The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the \(x\partial _{x}+{\sum }_{i=1}^{n}a_{i}(x,\mathbf {z})\partial _{z_{i}}\), where ai(x,z) is a germ of analytic function with ai(x,0) = 0. We prove, under some hypothesis, that these germs of singular foliations are

This paper deals with a class of fast diffusion p-Laplace equation with logarithmic non-linearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates and some ordinary differential inequalities, we study the conditions on extinction and non-extinction of global solutions. The results of this paper extend and complete the previous studies on this equation

A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed

Combining the method of upper and lower solutions with the topological degree, we show the existence of solutions satisfying periodic or Dirichlet boundary conditions, for the differential inclusion $$(\phi(x^{\prime}(t)))^{\prime} \in F(t,x(t)),$$ where F(⋅,⋅) is a compact lower semi-continuous multi-valued map and ϕ is an homeomorphism.

In this manuscript, the approximate controllability is encountered for a system involving instantaneous and non-instantaneous impulses. The theory of semigroup and Sadovskii’s fixed point theorem are used to prove our main results. In the direction of controllability for a functional differential equations, a problem consists of either instant impulses or non-instant impulses separately. This is the

A Correction to this paper has been published: https://doi.org/10.1007/s10883-021-09543-4

In this paper we extend the definitions of mean dimension and metric mean dimension for non-autonomous dynamical systems. We show some properties of this extension and furthermore some applications to the mean dimension and metric mean dimension of single continuous maps.

In this paper, we propose dynamical systems for solving variational inequalities whose mapping is paramonotone, strongly pseudomonotone or pseudomonotone and Lipschitz continuous, respectively. Solutions of these dynamical system are shown to converge to a desired solution of the variational inequalities. When the mapping is strongly pseudomonotone but is not necessarily Lipschitz continuous, the convergence

The Mangasarian-Fromovitz constraint qualification has played a fundamental role in mathematical programming problems involving inequality constraints. It is known to be equivalent to a normality condition (in terms of the positive linear independence of active gradients) which, in turn, implies regularity (the tangent and the linearizing cones coincide), a condition which has been crucial in the derivation

We classify the phase portraits in the Poincaré disc of the differential equations of the form \(x^{\prime } = -y + x f(x,y)\), \(\dot y =x + y f(x,y)\) where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2,3,4,5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2. All our results have been revised by the program P4; see

Our main aims in this paper are to investigate the regularity of inertial manifolds for non-autonomous semi-linear evolution equations and to give an application of inertial manifolds to a feedback control problem. We first prove that the inertial manifolds are smooth if the nonlinear term is smooth. Then, using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.

In this paper, we give the different topological types of phase portrait for Liénard system \(\dot {x}=y, \dot {y}=-g(x)\) in the case that \(\deg g(x)=7\) and the system have six and seven singular points, respectively. For its perturbed system, the expansion of the Melnikov function near any of the above closed orbits, except that the closed orbit is a compound loop passing through a nilpotent cusp

In this paper, various properties of non-autonomous dynamical systems having periodic shadowing property and local weak specification property are studied. The main theorem gives an interrelation among the shadowing property, periodic shadowing property and local weak specification property of an expansive non-autonomous system.

On an elliptic billiard, we study the set of the circumcenters of all triangular orbits and we show that this is an ellipse. This article follows Romaskevich (L’Enseig Math 60:247–255, 2014), which proves the same result with the incenters, and Glutsyuk (Moscow Math J 14:239–289, 2014), which among others, introduces the theory of complex reflection in the complex projective plane. The result we present

This paper deals with the optimal feedback control problem for the modified Kelvin-Voigt model. The considered model describes the motion of weakly concentrated aqueous polymer solutions. In our case, the control function (the external force) depends on the velocity of the fluid. In such a way, the control is not selected from a finite set of available controls, but belongs to the image of some multi-valued

The paper is devoted to the systems of equations \(A(x) \dot x = v(x)\) in real finite-dimensional phase space. The elements of the matrix A are real analytic functions, as well as the components of the vector function v. Generically, the matrix A degenerates on a certain hypersurface \({\Gamma } = \{ \det A(x)= 0\}\). Points of Γ are called singular (or impasse) points of the system. The local analytic

We consider a natural mechanical system on a Finsler manifold and study its curvature using the intrinsic Jacobi equations (called Jacobi curves) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential

In this paper, we consider planar magnetohydrodynamics (MHD) system when the viscous coefficients and heat conductivity depend on specific volume v and temperature 𝜃. For technical reasons, the viscous coefficients and heat conductivity are assumed to be proportional to h(v)𝜃α where h(v) is a non-degenerate smooth function satisfying some additional conditions. We prove the existence and uniqueness

