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

We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation $$\begin{aligned} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega }u,\quad (x,t)\in {\mathbb {R}}^{N}\times {\mathbb {R}}, \end{aligned}$$ where \(N=2\) or 3 and V(x) is an anisotropic harmonic potential. Here \(L_{\Omega }\) is the quantum mechanical angular momentum operator. We establish conditions

In this paper we study the maximum number of limit cycles of the discontinuous piecewise differential systems with two zones separated by the straight line \(y=0\), in \(y\ge 0\) there is a polynomial Hamiltonian system of degree m, and in \(y\le 0\) there is a polynomial Hamiltonian system of degree n. First for this class of discontinuous piecewise polynomial Hamiltonian systems, which are perturbation

In this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn

This paper provides an analysis on global dynamics of a diffusive cholera model. We formulate the model by a reaction–diffusion system with space-dependent parameters and bacterial hyperinfectivity, where susceptible and infectious humans disperse with distinct dispersal rates and the hyper-infectious and lower-infectious vibrios in contaminated water are assumed to be immobile in the domain. We first

In this paper, we prove the generic version of Cantor spectrum property for quasi-periodic Schrödinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency \(C^k\) long-range operators on \(\ell ^2({\mathbb Z}^d)\). These results are based on reducibility properties of finitely differentiable quasi-periodic \(SL(2,{\mathbb R})\) cocycles

We consider a class of one-dimensional absorbed diffusion processes with small singular noises that exhibit multi-scale dynamics in the sense that typical trajectories of the solution process first quickly approach to transient states, captured by quasi-stationary distributions (QSDs), then stay with transient states for a very long period, and finally deviate from transient states and slowly relax

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial

We consider the reducibility problem of quasi-periodic cocycles \((\alpha ,A)\) on \({\mathbb {T}}^d\times U(n)\) in \(C^k\) class, with k large enough and \(\alpha \) being a Diophantine vector. We show that if \((\alpha ,A)\) is conjugated to a constant cocycle \((\alpha ,C)\) via \((\theta ,X)\mapsto (\theta , B(\theta )X)\), with a map \(B:{\mathbb {T}}^d\rightarrow U(n)\) being measurable, then

The purpose of this paper is to present an example of an Ordinary Differential Equation \(x'=F(x)\) in the infinite-dimensional Hilbert space \(\ell ^2\) with F being of class \(\mathcal {C}^1\) in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator DF(0) intersects the half-plane \(\mathfrak {R}(z)>0\). The possible existence

In this paper, we investigate the asymptotic dynamics of a time-periodic parasitic–mutualistic model of mistletoes and birds in heterogeneous environment with the especial concerns over the spreading–vanishing scenarios, in which the Stefan class free boundary is introduced as the spreading frontier. By defining the ecological reproduction number and generalizing it as the spatial-temporal risk index

In this paper, we prove the existence of a random exponential attractor (a positively invariant, compact, random set with finite fractal dimension that attracts any trajectory exponentially) for the 3D non-autonomous damped Navier–Stokes equation with additive noise, which implies that the asymptotic behavior of solutions for the equation can be described by finite independent parameters. The key and

We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order \(n-1\) with \(n\ge 3\) and \(d\ge 2\). If \(\varepsilon \ll 1\) is the size of the initial datum, we prove that the lifespan \(T_\varepsilon \) of solutions is \(O(\varepsilon ^{-A(n-2)^-})\) where \(A\equiv A(d,n)= 1+\frac{3}{d-1}\) when n is even and \(A= 1+\frac{3}{d-1}+\max (\frac{4-d}{d-1}

For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.

In this paper, we show the existence of a Hopf bifurcation in a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the “local” Hopf bifurcation values over all

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier–Stokes equations, reaction-diffusion-advection systems,

We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on \({\mathbb {R}}\), we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality

We study the Cauchy problem for a class of nonlinear damped fractional Schrödinger type equation in a two dimensional unbounded domain. Then, we focus on long-time behaviour of the solutions proving that this behaviour is described by the existence of regular finite-dimensional global attractor in the energy space.

