For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the n-body is a (constant up to rotations and scalings) central configuration. For \(d\le 3\), the only possible homographic motions are those given

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant

Singular exponential nonlinearities of the form \(e^{h(x)\epsilon ^{-1}}\) with \(\epsilon >0\) small occur in many different applications. These terms have essential singularities for \(\epsilon =0\) leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization

The dynamics of a general reaction–diffusion–advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay \(\tau \) as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution

In this paper, we consider the initial-boundary value problem of Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{aligned}&n_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla n^{\kappa } =\varDelta n^{\kappa } - \nabla \cdot \big (n^{\kappa }(1+n^{\kappa })^{-\alpha }\nabla c^{\kappa } \big ),&\qquad x\in \varOmega ,\,t>0, \\&c_t^{\kappa } +\mathbf{u}^{\kappa } \cdot \nabla c^{\kappa

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar

We analyze the long-time behavior of the semigroup S(t) generated by a heat equation with degenerate past history and a nonlinear heat supply posed in a three dimensional bounded domain \(\Omega \). Assuming that the degeneracy occurs in a positive measure subset of \(\Omega \), we prove the existence and regularity of the global attractor associated to this semigroup.

In this paper, the multi-dimensional (M-D) isentropic compressible Navier–Stokes equations with degenerate viscosities (ICNS) is considered in the whole space. We show that for a certain class of initial data with local vacuum, the regular solution of the corresponding Cauchy problem will blow up in finite time, no matter how small and smooth the initial data are. It is worth pointing out that local

This paper deals with the limiting behavior of invariant measures of the stochastic delay lattice systems. Under certain conditions, we first show the existence of invariant measures of the systems and then establish the stability in distribution of the solutions. We finally prove that any limit point of a tight sequence of invariant measures of the stochastic delay lattice systems must be an invariant

This paper is devoted to the synchronization of stochastic lattice dynamical systems driven by fractional Brownian motion with Hurst parameter \(\frac{1}{2}

In this paper, we extend the classical atmospheric Ekman equations to discontinuous differential equations of Ekman flows by considering the eddy viscosity with piecewise form. Local and global analysis for the corresponding discontinuous ODEs are studied respectively. In the section of local analysis, all the possible crossings of solutions are presented over the certain discontinuity linear subspaces

This work deals with a general cross-diffusion system modeling the dynamics behavior of two predators and one prey with signal-dependent diffusion and sensitivity subject to homogeneous Neumann boundary conditions. Firstly, in light of some \(L^p-\)estimate techniques, we rigorously prove the global existence and uniform boundedness of positive classical solutions in any dimensions with suitable conditions

The paper is concerned with the exponential attractors for the viscoelastic wave model in \(\varOmega \subset \mathbb R^3\): $$\begin{aligned} u_{tt}-h_t(0)\varDelta u-\int _0^\infty \partial _sh_t(s)\varDelta u(t-s)\mathrm ds+f(u)=g, \end{aligned}$$ with time-dependent memory kernel \(h_t(\cdot )\) which is used to model aging phenomena of the material. Conti et al. (Am J Math 140(2):349–389, 2018a;

This paper concerns the effect of the (separated/connected) protection zone for the evolution of an endangered species on the reaction–diffusion equation with the strong Allee effect and the free boundary. First, we describe the long-time dynamical behavior of the system of two types protection zones with the same length. Furthermore, the asymptotic profiles of solutions and the asymptotic spreading

This paper focuses on the vector fields on the quasi-periodically forced circle flow $$\begin{aligned} \left\{ \begin{array}{l} {\dot{\varphi }}=\rho +f(\varphi ,\theta ),\\ \\ {\dot{\theta }}=\alpha \end{array} \right. \end{aligned}$$ where the forcing term f is real analytic in its arguments and small enough, the frequency vector \( \alpha \in {\mathbb {R}}^d (d\ge 2)\) is beyond Brjuno frequency

In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will prove the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing’s equation with an almost periodic external force.

We examine how climate change enhances the spreading of an invading species through a nonlinear diffusion equation of the form \(u_{t}=du_{xx}+A\left( x-ct\right) u-bu^{2}\) with a free boundary, where climate change causes more favourable environment shifting into the habitat of the species with a constant speed \(c>0\). The free boundary represents the invading front of the expanding population range

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov–Schmidt reduction together

A non-monotone time-delayed lattice equation in discrete periodic habitat is concerned. As a fundamental step, a principle eigenvalue problem is introduced, which viewed as a function on a parameter is proved to be strictly convex, piecewise monotone and positive. With the help of such properties, the existence of pulsating waves is obtained by first constructing super and sub solutions, then using

Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton’s inverse square law, that is under a ’cotangent’ potential. When the distance between the bodies is sufficiently small, the reduced equations of motion may be seen as a perturbation of an integrable system. We take suitable action-angle coordinates to average these perturbing

We consider one-dimensional Schrödinger operators with almost periodic potentials and \(\delta \)-interactions supported on an almost periodic point set and with almost periodic coefficients. For operators of this kind we introduce a rotation number in the spirit of Johnson and Moser.