In this paper, we continue the study of abelian semigroup actions of several stronger versions of sensitivity, such as syndetically sensitive, thickly sensitive and thickly syndetically sensitive. We derive some sufficient conditions for a dynamical system to have these sensitivities. Also, we prove that the minimal and sensitive system is syndetically sensitive and non-minimal M-system is thickly

We prove that, for any Fuchsian system of differential equations on the Riemann sphere, there exists a rational matrix function whose partial indices coincide with the splitting type of the canonical vector bundle induced from the Fuchsian system. From this, we obtain solution of the Riemann-Hilbert boundary value problem for piecewise constant matrix function in terms of holomorphic sections of vector

In this paper, we study the creation of zeros in a certain type of families of functions. The families studied are given by the difference of two basic functions with a translation made in the argument of one of these functions. The problem is motivated by applications in the 16th Hilbert problem and its ramifications. Here, we apply the results obtained to the study of bifurcations of critical periods

We consider a complex differential system with a weak saddle at the origin and we characterize the existence of a local analytic first integral around the weak saddle. If the system does not have a fixed degree and instead the degree is arbitrarily large, the family can have a numerable infinite number of integrability cases.

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely,

The main purpose of this paper is to establish the machinery for doing homotopy of (regular) trajectories of control systems. In a mildly different setting than our earlier work in Colonius et al. (J Differ Equ. 2005; 216:324–53), we require this time two trajectories of a (conic) control system to be homotopic by means of their control parameters and simply call them control homotopic. More precisely

Non-local properties of non-linear systems, linear with respect to control function, are studied. We consider a class of such systems that are homogeneous at the origin. At a point different from the origin the system may be non-homogeneous, so its homogeneous approximation is of interest. However, at such points, the system may become less singular than at the origin. We describe a set of points where

In this paper, we follow an approach which considers the entropy of a dynamical system as a linear operator. We assign a linear operator on Lp spaces using a kernel entropy function. The case p = 2 is of special interest, since we may relate the entropy of the system in terms of the eigenvalues of the operator. The special case p = 1 also results in a local entropy map.

In the present paper, we obtain explicit formulae for geodesics in some left–invariant sub–Finsler problems on Heisenberg groups \(\mathbb {H}_{2n+1}\). Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2–dimensional sets, all left–invariant sub–Riemannian structures

The original version of this article unfortunately contained a mistake.

For a complete noncompact Riemannian manifold with bounded geometry, we prove a “generalized” compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extend previous results contained in Nardulli (Asian J Math 18(1):1–28, 2014), in such a way that the main theorem is a generalization of

A hydrodynamic-type system taking into account nonlocal effects is investigated. The exact traveling wave solutions including smooth solitary waves solutions, pseudo-peakons, periodic peakons, compactons, kink, and anti-kink wave solutions and so on are derived via the method of dynamical systems and the theory of singular traveling wave systems. It is worth pointing out that the uncountably infinitely

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint

In this paper, we study the limit behavior of trajectories of a nonlinear and discontinuous model of opinion dynamics based on the notion of bounded confidence. This model was previously studied in the case where the influence function has the form i(v) = v. It was shown that, under a particular condition on parameters of the system, any its trajectory tends to a fixed point. In this paper, we prove

The upper bound on the degrees of irreducible Darboux polynomials associated to the ordinary differential equations \( x_{tt}+\varepsilon {x_{t}}^{2}+\eta x_{t}+f(x)=0 \) with \( f(x)\in \mathbb {C}[x]\setminus \mathbb {C} \) and ε ≠ 0 is derived. The availability of this bound provides the solution of the Poincaré problem. Results on uniqueness and existence of Darboux polynomials are presented. The

In this paper, when the predator breeds only by the prey, the relationship among the two classes of prey (susceptible and infected prey) and the predator is represented by using an epidemic model with nonlinear incidence and the response function depending on the density of the three individuals under homogeneous Dirichlet boundary conditions describing a hostile environment at its boundary. For this

A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In these cases, the envelopes are always fronts. We investigate singular points of envelopes for first-order ordinary differential

One of the singularities of the convex hull of a generic hypersurface in \(\mathbb {R}^{4}\) leads to a generic sewing of two famous surfaces, the swallowtail and the Whitney umbrella, along their self-intersection lines. We prove that germs of all such sewings at the common endpoint of the self-intersection lines are diffeomorphic to each other with respect to diffeomorphisms of the ambient space

This paper is concerned with the (p, q)-Kirchhoff type equations, for which the existence of infinitely many high energy solutions is proved by employing the symmetric mountain pass lemma. Furthermore, an \(L^{\infty }\) estimate for the solutions is given by applying the Moser iteration technique.

