is also described. The forward modeling scheme is validated against analytic solutions of the Helmholtz equation, showing good agreement between the two. Full waveform inversion is tested by inverting a synthetic dataset consisting of flat layers with embedded velocity anomalies. Important features are recovered after inversion at a small number of complex frequencies, providing a detailed initial model for successive 3D full waveform inversion.

Memristors are often utilized in circuit model analysis as one of the fundamental circuit components. In this paper, a five-dimensional conservative memristor chaotic system is built after the introduction of the memristor into a four-dimensional conservative chaotic system. The dynamic changes of the system are examined using phase diagram, bifurcation diagram, and the Lyapunov exponent spectrum. A line equilibrium point, symmetry and multi-stability are characteristics of the system, the phase trajectory can also produce shrinking and structure transformation behavior with the change of parameters. Furthermore, the system has initial offset boosting behaviors, conservative flows of it can be altered in position by changing two initial values, respectively. Most notably, we discover that the complexity of the system rises with the inclusion of memristor and again with the addition of fractional differential operators. It is shown that the complexity of chaotic systems may increase with the addition of memristors and fractional-order differential operators. At last, the NIST is used to test the randomness of the sequence, and the system’s physical realizability is confirmed by the DSP platform.

Organic semiconductor CuPc thin films were deposited on different substrates with OAD CSP technique. The effects of substrate type, glancing angle and deposition temperature on surface and structural properties of CuPc thin films was reported in this study. The structural and morphological properties of CuPc thin films were investigated by XRD, RAMAN and FEG-SEM analysis. Dendritic nanostructures and nanoporous-straticulate sponge like structures were determined at surface area of CuPc thin films. XRD patterns showed that these CuPc thin films have orthorhombic α-phase structure with 2θ peak at 6.89˚, which is %100. The optical properties of all CuPc films which were determined from transmittance and reflectance measurements. The increasing of spray glancing angle and substrate temperature were decreased the UV-VIS absorption intensities at S-band and Q-band. However at 200°C and 250°C deposition temperatures at 45° spray glancing angle affected the Q-band absorption acerbically and red shift observed in here.

Laser shock peening has been widely used in anti-fatigue strengthening of metals. The different types of lasers (Flattop laser and Gaussian laser) used have influence on the pressure characteristics of shock wave and the strengthening effect. In this work, using PDV (photonic doppler velocimetry) system, the difference of peak pressure of Flattop laser-induced shock wave and Gaussian laser-induced shock wave is studied experimentally and theoretically. Research shows that the peak pressure of shock wave induced by Gaussian laser is higher than that of Flattop laser, because of the infinitesimal power density in the center of Gaussian laser spot is higher than that of Flattop laser, which also causes the Gaussian laser to break down the confinement layer more easily. Furthermore, at the same power density, Gaussian laser can implant a larger and deeper residual stress field in the material.

The high quality Ba2NdFeNb4O15 epitaxial films with thicknesses varying from 10 nm to 1μm have been fabricated by RF cathode sputtering. The films were characterized by the formation of orientational domains with 18.4° in-plane rotation and tetragonal unit cell. The films have a smooth surface, while the presence of orientational domains does not manifest itself in any way on the surface relief. The obtained films have no magnetic response due to the absence of magnetic inclusions of barium hexaferrite. The absence of any impurities has been confirmed by X-ray diffraction, atomic force microscopy and energy dispersive elemental analysis.

In this manuscript, we have theoretically studied the four level closed loop atomic systems in the presence of two vector beams. A spatially dependent transparency for the probe vector beam is obtained based on the semiclassical model. We have explicitly shown that the number of petals formed for probe absorption depends on the value of orbital angular momentum (OAM) of the constituting beams. A detailed study for absorption and dispersion of right circularly polarized (RCP) and left circularly polarized (LCP) components of the probe beam is carried out and the importance of the polarization state of the beams on Higher Order Poincare Sphere (HOPS) is highlighted. An explicit effect of the interferometer phase of the vector beam which is geometric in nature, is shown for probe beam response. Three types of four level closed loop atomic system is studied with particular emphasis given for double Λ and Diamond atomic system. A dark state analysis of the atomic system is carried out which facilitate a physical understanding of the obtained results. Our study has explored the effects of inhomogeneity in both polarization and intensity for probe and coupling beam in a closed loop atomic system which is phase dependent.

Pest issues have always been the focus of attention in agriculture. The integrated Pest Management(IPM) method is currently the most popular way to be applied for pest control. In this study, according to the IPM strategy, we regard pest quantity as a threshold index and extend the Leslie-Gower model into a non-smooth Filippov system through combining chemical and biological control. To maintain the pest population at or below the given economic threshold(ET), we investigate the global dynamics of the proposed model, including the existence of sliding mode and various equilibria, sliding dynamics and bifurcations, and global stability of equilibria. The result shows that desired equilibria can be globally stable under some conditions, meaning that our control tactics work. In particular, the case where our strategy fails to be effective arouses interest. In the end, the biological implications of the results are discussed and given in detail.

Van der Waals materials, such as α-phase molybdenum trioxide (α-MoO3), have promising prospects in modern optics technologies, such as nano-imaging, negative refraction, and infrared detection. Particularly, the natural hyperbolic properties of α-MoO3 make it an excellent candidate for perfect absorber. Here, we propose a design method for achieving broadband absorption based on van der Waals material (α-MoO3) in the mid-infrared band. The segmented cubic Hermite interpolation is used to generate various geometric structures. Numerical results show that the average spectral absorptance of the optimized structure is up to 0.993 in the wavelength range of 10.4-12.7 μm. The high absorption performance can be explained as the slow-light effect. The impact of incident angle on absorption performance is also investigated. Finally, we calculate the spectral absorptance of the proposed absorber when the crystal axes of α-MoO3 are rotated in the x-y plane. Our findings pave a novel path for designing broadband absorbers based on van der Waals materials, particularly in the mid-infrared band.

A many-body classical trajectory Monte Carlo (CTMC) method is applied in the study of scattering probabilities of electrons and positrons after interacting with a two-dimensional (2D) artificially generated, uniform lattice, composed of fixed protons. We used different lattice parameters between the protons for different simulations, where the projectiles have kinetic energies of 500 and 1000 eV. We found a very strong focusing of electrons and a very strong defocusing of the positrons at lower lattice parameters. Furthermore, we found, these effects get weaker with increasing lattice parameters. Interesting changes took place at lattice parameters 2 and 3 au. which are close to the lattice parameter between carbon atoms in graphene of value 2.68 au. We also performed a simulation of a defective lattice by removing some protons and noticed distinguishable changes in the spectra compared to the spectra of an ideal lattice. This comparison may open a way for the detection of lattice defects in real samples.

Numerical modelling on functional Sn-based perovskite solar cells (PSCs) has been performed and compared with Pb-based PSCs by using general-purpose photovoltaic device model software. The effect of variation in active layer thickness and various electron transport layers (ETLs), including tin oxide (SnO2), zinc oxide, C60, titanium dioxide, phenyl-C61- butyric acid methyl ester, on the photovoltaic parameters of Sn-based PSCs has been investigated. The active layer thickness was observed to be 500 nm, and SnO2 as ETL material resulted in the most efficient PSC. The optimized Sn-based device with formamidinium tin iodide as perovskite active layer shows promising results with a maximum power conversion efficiency of 24.41% compared to 27.49% for formamidinium lead iodide-based device. Further, other photovoltaic parameters for lead free PSC devices are quite comparable as for lead-based devices, showing the potential of Sn-based perovskite material as a fair candidate to replace toxic Pb-based-PSCs.