The presence of slow–fast Hopf (or singular Hopf) points in slow–fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In De Maesschalck et al. (Canards from birth to transition, 2020), Wechselberger (Geometric singular perturbation theory beyond the standard form, Springer, New York,

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on \(\mathbb {T}^\infty

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source, $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in {\mathbb R}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in {\mathbb

Consider the Kirchhoff equation $$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$ on the d-dimensional torus \(\mathbb {T}^d\). In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads

In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of \(\mathbb {K}^n (\mathbb {K}={\mathbb {C}}\text {~or~} {\mathbb {R}})\). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp.

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space \(\mathbb {R}^n\), where \(n \ge 2\), assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as \(|x| \rightarrow \infty \). Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We

In this paper, we propose a mild condition, named Condition \((**)\), for collections of sequence of integers and show that for any measure preserving system the Pinsker \(\sigma \)-algebra is a characteristic \(\sigma \)-algebra for the averages along a collection satisfying Condition \((**)\). We introduce the notion of \(\Delta \)-weakly mixing subsets along a collection of sequences of integers

We study existence and multiplicity of positive ground states for the scalar curvature equation $$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$ when the function \(K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+\) is bounded above and below by two positive constants, i.e. \(0<\underline{K} \le K(r) \le \overline{K}\) for every

We consider a periodic function \(p:{\mathbb {R}}\rightarrow {\mathbb {R}}\) of minimal period 4 which satisfies a family of delay differential equations $$\begin{aligned} x'(t)=g(x(t-d_{\Delta }(x_t))),\quad \Delta \in {\mathbb {R}}, \end{aligned}$$(0.1) with a continuously differentiable function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and delay functionals $$\begin{aligned} d_{\Delta }:C([-2

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that

Let \(h:V\subset {\mathbb {R}}^{2}\longrightarrow {\mathbb {R}}^{2}\) be an embedding. The aim of this paper is to analyze the dynamical behavior of h depending on the number of fixed points and 2-cycles, their local behaviors and the features of V. Our approach allows us to extend some celebrated results of the theory of monotone flows, namely the order interval trichotomy, for non-monotone maps.

We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known

Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. In addition we show

We study a Lotka–Volterra competition–diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Some results have been obtained previously for some cases for the diffusion rates and competitions rates, and in this paper we continue to explore the remaining complementary case for the spatial dynamics of the system. Our main result in this paper reveals an

The Cauchy problem in \({\mathbb {R}}^n\), \(n\ge 1\), for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$ is considered in the strongly degenerate regime \(p\ge 1\). The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback

We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential

This paper establishes the global stability for the 2D Boussinesq system with partial dissipation and horizontal thermal diffusion. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. We extend the \(H^1\)-stability in [7] to \(H^2\)-stability which we care about and obtain the large time behavior of the linearized system.

For each given \(n \ge 2\), we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body

The mass-in-mass (MiM) lattice consists of an infinite chain of identical beads that are both nonlinearly coupled to their nearest neighbors and linearly coupled to a distinct resonator particle; it serves as a prototypical model of wave propagation in granular crystals and metamaterials. We study traveling waves in an MiM lattice whose bead interaction is governed by the cubic Fermi–Pasta–Ulam–Tsingou

We first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of the paper is to study the 2-body problem on a surface

This paper is devoted to studying wave breaking phenomena for the Fornberg–Whitham equation. Making use of the structure and the conservation quantity of the Fornberg–Whitham equation, we give a sufficient condition on the initial data to ensure wave breaking.

