In this article we investigate polynomial vector fields that exhibit certain invariant curves, surfaces or manifolds. More precisely, we will provide an upper bound for the maximum number of conics, quadrics and Clifford tori that a polynomial vector field $ \ensuremath {\mathcal {X}} $ X can have in $ \ensuremath {\mathbb {C}}<^>2 $ C2, $ \ensuremath {\mathbb {C}}<^>3 $ C3 and $ \ensuremath {\mathbb {R}}<^>{2n} $ R2n respectively, in function of the degree of the polynomial vector field. The main tool for proving these results is the Darboux theory of integrability.
In this paper, a class of nonautonomous neutral differential equations with impulsive effects is investigated. The proofs of the existence of positive periodic solutions rely on the well-known Guo-Krasnoselskii's fixed point theorem together with the Green functions method. We derive new criteria that guarantee the existence of at least single and multiple positive periodic solutions, thereby extending and improving several known results in the literature. Furthermore, by employing the Banach fixed point theorem and imposing additional conditions, we prove the existence and uniqueness of a periodic solution. Finally, two examples are exhibited to show the efficiency and application of our findings, which are completely new and enrich the existing literature.
This paper analyzes a general discrete neural network with two arbitrary delays. Three criteria of both Devaney and Li-Yorke chaos are given. Furthermore, the lower bounds of the parameters of making the network chaotic are obtained. These results can be viewed as an extension of the results in Huang and Zou [Co-existence of chaos and stable periodic orbits in a simple discrete neural network. J. Non. Sci. 2005;15(5):291-303], Kaslik and Balint [Chaotic dynamics of a delayed discrete-time Hopfield network of two nonidentical neurons with no self-connections. J. Non. Sci. 2008;18(4):415-432], Chen et al. [Chaotic invariant sets of a delayed discrete neural network of two non-identical neurons. Sci. China Math. 2013;56(9):1869-1878]. To understand the complex chaotic behaviours of the network better, computer simulations are provided and the largest Lyapunov exponents are computed.
In this paper, we investigate the existence and multiplicity of positive solutions for a semipositone nonlinear elastic beam equation with Neumann boundary conditions. The main tools are based on the Krasnosel'skii fixed point theorem.
In this paper, a reaction-diffusion epidemic model with general nonlinear incidence and treatment is proposed. The basic reproduction number R-0 and the critical wave speed c* are defined, and existence and nonexistence of travelling wave solutions satisfying certain boundary conditions is investigated by some fixed point theorems, two-sided Laplace transform method and Stable Manifold Theorem. Specifically, when R-0 > 1 and c > c*, then the existence of the travelling wave solutions is established. When R-0 > 1 and 0 < c < c* or R-0 <= 1 and c > 0 the non-existence of travelling wave solutions also is established. Finally, we present the numerical simulations and investigate the effect ofthe critical wave speed on disease propagation.
In this paper, we establish a connection between the centre problem for Abel equations and that for resonant saddle points, showing that the solution to the latter implies the solution to the former. Furthermore, it is shown that for complex Abel equations, these two centre problems are equivalent.
We extend to the n-dimensional hyperboloid contained in Rn+1, the Darboux theory of integrability for polynomial vector fields. Additionally, new results are provided on the maximum number of invariant parallels and meridians that polynomial vector fields having invariant the n-dimensional hyperboloid can exhibit in function of the degree of the polynomial vector field. In some sense, our results extend the known results on the upper bound for the number of invariant hyperplanes that a polynomial vector field in R-n can have in function of its degree.
We will introduce the notion of lag map of one-sided topological Markov shifts to investigate eventual conjugacy of one-sided topological Markov shifts. We will then clarify structure of one-sided eventual conjugacy of topological Markov shifts from the three viewpoints: (1) isomorphisms of Cuntz-Krieger algebras, (2) homeomorphisms between one-sided topological Markov shifts, (3) matrix relations defining the eventual conjugate one-sided topological Markov shifts. As a result, we will characterize eventual conjugacy of one-sided topological Markov shifts in terms of the above three viewpoints by using lag maps.
This paper considers the number of limit cycles of a piecewise cubic Hamiltonian system with general polynomial perturbations. We express the first-order Melnikov function as a linear combination of several generating integrals with polynomial coefficients, and prove that the coefficients of these polynomials are independent using mathematical induction. Then we obtain lower bounds for the number of limit cycles near the centre and heteroclinic loop by applying asymptotic expansions.
In this paper, inspired by the published work (Hiroshima Math. J. 54(2) (2024), pp. 219-232), we continue to explore the exponential instability of stochastic dynamical systems described by cocycles on Banach spaces. From the perspectives of both continuous time and discrete time, new Datko-type conditions for the cocycles satisfying uniform exponential instability in mean are obtained respectively.