In this work, we deal with the initial value problem of the 5th-order Gardner equation in \({\mathbb {R}}\), presenting the local well-posedness result in \(H^2({\mathbb {R}})\). As a consequence of the local result, in addition to \(H^2\)-energy conservation law, we are able to prove the global well-posedness result in \(H^2({\mathbb {R}})\). Finally as a direct application, we prove that some globally

We consider the propagation dynamics of a general heterogeneous reaction-diffusion system under a shifting environment. By developing the fixed-point theory for second order non-autonomous differential system and constructing appropriate upper and lower solutions, we show there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed. We further show the uniqueness

We study the continuity of pullback random attractors \(A_\lambda \), where the parameter belongs to a complete metric space \(\Lambda \). By a Hausdorff sub-norm space, we mean the collection of all nonempty compact subsets of the state space, equipped by the Hausdorff metric as well as a sub-norm. Under weaker conditions, we prove that the binary map \((\lambda , s)\rightarrow A_\lambda (s,\theta

This article studies the Stochastic Degasperis-Procesi equation on \( \mathbb {R}\) with an additive noise. Applying the kinetic theory, and considering the initial conditions in \(L^2(\mathbb {R})\cap L^{2+\delta }( \mathbb {R})\), for arbitrary small \(\delta >0\), we establish the existence of a global pathwise solution. Restricting to the particular case of zero noise, our result improves the deterministic

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation $$\begin{aligned} u_{tt}(t)+ Au(t) -\int ^\tau _0 k(s) Au(t-s)ds+ f(u)=g(t), \ \tau \in \{t, \infty \}, \end{aligned}$$ with finite (\(\tau =t\)) or infinite (\(\tau =\infty

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems (Cornwell and Jones in SIAM J Appl Dyn Syst 17(1):754–787, 2018). In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These

This paper is concerned with the nonlocal dispersal problem in inhomogeneous media. Our goal is to show the limiting behavior of perturbation equation with parameters. By analyzing the asymptotic behavior of solutions when the parameter is small, we find that convection appears in inhomogeneous media. Moreover, if the effect of inhomogeneous media changes, then we prove a convergence result that convection

Given an n-dimensional, connected, analytic manifold M and analytic vector fields \(\{f_i\}_{i=0}^m\) on M consider the affine control process $$\begin{aligned} {\xi }' = f_0(\xi ) + \sum \limits _{i = 1}^mu_if_i(\xi )\,. \end{aligned}$$(0.2) Let \(t\rightarrow \xi _{x,\overline{u}}(t)\) be the solution curve satisfying the above differential equation with initial condition \(\xi (0) = x\) and \(\overline{u}

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$\begin{aligned} \partial ^2_tu(t,x)- a(x,\lambda )^2\partial _x^2u(t,x)= b(x,\lambda ,u(t,x),u(t-\tau ,x),\partial _tu(t,x),\partial _xu(t,x)), \; x \in (0,1) \end{aligned}$$ with smooth coefficient functions a and b such that \(a(x,\lambda )>0\) and \(b(x,\lambda ,0,0,0,0) = 0\) for all

Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities

In this paper, a single reaction-diffusion population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of super and subsolutions, which is linearly stable for relatively small memory-induced diffusion. However, after the memory-induced diffusion

A correction to this paper has been published: https://doi.org/10.1007/s10884-021-10016-2

This paper considers small perturbations of an integrable reversible system which has a degenerated lower dimensional invariant torus in some sense. In the presence of some higher-order terms, by some KAM technique and the stability of critical points of real analytic functions developed for hamiltonian systems, we prove the persistence of the degenerate lower dimensional invariant torus with prescribed

In this paper, a neutral functional differential equation with multiple delays is considered. In a first step, we assumed some sufficient hypotheses to guarantee the existence of the Bogdanov–Takens and the triple-zero bifurcations. In a second step, the normal form of the two bifurcations is obtained by using the reduction on the center manifold and the theory of the normal form. Finally, we applied

We address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain \([0,L_1]\times [0,L_2]\) with initial data satisfying \(\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}\), where C is a constant. We prove that the global solution exists under the condition \(L_2\le 1/C L_1^{3/5}\), improving earlier results. The solutions are smooth and decrease energy until they

In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general

The “N-Body Problem” refers to the study of the dynamics of a system of N point masses, moving under their mutual gravitational attraction. For such a system, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets \({\mathfrak {M}}(c,h)\) of these conserved quantities are parameterized by the angular momentum c and the energy

First some sufficient conditions are presented for the existence and the construction of pullback exponential attractors for infinite dimensional non-autonomous dynamical systems with delays. This abstract result is then used to establish the existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neural networks with discrete and distributed time-varying

A correction to this paper has been published: https://doi.org/10.1007/s10884-021-09990-4

In this paper, we propose a method to handle the problem of complete degeneracy in the study of lower dimensional invariant tori surviving from destructed resonant torus. As its application, we obtain a different proof of the existence of co-dimension one invariant tori. No other condition is assumed on the perturbation except for the smallness.