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different

Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed differential equation has a solution of the same kind. By a suitable, nonlinear, change of coordinates these kind of solutions are associated to homoclinic solutions to a fixed point of an ordinary differential equation with a one-dimensional centre manifold

We study a quasi-one-dimensional classical Poisson–Nernst–Planck model for ionic flow through a membrane channel with two positively charged ion species (cations) and one negatively charged, and with zero permanent charges. We treat the model problem as a boundary value problem of a singularly perturbed differential system. Under the framework of the geometric singular perturbation theory, together

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate \(\frac{kIS}{1+\beta I+\alpha I^2}\) (\(\beta >-2 \sqrt{\alpha }\) such that \(1+\beta I+\alpha I^{2}>0\) for all \(I\ge 0\)) is considered in this paper. It is shown that the basic reproduction number \(R_0\) does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold

Most epidemiological models for mosquito-borne disease assume exponentially distributed residence times in disease stages in order to simplify the model formulation and analysis. However, these models do not allow for accurate description of the interaction between drug concentration and pathogen load within hosts. To improve this, we formulate a model by considering arbitrarily distributed sojourn

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index \(c>0\) and homoclinic tangencies of arbitrary codimension \(c>0\). These type of sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite, we also construct robust homoclinic tangencies

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured

We consider a scalar delay differential equation \(\dot{x}(t)=-dx(t)+f((1-\alpha )\rho x(t-\tau )+\alpha \rho x(t-2\tau ))\) with an instant mortality rate \(d>0\), the nonlinear Rick reproductive function f, a survival rate during all development stages \(\rho \), and a proportion constant \(\alpha \in [0, 1]\) with which population undergoes a diapause development. We consider global continuation

In this paper we focus on a class of symmetric vector fields in the context of singularly perturbed fast-slow dynamical systems. Our main question is to know how symmetry properties of a dynamical system are affected by singular perturbations. In addition, our approach uses tools in geometric singular perturbation theory [8], which address the persistence of normally hyperbolic compact manifolds. We

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite

In this paper, the speed sign of the traveling wave to the bistable Lotka–Volterra competitive lattice system is investigated via the upper–lower solution method as well as the comparison principle. We provide an interval estimation for the bistable speed firstly. Two comparison principles are further established to obtain new conditions to the determinacy of the sign of the bistable speed. To our

For the curved n-body problem in \(\mathbb {S}^3\), we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium if and only if n is odd and the masses are equal. The equilibrium is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on \(\mathbb {S}^1\) embedded in \(\mathbb {S}^2\). We then study the stability

In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation, $$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$(1) where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}})\) is a given probability space, \(\theta _t\) is an ergodic metric dynamical system on

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \end{aligned}$$(*) on the unit interval \(0< x<1\) with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria \(v\in {\mathcal {E}}\). The global attractor \({\mathcal {A}}\) of (*), also called

We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy

We study a free boundary problem for Fisher-KPP equation: \(u_t=u_{xx}+f(u)\) (\(g(t)< x < h(t)\)) with free boundary conditions \(h'(t)=-u_x(t,h(t))-\beta \) and \(g'(t)=-u_x(t,g(t))-\alpha \) for \(\alpha >0\) and \(\beta \in \mathbb {R}\). Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. \(\beta >0\) means that there is

A compact space X is said to be minimal if there exists a map \(f:X\rightarrow X\) such that the forward orbit of any point is dense in X. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha and Tywoniuk (J Dyn Differ Equ, 29:243–257, 2017) on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha

In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen’s formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein–Weiss lemma is valid for all

We show that for any uniformly bounded in time \(H^1\cap L^1\) solution of the dispersion generalized Benjamin–Ono equation, the limit infimum, as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order \(t/\log t\). This result is in accordance with the one established by Muñoz and Ponce (Proc Am Math Soc 147(12):5303–5312, 2019) for solutions of the Benjamin–Ono

We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected media with Dirichlet conditions outside the domain. After showing the existence and uniqueness of a non-trivial bounded stationary state \(n_+\), we prove that it invades the unstable state zero exponentially fast in time.