In this work, we have discussed the dynamics of Steffensen's family of multipoint iterative root-finding methods for rational functions including polynomials and Mobius maps. First, we have shown that simple zeroes of a rational function are superattracting fixed points of the beta-Steffensen method of the function. We also describe the dynamics of the beta-Steffensen method for polynomials with simple zeroes and show that the Julia set is connected in this case. The dynamics of the beta-Steffensen method applied to Mobius maps, have also been studied. We show that for Mobius maps without any zero, the Fatou set is the attracting basin of infinity. For Mobius maps with zeroes, we give the expressions of the extraneous fixed points of the beta-Steffensen method and show that infinity is a superattracting fixed point. The existence of attracting domains, parabolic domains and Siegel discs for different parameters have been shown under various conditions. A detailed description of the beta-Steffensen method for Mobius maps with zeroes is provided when all the parameters are real.
This paper provides global attractivity results for the interior equilibrium point of a general Lotka-Volterra system with no restriction on the dimension of the system and with no special structure or properties of the interaction matrix. The main result contains as special cases all known general results, including the Volterra-Lyapunov theorem and the recently proposed eigenvector conditions. Moreover, global attractivity of the interior equilibrium point is shown for a three-dimensional example, where none of the existing general results can be applied.
In this paper we consider the weak Gibbs measures for (α, β)-shifts. In the case of α=0, Pfister and Sullivan have given a necessary and sufficient condition on β such that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure in the natural extension of a β-shift. So it is natural to ask what happens when α>0. However, their proof cannot be applied to general (α, β)-shifts in a similar way. In this paper we consider the case of α=κ/β and give a criterion for the weak Gibbs property of equilibrium measures for (κ/β, β)-shifts.
In a hyperbolic polygon any finite collection of closed billiard trajectories can be assigned an average length function. In this paper, we consider the average length of the collection of cyclically related closed billiard trajectories in even-sided right-angled polygons and the collection of reflectively related closed billiard trajectories in Lambert quadrilaterals with acute angle $ \pi /k $ pi/k. We show that in the former case the average length is minimized by the regular even-sided right-angled polygon, and in the latter case it is minimized by the Lambert quadrilateral with a reflective symmetry about its long axis. We use techniques from Teichm & uuml;ller theory to prove the main theorems.
In this paper, we show the uniqueness of equilibrium state for a family of partially hyperbolic horseshoes, introduced in [12] for some classes of continuous potentials. For the first class, the method used here is making use of the Sarig's theory for countable shifts. For this purpose, we study the dynamics of an induced map associated to the horseshoe map, we build a symbolic system with infinitely many symbols that is topologically conjugated to this induced map and we show that the induced potential is locally Hölder and recurrent. For the second class, by following ideas of [29] and [31], we prove that uniqueness for the horseshoe is equivalent to uniqueness for the restriction to a non-uniformly expanding map, which is a hyperbolic potential and then has uniqueness. Both classes include potentials with high variation, differently from previous results for potentials with low variation. We also prove uniqueness when the potential presents its supremum at a special fixed point and less than the pressure.
We derive sufficient conditions for a homeomorphism of a compact metric space without isolated points to be topologically transitive. In particular, we prove that a homeomorphism with the shadowing property has a dense orbit if either it has the su-intersecting property in the whole space; or it has the barycentre property in the non-wandering set; or else it is accessible in the recurrent set. To elucidate their dynamical nature, we compare these three conditions, discuss examples with a variety of dynamics and present some applications of interest.
We study the long-time behavior of non-autonomous stochastic FitzHugh-Nagumo systems on thin domains. As the (n+ 1)-dimensional thin domains collapses onto an n-dimensional domain, an n-dimensional limiting FitzHugh-Nagumo system is derived. This n-dimensional limit?ing system encodes the defining geometry of the (n+1)-dimensional system. To justify this limiting process, we show that the pullback measure attractors of the FitzHugh-Nagumo systems on thin domains are upper semi-continuous as the height of thin direction tends to zero.
Hitting rate and escape rate are two examples of recurrence laws for a dynamical system, and a general limit connects them. We show that for both Gibbs-Markov systems or any systems with a phi-mixing measure, for a sequence of nested sets whose intersection is a measure zero set, this general limit equals one in the absence of short returns and less than one otherwise, which is given by an explicit formula called extremal index. One of the applications of this result is to dynamical systems on Riemannian manifolds such as hyperbolic maps and expanding maps, and it can be applied to any system with a suitable Young tower.
In this article we study orbits of proximal pairs in almost automorphic subshifts. The corresponding orbits in the maximal equicontinuous factor are precisely those orbits that intersect the boundary of the subshift's separating cover. We impose certain finiteness conditions on this boundary and explore the resulting consequences for the subshift, for instance in terms of complexity or the relations between proximal and asymptotic pairs. The last part of our article deals with Toeplitz subshifts without a finite boundary. There we treat the question of necessary conditions and sufficient conditions for the existence of a factor subshift with a finite boundary. Throughout the whole article, we provide explicit constructions for Toeplitz subshifts to illustrate our findings and the necessity of our assumptions.
In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map with a countably infinite number of branches on the interval $ I=[0,1] $ I=[0,1]. We establish that for sufficiently high iterates of the map, the iterations are piecewise expanding. By adapting the Lasota-Yorke inequality to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map, the exactness of the dynamical system and the quasi-compactness of the Frobenius-Perron operator induced by the map. These findings imply a multitude of strong ergodic properties for the system.
