Journal of Dynamical and Control Systems

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Article
This paper deals with a p(x)-Laplacian parabolic problem involving positive and negative sources, subject to a homogeneous Dirichlet boundary condition. We classify the initial energy and the Nehari energy completely to show global existence and blow-up of weak solutions. More- over, we obtain non-extinction and extinction properties of weak solutions.
 
Article
Based on two different results of the averaging theory of the first order, we provide sufficient conditions for the existence of periodic solutions for two differential systems, the first one in ℝ5 of the form x.=y,y.=z,z.=u,u.=v,v.=−αβμx−βμy−α(β+μ)z−(β+μ)u−αv+εf(t,x,y,z,u,v), where α,β and μ are rational numbers different from 0 such that α≠ ± β,α≠ ± μ, and β≠ ± μ with |ε| sufficiently small, and f is non-autonomous periodic function. And the second differential system in ℝ6 given by x.=y,y.=−x−εF(t,x,y,z,u,v,w),z.=u,u.=−z−εG(t,x,y,z,u,v,w),v.=w,w.=−v−εH(t,x,y,z,u,v,w), where F,G and H are 2π-periodic functions in the variable t, and |ε| is a small parameter. Moreover we provide some applications.
 
Article
In this work, we introduce a concept of expansiveness for actions of connected Lie groups. We study some of its properties and investigate some implications of expansiveness. We study the centralizer of expansive actions and introduce CW -expansiveness for pseudo-group actions. As an application, we prove positiveness of geometric entropy for expansive foliations and expansive group actions.
 
Article
In this paper, we introduce a notion of shadowing property for a free semigroup action on a compact metric space, which is different of the notion of the shadowing property introduced by Bahabadi, called chain shadowing property. We study the relation between the shadowing property of a free semigroup action on a compact metric space X and the shadowing property of the induced free semigroup action on the hyperspace 2X. Specially, we not only theoretically prove that (Fm+,F)↷X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(F_{m}^{+},\mathcal {F})\curvearrowright X$\end{document} has the (chain) shadowing property if and only if (Fm+,F)↷2X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(F_{m}^{+},\mathcal {F})\curvearrowright 2^{X}$\end{document} has the (chain) shadowing property, but also give examples to illustrate it. Finally, we compare the two notions of shadowing for free semigroup actions and obtain an interesting result that if (Fm+,F)↷X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(F_{m}^{+},\mathcal {F})\curvearrowright X$\end{document} has the shadowing property, then it has the chain shadowing property, but not vice versa.
 
Article
In this paper, practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion (G-SDEs) is studied. The analysis of the global practical uniform p th moment exponential stability, as well as the global practical uniform exponential stability with respect to a part of the variables of G-SDEs, is investigated by means of the G-Lyapunov functions. An illustrative example to show the usefulness of the practical stability with respect to a part of the variables notion is also provided.
 
Article
In this paper, we extend the notion of asymptotic measure expansivity for diffeomorphisms to flows on a compact metric space, and prove that there exists an asymptotic measure expansive flow in that space. Then we consider the hyperbolicity of the asymptotic measure expansive vector fields on a closed smooth Riemannian manifold M. More precisely, we prove that any C¹ stably asymptotic measure expansive vector field on M satisfy Axiom A without cycles, and it is quasi-Anosov. On the other hand, we show that C¹ generically, every asymptotic measure expansive vector field on M satisfies Axiom A without cycles. Moreover, we study the asymptotic measure expansivity of the homoclinic classes which are the delegate of the invariant subsets for given systems, prove that C¹ stably and C¹ generically, the asymptotic measure expansive homoclinic class is hyperbolic. Furthermore, we consider the hyperbolicity of asymptotic measure expansive divergence-free vector fields for C¹ stably and C¹ generic point of view. We also apply the our results to the divergence-free vector fields.
 
Electric circuit with Josephson junction, [35]
Article
We study linear and semi-linear differential-algebraic equations (DAEs) on the half-line ℝ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}_{+}$\end{document}. In our strategy, we first show a characterization for the existence of exponential dichotomy for linear DAEs based on the Lyapunov-Perron method. Then, we prove the existence of invariant stable manifolds for semi-linear DAEs in the case that the evolution family corresponding to a linear DAE admits an exponential dichotomy and the non-linear forcing function fulfills the non-uniform φ-Lipschitz condition where the Lipschitz function φ belongs to wide classes of admissible function spaces such as Lp, 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq p \leq \infty $\end{document}, and Lp,q.
 
Article
In this paper, mean Li-Yorke chaos and mean sensitivity are investigated in non-autonomous discrete systems (X,f1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(X, f_{1, \infty })$\end{document}, where f1,∞={fi}i≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1, \infty } = \{f_{i}\}_{i \geq 1}$\end{document} is a sequence of self-maps on a metric space X. It is shown that mean Li-Yorke chaos is preserved under iteration for a non-autonomous discrete system with certain continuity. Moreover, sensitivity, Banach mean sensitivity and mean sensitivity of non-autonomous linear discrete systems are characterized, respectively. We provide sufficient conditions under which a mean sensitive non-autonomous discrete system is mean Li-Yorke chaotic.
 
Article
This paper deals with the fourth-order parabolic equation ut+Δ2u=up(x)logu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{t}+{\varDelta }^{2}u=u^{p(x)}\log u$\end{document} in a bounded domain, subject to homogeneous Navier boundary conditions. For subcritical and critical initial energy cases, we combine the Galerkin’s method with the generalized potential well method to prove the existence of global solutions. By the concavity arguments, we obtain the results about blow-up solutions. For super critical initial energy case, we use some ordinary differential inequalities to study the extinction of solutions. Moreover, extinction rate, blow-up rate and time, and decay estimate of solutions are discussed.
 
Article
This paper is devoted to the investigation of fiber measure-theoretical restricted sensitivity and fiber topological restricted sensitivity by restricting the first sensitive time of random dynamical systems. Under the actions of a random dynamical system, we define fiber measure-theoretical restricted asymptotic rate with respect to sensitivity, and show that it is equal to the reciprocal of the Brin-Katok local entropy for almost every point. For topological version, we define fiber topological restricted asymptotic rate with respect to sensitivity, and establish the relation between fiber topological restricted asymptotic rate and fiber topological entropy of random dynamical systems.
 
Article
In the present paper, we investigate the existence of positive periodic solutions for an n-species Lotka-Volterra system with distributed delays and nonlinear impulses. In the process, we convert the given system into an equivalent integral equation. Then, we construct appropriate mappings and use Krasnoselskii’s fixed point theorem in a cone of a Banach space to show the existence of a positive periodic solution of the system. Easily verifiable sufficient conditions are established. We discuss our problem in two situations: when impulse functions are subquadratic and when they are sublinear. The technique to deal with the impulsive term is different from earlier approaches. In particular, the results improve some previous ones in the literature. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.
 
Article
Oncological therapies usually are applied intermittently, i.e., not continuously over time. In the periods in between, though, the cancer cells are usually left free to grow. This intermittency is a key issue in prostate cancer hormonal therapies based on androgen suppression. Here, we address this treatment modality by analyzing a piecewise smooth vector field approach. In fact, using the PSA (prostate-specific antigen) serum level as a control variable to switch between treatment and no-treatment periods of hormone therapy (androgen withdrawal), by means of typical parameter values, our theoretical analysis supports the idea that intermittent androgen suppression may prevent a prostate cancer relapse for a specific class of patients.
 
Article
The main objective of this paper is to study the optimal distributed control of the three-dimensional planetary geostrophic equations. We apply the well-posedness and regularity results proved in Cao and Titi (Commun Pure Appl Math 56:198–233, 2003) to establish the existence of an optimal control as well as the first-order necessary optimality condition for an associated optimal control problem in which a distributed control is applied to the temperature.
 
Article
We extend the definition of topological entropy for any (not necessarily continuous) amenable groups acting on a compact space by defining entropy of arbitrary subsets of a product space. We investigate how this new notion of topological entropy for amenable group actions behaves and some of its basic properties; among them are the behavior of the entropy with respect to disjoint union, Cartesian product, and some continuity properties with respect to Vietoris topology. As a special case for 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq p\leq \infty $\end{document}, the Bowen p-entropy of sets is introduced. It is shown that the notions of generalized topological entropy and Bowen ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\infty $\end{document}-entropy for compact metric spaces coincide.
 
The cups at infinity
Phase portraits of the polynomial differential system (2) with n = 2
Phase portrait of the polynomial differential system (2) with n = 3, and also with n = 5
Phase portraits of the polynomial differential system (2) with n = 4
Phase portraits of the polynomial differential system (2) with n = 5
Article
We classify the phase portraits in the Poincaré disc of the differential equations of the form x′=−y+xf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\prime } = -y + x f(x,y)$\end{document}, ẏ=x+yf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot y =x + y f(x,y)$\end{document} 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 Chaps. 9 and 10 of Dumortier et al. (UniversiText, Springer-Verlag, New York, 2006).
 
Article
In this paper, we give the different topological types of phase portrait for Liénard system ẋ=y,ẏ=−g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot {x}=y, \dot {y}=-g(x)$\end{document} in the case that degg(x)=7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\deg g(x)=7$\end{document} 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 and two hyperbolic saddles or passing through three hyperbolic saddles, has been studied. In this paper, as one of main results, for a near-Hamiltonian system, we give the expansion of the Melnikov function near a compound loop with a nilpotent cusp and two hyperbolic saddles. Based on this, we present the conditions to obtain limit cycles.
 
Article
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.
 
Article
We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{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 w.r.t. a Riemannian metric induced from the Finslerian metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show new examples of Anosov flows.
 
Article
The skew tent map T is a chaotic map with rich dynamic properties. We present some interesting dynamical properties of the forts and the index function ε of skew tent maps. Using the coordinates of of forts of Tⁿ, we present the exact expression of Tⁿ(x) for n∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\in \mathbb {N}$\end{document}. Further, we obtain the coordinates of all fixed points of Tⁿ, and give a necessary and sufficient condition for the fixed point of Tⁿ to be a p-periodic point of T for any p|n. This condition is so simple that we can easily present the exact expressions of all periodic points of T. We give the exact expressions of the coordinates of all forts and periodic points of this map.
 
2-dimensional example of Ω~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde {\Omega }$\end{document}, Ω, and Γ
Positions of parallelepipeds in dimension d = 2: (a) y+Qa⊂x+Qã\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y+Q_{a} \subset x+Q_{\tilde a}$\end{document} or M(y+Qa)⊂x+Qã\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M(y+Q_{a}) \subset x+Q_{\tilde a}$\end{document} is for some y∈ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y\in \mathbb {R}^{d}$\end{document} contained in one of the two half spaces. (b) A parallelepiped y + Qa ⊂ x + Q2a is contained in one of the orthants. (c) The ball of radius a12+a22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt {{a_{1}^{2}} + {a_{2}^{2}}}$\end{document} at the origin contains both a parallelepiped of sides a1 and a2 and its reflection with respect to M𝜃
(a) For every reflection hyperplane cutting a 4G-cell, there is a G-subcell in the lower or a 3G-subcell in the upper half-space. (b) The blue 3G-subcell contains the reflection of the red cube Q
Article
We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle π/2ⁿ. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms in terms of geometric parameters of the control set.
 