We improve the Gagliardo–Nirenberg inequality $$\begin{aligned} \Vert \varphi \Vert _{L^q({\mathbb {R}}^n)} \le C \Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)} {\mathcal {L}}^{-(\frac{1}{q} - \frac{n-2}{2n})} (\Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)}), \end{aligned}$$ \(r=2\), \(0

We are interested in the global-in-time existence results for quasilinear wave equations satisfying null condition in one dimension. The main difficulty is that waves do not decay in one dimension space, and we rely on the new type of weighted energy estimates introduced in Luli et al. (Adv Math 329:174–188, 2018), where the authors proved that small data lead to global-in-time solutions for semilinear

We consider the nonlinear Schrödinger equation on \({\mathbb R}^N \), \(N\ge 1\), $$\begin{aligned} \partial _t u = i \varDelta u + \lambda | u |^\alpha u \end{aligned}$$ with \(\lambda \in {\mathbb C}\) and \(\mathfrak {R}\lambda >0\), for \(H^1\)-subcritical nonlinearities, i.e. \(\alpha >0\) and \((N-2) \alpha < 4\). Given a compact set \(K \subset {\mathbb {R}}^N \), we construct \(H^1\) solutions

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The

The original version of this article unfortunately contained a mistake in Eq. 1.4. The correct equation is given below.

In this paper, we will define the index pair \((i_A(B),\nu _A(B))\) by the dual variational method, and show the relationship between the indices defined by different methods. As applications, we apply the index \((i_A(B),\nu _A(B))\) to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations.

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary

About a fixed point the action of a continuous map can be decomposed into a rotation and an expansion. Here it is shown how the stability of a fixed point can be determined by averaging the expansion over invariant measures of the rotation. This is useful for fixed points on switching manifolds of piecewise-smooth maps where stability cannot be determined in the usual way from the eigenvalues of a

This paper deals with a two-competing-species chemotaxis-fluid system with two different signals$$\begin{aligned} \left\{ \begin{aligned}&(n_{1})_{t}+\mathbf{u }\cdot \nabla n_{1}=d_{1}\Delta n_{1}-\chi _{1}\nabla \cdot (n_{1}\nabla c) +\mu _{1} n_{1}(1-n_{1}-a_{1}n_{2}),&\text {in}\; \Omega \times (0,\infty ), \\&c_{t}+\mathbf{u }\cdot \nabla c=d_{2}\Delta c-\alpha _{1} cn_{2},&\text {in}\; \Omega

When the Favard separation condition fails, a linear almost periodic equation possessing bounded solutions may have no almost periodic solutions, or equivalently, no continuous solutions in hull. Almost automorphic solutions are however known to persist and, in the scalar case, the same happens to semi-continuous solutions in the hull. The aim of the present paper is twofold: extending the second type

A time-dependent determining wavenumber was introduced in Cheskidov and Dai (Phys D Nonlinear Phenom 376–377:204–215, 2018) to estimate the number of determining modes for the surface quasi-geostrophic equation. In this paper we continue this investigation focusing on the subcritical case and study trajectories inside an absorbing set bounded in \(L^\infty \). Utilizing this bound we find a time-independent

We give sufficient conditions for the boundedness of the blow-up set and no boundary blow-up for type I blowing up solutions to a nonlinear parabolic system with unequal elliptic operators. We introduce a new method to simplify the nonlinear parabolic system into a scalar nonlinear parabolic inequality and investigate qualitative properties of the blow-up set.

Due to the intractability of the Navier–Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-\(\alpha \) equation (which replaces the solution u with \((1-\alpha ^2{\mathcal {L}}_2)u\) for a Fourier Multiplier \({\mathcal {L}}_2\)) and the generalized Navier–Stokes equation (which replaces the viscosity term \(\nu \triangle \) with \(\nu {\mathcal

In order to prove numerically the global existence and uniqueness of smooth solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori upper bounds on the supremum of the numerical range of the linearized operator. These bounds also have to be easily computable in order to be applicable to our rigorous a-posteriori methods, as we use them in each time-step of the numerical discretization

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations—ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse–Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still