We consider a nonnegative potential W that vanishes on a finite set and study the existence of periodic orbits of the equation $$\begin{aligned} \ddot{u}=W_u(u),\;\;t\in {\mathbb {R}}, \end{aligned}$$ that have the property of visiting neighborhoods of zeros of W in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable \(x=\epsilon t\), \(\epsilon

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form \({\mathbb {R}}^N\setminus K\), where \(K\subset {\mathbb {R}}^N\) is a compact “obstacle”. The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this

We consider a hyperbolic quasilinear version of the Navier–Stokes equations in \({\mathbb {R}}^2\), arising from using a Cattaneo type law instead of a Fourier law. These equations, which depend on a parameter \(\varepsilon \), are a way to avoid the infinite speed of propagation which occurs in the classical Navier–Stokes equations. We first prove the existence and uniqueness of solutions to these

We prove the existence of domain walls for the Bénard–Rayleigh convection in the case of “rigid–free” boundary conditions. In the recent work (Haragus and Iooss in Arch Ration Mech Anal 239:733–781, 2021), we studied this bifurcation problem in the cases of “rigid–rigid” and “free–free” boundary conditions. In the three cases, for the existence proof we use a spatial dynamics approach in which the

We study the problem of existence/nonexistence of limit cycles for a class of Liénard generalized differential systems in which, differently from the most investigated case, the function F depends not only on x but also on the y-variable. In this framework, some new results are presented, starting from a case study which, actually, already exhibits the most significant properties. In particular, the

We consider the Euler–Korteweg system with space periodic boundary conditions \( x \in {\mathbb {T}}^d\). We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous

The applicability of Bowen’s equation to an arbitrary subset Z of a compact metric space has been well-studied by Climenhaga (Ergod Theory Dyn Syst 31(4):1163–1182, 2011). This paper aims to generalize the main results obtained by Climenhaga to free semigroup actions. To this end, we adopt the Carathéodory–Pesin structure (C–P structure) and introduce the notions of the topological pressure and lower

The work is about multiscale stochastic dynamical systems driven by Lévy processes. We prove that such a system can be approximated by a low-dimensional system on a random invariant manifold, and the original filter can be also approximated by the reduced low-dimensional filter. Finally, we investigate the reduction for \(\varepsilon =0\) and obtain that these reduced systems does not approximate these

The current paper is concerned with the forced waves of Keller–Segel chemoattraction systems in shifting environments of the form, $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=u_{xx}-\chi (uv_x)_x +u(r(x-ct)-bu),\quad x\in \mathbb {R}\\ 0=v_{xx}- \nu v+\mu u,\quad x\in \mathbb {R}, \end{array}\right. } \end{aligned}$$(0.1) where \(\chi \), b, \(\nu \), and \(\mu \) are positive constants, \(c\in

we investigate the existences of KAM tori for the infinite dimensional Hamiltonian system with finite number of zeros among normal frequencies. By constructing a constant vector we show that, for “most” tangent frequencies in the sense of Lebesgue measure, either if the vector is zero, there is a KAM torus or if the vector is not zero, there is no KAM torus in some domain. As an application, we show

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem $$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$ where \(\lambda >0\) is a bifurcation parameter and \(L>0\)

We characterize a criterion for the existence of relaxation oscillations in planar systems of the form $$\begin{aligned} \frac{du}{dt}= u^{k+1} g(u,v,\varepsilon ), \qquad \frac{dv}{dt}=\varepsilon f(u,v,\varepsilon ) + u^{k+1} h(u,v,\varepsilon ), \end{aligned}$$ where \(k\ge 0\) is an arbitrary constant and \(\varepsilon >0\) is a sufficiently small parameter. Taking into account of possible degeneracy

We consider a system of first order coupled mode equations in \({\mathbb {R}}^d\) describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree \(f :X\rightarrow Y\) between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over Y. If the structure groups of f are connected then the fibers are (uniformly) isomorphic (in a strong sense) to

We study a discrete dynamical Schrödinger bridge problem (SBP) as a dynamical variational problem on a finite graph. We prove that the discrete SBP exists a unique minimizer, which satisfies a boundary value Hamiltonian flow on probability simplex equipped with \(L^2\)-Wasserstein metric. In our formulation, we establish the connection between discrete SBP problems and Hamiltonian flows.

Given a possibly discontinuous, bounded function \(f:{{\mathbb {R}}}\mapsto {{\mathbb {R}}}\), we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE \(\dot{x} = f(x)\). The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.

We establish a general version of the Siegel-Sternberg linearization theorem for ultradiffentiable maps which includes the analytic case, the smooth case and the Gevrey case. It may be regarded as a small divisior theorem without small divisor conditions. Along the way we give an exact characterization of those classes of ultradifferentiable maps which are closed under composition, and reprove regularity