Our concern is to consider linear integral equations with two delays. A new result on necessary and sufficient conditions for the exponential stability is presented by careful study of the characteristic equation. Explicit expressions of limits of solutions in the critical case where integral equations lose exponential stability are also given by use of an asymptotic formula for solutions.

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus \(\mathbb T^d\), with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in \(H^s\) (for s large enough). More precisely, for any initial datum which is close to the invariant torus

In a previous paper we studied parametrized autonomous systems and gave a computable criterion that an approximate orbit connecting hyperbolic equilibria is shadowed by a true connecting orbit. This criterion was used to give rigorously verified examples of Shilnikov saddle-focus homoclinic orbits in three dimensions. This involved verifying a condition on the eigenvalues of the linearization at the

We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Our result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrödinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus.

In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger flow on a cylindrical domain \({\mathbb {R}}\times {\mathbb {T}}\). When the nonlinearity is of power type \(|\psi |^{p-1}\psi \) with \(p>1\), we show that there exists a critical frequency \(\omega _{p} >0\) such that the line standing wave is stable for \(0<\omega < \omega _{p}\)

In this paper we investigate the effect of nonlinear damping on the Lugiato–Lefever equation \(\mathrm {i}\partial _t a = -(\mathrm {i}-\zeta ) a - da_{xx} -(1+\mathrm {i}\kappa )|a|^2a +\mathrm {i}f\) on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping \(\kappa >0\) stationary spatially periodic solutions exist on branches that bifurcate from constant

In this paper, we consider dynamics defined by the Navier–Stokes equations in the Oberbeck–Boussinesq approximation in a two dimensional domain. This model of fluid dynamics involves fundamental physical effects: convection, and diffusion. The main result is as follows: local semiflows, induced by this problem, can generate all possible structurally stable dynamics defined by \(C^1\) smooth vector

This paper concerns the global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free boundaries. Given a suitable compatible condition for initial values, we establish the global well-posedness of solutions and provide some sufficient conditions for spreading and vanishing. When spreading occurs, we prove that the asymptotic spreading speed is uniquely

This paper is concerned with the spatio-temporal dynamics of nonnegative bounded entire solutions of some reaction–diffusion equations in \(\mathbb {R}^N\) in any space dimension N. The solutions are assumed to be localized in the past. Under certain conditions on the reaction term, the solutions are then proved to be time-independent or heteroclinic connections between different steady states. Furthermore

We consider the equation $$\begin{aligned} \Delta _x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots ,x_N)\in {{\mathbb {R}}}^N,\ y\in {{\mathbb {R}}}, \end{aligned}$$(1) where \(N\ge 2\) and f is a sufficiently smooth function satisfying \(f(0)=0\), \(f'(0)<0\), and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are

The time-dependent restricted \((n+1)\)-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by n primary bodies following a periodic solution of the n-body problem. We prove that the satellite has periodic solutions close to the large-amplitude circular orbits of the Kepler problem (comet solutions), and in the case that the primaries

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same spatial decay. In the presence of resonances the conjugations are to a polynomial normal form that also has decaying properties.

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carathéodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained

In Treschev (Proc Steklov Math Inst 310:262–290, 2020) we introduce the concept of a \(\mu \)-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from Treschev (2020) and present further results on the \(\mu \)-norm. More precisely, we specify three classes of unitary operators

We start to discuss some aspects of the scattering theory for the Sturm–Liouville operator \(L:\dfrac{1}{y}\left[ -D^2+q\right] \). In particular, we pose and solve the problem of reconstructing the function q when y is fixed and when a set \({\mathcal {S}}\) of scattering data is given. In the meanwhile, several relations concerning the spectral properties of L and the solutions of the related eigenvalue