Article
For a complete Riemannian manifold with bounded geometry, we prove a generalized compactness theorem for sequences of clusters (with uniformly bounded perimeter and volume) in a larger space obtained by adding at most countable many limit manifolds at infinity, in the spirit of Muñoz Flores and Nardulli (Journal of Dynamical and Control Systems 1–11, 2020). The arguments presented in the proof of this generalized compactness theorem when applied to minimizing sequences of clusters give a generalized existence theorem for isoperimetric clusters in a larger space obtained by adding, in this case, only finitely many limit manifolds at infinity, as in Nardulli (Asian J Math 18(1):1–28, 2014). To achieve this goal, we show that isoperimetric clusters are bounded and also we prove the continuity of the multi-isoperimetric profile. In fact, we prove a stronger continuity property that is the Hölder continuity of the multi-isoperimetric profile. The multi-isoperimetric profile has been introduced recently in Milman and Neeman (2018) in the context of smooth metric measured spaces with a Gaussian-weighted notion of perimeter. This work generalizes to the context of Riemannian isoperimetric clusters some previous results about the classical Riemannian and sub-Riemannian isoperimetric problem, see Galli and Ritoré (Jour Math Anal Applic, 2012), Morgan (Trans Amer Math Soc 355(12), 2003), Mondino and Nardulli (Commun Anal Geom, 24(1):115–138, 2016), Nardulli (Glob Anal Geom, 36(2):111–131, 2009), Nardulli (Asian J Math 18(1):1–28, 2009), and Nardulli Bull Braz Math Soc (N.S.) 49(2):199–260, 2018), as well as results from clusters theory in the Euclidean setting, see Maggi (2012) and Morgan (Math Ann 299:697–714, 1994). In particular, as a consequence of our generalized existence results, we prove an existence theorem (in the classical sense) for isoperimetric clusters in a quite large class of noncompact Riemannian manifolds (the same considered in Mondino and Nadulli (Commun Anal Geom 24(1):115–138, 2016)) which includes, for instance, the space forms.
 
Article
We consider a time-optimal problem for the classical “double integrator” system under an arbitrary linear state constraint. Using the maximum principle, we construct the full synthesis of optimal trajectories and provide a qualitative investigation of the measure, the Lagrange multiplier corresponding to the state constraint.
 
Article
The purpose of this paper is to present explicitly the solution curve for commutative affine control systems on Lie groups under the assumption that the automorphisms associated with the linear vector fields commutes. If we assume that the derivations associated with the linear vector fields of the system are inner, we obtain a simpler solution and we show some results of controllability. To finish, we work with conjugation by homomorphism of Lie groups between affine systems.
 
Article
In this work, we are concerned with the study of stabilization of one-dimensional weakly degenerate wave equation utt − (xγux)x = 0 with x ∈ (0, 1) and γ ∈ [0, 1), controlled by a fractional boundary feedback acting at x = 0. Strong, uniform, and nonuniform stabilization are obtained with explicit decay estimates in appropriate spaces. The results are obtained through an estimate on the resolvent of the generator associated with the semigroup.
 
Article
A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on any cross section of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbit of a Poincaré map.
 
Article
In this paper, we consider the feedback boundary stabilization of a flexible beam clamped at one end to a rigid disk, and attached at the other end to a tip mass. The rigid disk is assumed to rotate freely with a non-constant angular velocity. To stabilize the system, we propose a feedback control which consists of a boundary control force and a boundary control moment applied at the tip mass, and a torque control applied on the disk. The well-posedness of the system is proved using the semigroup theory. In the case where the angular velocity of the disk is constant, the system is a Riesz spectral system by the method of spectral analysis. Consequently, the spectrum-determined growth condition and the exponential stability are held. Moreover, for the homogeneous beam, if the control force is the only control applied, the lack of exponential stability is proved and the polynomial stability is established using Borichev-Tomilov’s Theorem. The numerical simulations of the solution, the distribution of the eigenvalues, and the energy for the linear subsystem are presented to support the theoretical results.
 
The one-dimensional Lorenz-like map T0
Article
We study rates of mixing for small random perturbations of one-dimensional Lorenz maps. Using a random tower construction, we prove that, for Hölder observables, the random system admits exponential rates of quenched correlation decay.
 
Article
In this note, a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space, the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen’s entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq p\leq \infty $\end{document}, the Bowen p-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\infty $\end{document}-entropy for compact metric spaces coincide.
 
The definition of the loops γL and γR
The sectors Ωj,0 and Ωj,0(ε),j = 1,2 when − β < 0
The monodromy operators MR(ε) when − β > 0 and ML(ε) when − β < 0
Article
We study the effect of the unfolding of a reducible double confluent Heun equation from the point of view of the Stokes phenomenon. We introduce a small complex parameter ε that splits together the non-resonant singular points x = 0 and \(x=\infty \) into four different Fuchsian singularities \(x_{L}=-\sqrt {\varepsilon }, x_{R}=\sqrt {\varepsilon }\), and \(x_{LL}=-1/\sqrt {\varepsilon }, x_{RR}=1/\sqrt {\varepsilon }\), respectively. The perturbed equation is a symmetric general Heun equation and its general solution depends analytically on \(\sqrt {\varepsilon }\). Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial double confluent Heun equation are realized as a limit of the upper-triangular parts of the monodromy matrices of the perturbed equation when \(\sqrt {\varepsilon } \rightarrow 0\). To establish this result we combine a direct computation with a theoretical approach.
 
Article
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 of the global-in-time classical solution to the Cauchy problem with general large initial data when |α| is sufficiently small and the coefficient of magnetic diffusion ν is suitably large. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.
 
Article
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 syndetically sensitive. Some other significant properties of this new class are obtained.
 
In the first row, phase portrait of (5), with D < − 1 and F > 1 in the Poincaré disc, where we use thick lines to draw the conic. In the second row, the phase portrait near the period annulus after blowing-up the saddle point SD = (− 1/D, 0)
Bifurcation diagram of the period function at the polycycle according to [9] and, in color, the subsequent improvements due to [5, 10, 11, 25, 26], where μ⋆ = (−F⋆, F⋆) with F⋆ ≈ 2.34. The curve that joins −32,32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left (-\frac {3}{2},\frac {3}{2}\right )$\end{document} and −12,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left (-\frac {1}{2},1\right )$\end{document} is the graphic of an analytic function D=G(F).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D=\mathcal G(F).$\end{document} The double bifurcation curve Γ of Theorem C appears starting from the point (− 2,2)
Article
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 in the Loud family of quadratic centers.
 
Article
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 bundle and calculate the partial indices of the problem.
 
Article
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, g′(t)≤−ξ(t)G(g(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g^{\prime }(t)\le -\xi (t) G(g(t))$\end{document}. We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.
 
Article
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.
 
Article
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 model, the coexistence steady state of three interacting species is mainly investigated, and the global stabilities of the extinction of all species and the survival of only healthy prey are discussed. Additionally, the conditions of the non-coexistence of three species are established. Finally, an ecological interpretation is also presented based on the results.
 
Singularities
Projection of classical complete solutions and the singular solution of the Clairaut equations
Article
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 equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type, respectively.
 
The sailboat
A sewing of the swallowtail and the Whitney umbrella
a) The swallowtail; b) the cut swallowtail
a) The Whitney umbrella; b) the Whitney umbrella with handle
The Whitney umbrella chamber
Article
One of the singularities of the convex hull of a generic hypersurface in ℝ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{4}$\end{document} 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.
 
Article
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a 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 extends previous results contained in [Nar14a], in such a way that the generalized existence theorem, Theorem 1 of [Nar14a] is actually a generalized compactness theorem. The suitable modifications to the arguments and statements of the results in [Nar14a] are non-trivial. As a consequence we give a multipointed version of Theorem 1.1 of [LW11], and a simple proof of the continuity of the isoperimetric profile function.
 
Hyperbolic cylinder
Self-intersecting cylinder: the segments I+ and I− are identified by an affine map with coefficient 12i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {1}{2i}$\end{document}. Circular arrows identify adjacent segments
Quartic differential on ℂP1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb C P^{1}$\end{document}
Quartic differential on a torus
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Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behavior of geodesics of such connections has been studied by, e.g., Abate and Bianchi (Math Z 282:247–272, 2016) and Abate and Tovena (J Differ Equ 251(9):2612–2684, 2011) in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviors of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar ω-limit sets.
 
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In this work, the null controllability problem for a linear system in ℓ ² is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤− 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $\ell ^{\infty }$ ℓ ∞ is not asymptotically stable if λ = − 1.
 
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We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler’s circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed linear combination of barycenter and circumcenter, its locus over the family is an ellipse.
 
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In this paper we show that smooth TA-endomorphisms of compact manifolds with c-expansivity (that is, expansive in the inverse limit) and $C^1$-stable shadowing are Axiom A.
 
Article
We study a critical Grushin-type equation by applying Lyapunov-Schmidt reduction method and obtain the existence of infinitely many positive multi-bubbling solutions with cylindrical symmetry. Particularly, since the concentration points here can be saddle points of some function related to the potential, we locate the concentration points of these solutions by local Pohozaev identities rather than estimating the derivatives of the reduced functional as usual.
 
Article
The paper continues the authors’ study of the linearizability problem for nonlinear control systems. In the recent work (Sklyar, Syst Control Lett 134:104572, 2019), conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper, we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear nonautonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.
 
Article
Let a flow ϕt be partially hyperbolic on Λ. If Λ has a local product structure, then ϕt has the quasi-shadowing property on Λ in the following sense: for any 𝜖 > 0, there exists constant δ > 0 such that for any (δ,1)-pseudo orbit \(\{x_{k}, t_{k}\}_{k\in \mathbb {Z}}\) of ϕt with 1 ≤ tk ≤ 2 for all \(k\in \mathbb {Z}\), there exist a sequence of points \(\{y_{k}\}_{k\in \mathbb {Z}}\) and a reparametrization \(\alpha \in Rep(\mathbb {R},\epsilon )\) such that \(\phi _{\alpha (t)-\alpha ({\Sigma }_{k})}(y_{k})\) trace \(\phi _{t-{\Sigma }_{k}}(x_{k})\) in which yk+ 1 lies in the local center leaf of \(\phi _{\alpha ({\Sigma }_{k+1})-\alpha ({\Sigma }_{k})}(y_{k})\) for k ≥ 0 t ≥ 0 and \(\phi _{\alpha (t)-\alpha (-{\Sigma }_{k})}(y_{k})\) trace \(\phi _{t-(-{\Sigma }_{k})}(x_{k})\) in which yk+ 1 lies in the local center leaf of \(\phi _{\alpha (-{\Sigma }_{k+1})-\alpha (-{\Sigma }_{k})}(y_{k})\) for k < 0, t < 0.
 
Article
A correction to this paper has been published: https://doi.org/10.1007/s10883-021-09567-w
 
Article
In the present work, we obtain rigidity results analyzing the set of regular points, in the sense of Oseledec’s Theorem. It is presented a study on the possibility of Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus \(\mathbb {T}^{d}, d \geq 3,\) C1 −close to a linear automorphism diagonalizable over \(\mathbb {R}\) and such that its characteristic polynomial is irreducible over \(\mathbb {Q}.\)
 
Article
A correction to this paper has been published: https://doi.org/10.1007/s10883-021-09569-8.
 
Top-cited authors
Andrei Agrachev
  • Scuola Internazionale Superiore di Studi Avanzati di Trieste
Bao-Zhu Guo
  • Chinese Academy of Sciences
Davide Barilari
  • University of Padova
Zhiliang Zhao
  • Shaanxi Normal University