Global solvability of a boundary value problem for a generalized Boussinesq model is proved in the case, when reaction coefficient depends nonlinearly on concentration of substance. Maximum principle is stated for substance’s concentration. Solvability of control problem is proved, when the role of controls is played by diffusion and mass exchange coefficients from the equations and from the boundary

The main objective of this paper is to study the optimal distributed control for a model of homogeneous incompressible two-phase flows. We apply the well-posedness and regularity results to establish the existence of optimal controls as well as the first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid.

We concern the asymptotic behaviour of the dynamical systems induced by nonlinear Lasota equation. We study chaoticity in the sense of Devaney and strong stability of the system. In many articles authors consider the properties of the linear version of the equation. By the construction of the operator in the separable space, we can formulate the relations between the solutions of the linear equation

The goal of this paper is to investigate Gevrey properties of formal solutions of certain generalized linear partial differential equations with variable coefficients. In particular, we extend the notion of moment partial differential equations to include differential operators that are not connected with kernel functions. Using the modified version of Nagumo norms and the properties of the Newton

This paper studies the concept of prolongational controllability for control affine system with piecewise constant controls. The purpose is to show the relationship between the controllability by prolongations and the notion of weak uniform attraction, presenting criteria for controllability and controllability by prolongations. The central result assures that a control affine system is controllable

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space X with at least two elements, a nonempty set Γ and a self-map φ :

In this paper, we introduce geometric tools to study the families of rational vector fields of a given degree over \(\mathbb {C}\mathbb {P}^{1}\). To a generic vector field of such a parametric family, we associate several geometric objects: a periodgon, a star domain, and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields

Using the averaging theory of second order, we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in \(\mathbb {R}^{n}\). We prove that there are at least 3n− 2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover, we provide an example in dimension 6 showing that this number of limit cycles is reached

This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter’s stability theorem to group actions on locally compact metric spaces. We show that action

We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. Then using the averaging theory, we prove

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation \( x^{\prime }= Ax(t) + Lx_t + f(t,x_t, \lambda ) \), with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained

We consider the existence of affine-periodic solutions to the nonlinear ordinary differential equation: $\label {ae} x^{\prime }=f(t,x) $(0.1) in ℝn, where f is continuous and ensures the existence and uniqueness of solutions with respect to initial conditions, and there exist T > 0 and Q ∈ GL(n) such that: $\label {me} f(t+T,x)=Qf(t,Q^{-1}x) \quad \forall (t,x)\in \mathbb R\times \mathbb R^{n}. $(0

A quenching phenomenon of a degenerate parabolic equation with nonlinear source and singular boundary condition in one-dimensional space is investigated. We establish the results that quenching will occur in a finite time on the boundary x = 0 or x = 1, respectively. And the blowing up of ut at the quenching point and quenching rate estimates are discussed.

We address the output feedback stabilization for a Euler–Bernoulli beam equation with boundary moment control and disturbance. The stabilization of this system has been studied in Guo et al. (J Dyn Control Syst. 2014;20:539–58), where the controller is based on full state feedback. In order to derive the output feedback controller, we design a new disturbance estimator to estimate the total disturbance

This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a fourth-order parabolic equation with a general nonlinearity. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, using differential inequalities. Moreover, upper and lower bounds for the blow-up time are derived when blow-up occurs

In this paper, we study an optimal control problem for nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. The system evolves with regular potential in a two-dimensional bounded domain. We extend the existence of weak solution results for the system to prove the existence of strong solution under extra assumptions on the forcing term and initial datum

The well-known Poincaré-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit

In this paper, practical stochastic uniform input-to-state stability (PSUISS) for perturbed and perturbed triangular systems depending on parameter is investigated. We present sufficient conditions for which each of this notion is preserved under triangular interconnection. Finally, an example with simulation is provided to demonstrate the applicability of our results.

We describe a method to analyze and decompose the dynamics of a bilinear control system subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This

This paper is concerned with the front-like entire solutions for a Lotka-Volterra weak competition system with nonlocal dispersal. Here, an entire solution means a classical solution defined for all space and time variables. This system has traveling wavefronts and enjoys the comparison principle. Based on these traveling wavefronts, we construct some super- and sub-solutions. Then, by using the comparison

We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable.