Autonomous Case Research Articles

Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization

In this paper, we extend the one-component Gross–Pitaevskii (GP) equation to the two-component coupled GP system including damping term, linear and parabolic density profiles. The Lax pair with nonisospectral parameter and infinitely-many conservation laws of this coupled GP system are presented. Actually, the Darboux transformation (DT) for this kind of nonautonomous system is essentially different from the autonomous case. Consequently, we construct the DT of the coupled GP equations, besides, nonautonomous multi-solitons, one-breather and the first-order rogue wave are also obtained. Various kinds of one-soliton solution are constructed, which include stationary one-soliton and nonautonomous one-soliton propagating along the negative (positive) direction of x-axis. The interaction of two solitons and two-soliton bound state are demonstrated respectively. We get the nonautonomous one-breather on a curved background and this background is completely controlled by the parameter β. Using a limiting process, the nonautonomous first-order rogue wave can be obtained. Furthermore, some dynamic structures of these analytical solutions are discussed in detail. In addition, the multi-component generalization of GP equations are given, then the corresponding Lax pair and DT are also constructed.

Permanence and extinction of stochastic competitive Lotka–Volterra system with Lévy noise

This paper derives sufficient conditions for stochastic permanence and extinction of a stochastic non-autonomous competitive Lotka–Volterra system with Levy noise. For the autonomous case, the results show that stochastic permanence and extinction are characterized by two parameters $$\mathcal {B}_{1}$$ and $$\mathcal {B}_{2}$$ : if $$\mathcal {B}_{1}\mathcal {B}_{2} \ne 0$$ , then the system is either stochastically permanent or extinctive. That is, it is extinctive if and only if $$\mathcal {B}_{1}<0$$ and $$\mathcal {B}_{2}<0$$ ; otherwise, it is stochastically permanent. Some existing results are included as special cases. An example and its simulations are given to support our theoretical results.

A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians

Let M be a closed and connected manifold, $$H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}$$ a Tonelli 1-periodic Hamiltonian and $${\mathscr {L}}\subset T^*M$$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $${\mathscr {L}}$$ is invariant by the time-one map of H, then $${\mathscr {L}}$$ is a graph over M. An interesting consequence in the autonomous case is that in this case, $${\mathscr {L}}$$ is invariant by all the time t maps of the Hamiltonian flow of H.

Uniqueness of large positive solutions

We establish the uniqueness of the positive solution of the singular problem (1.1) through some standard comparison techniques involving the maximum principle. Our proofs do not invoke to the blow-up rates of the solutions, as in most of the specialized literature. We give two different types of results according to the geometrical properties of $$\Omega $$ and the regularity of $$\partial \Omega $$ . Even in the autonomous case, our theorems are extremely sharp extensions of all existing results. Precisely, when $$\mathfrak {a}(x)\equiv 1$$ , it is shown that the monotonicity and superadditivity of f(u) with constant $$C\ge 0$$ entail the uniqueness; f is said to be superadditive with constant $$C\ge 0$$ if $$\begin{aligned} f(a+b) \ge f(a) + f(b) - C \quad \hbox {for all}\;\; a, b \ge 0. \end{aligned}$$ This condition, introduced by Marcus and Veron (J Evol Equ 3:637–652, 2004), weakens all previous sufficient conditions for uniqueness, as it will become apparent in this paper.

Effects of thermo hydrodynamic (THD) floating ring bearing model on rotordynamic bifurcation

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.

Heteroclinic connections for a double-well potential with an asymptotically periodic coefficient

We prove the existence of monotone heteroclinic solutions to a scalar equation of the kind u″=a(t)V′(u) under the following assumptions: V∈C2(R) is a non-negative double well potential which admits just one critical point between the two wells, a(t) is measurable, asymptotically periodic and such that inf⁡a>0, sup⁡a<+∞. In particular, we improve earlier results in the so called asymptotically autonomous case, when the periodic part of a, say a˜, is constant, i.e. a(t) converges to a positive value l as |t|→+∞. Furthermore, whenever a˜ fulfils a suitable non-degeneracy condition, the solutions are shown to be infinitely many.

Characteristics and application of modern distance education teaching model

Teaching mode of modern distance medical education mainly includes teacher-centered teaching mode and the student-centered teaching model, while the latter derivatives autonomous planning learning, collaborative learning and case learning, etc. Student-centered teaching mode embodies the teaching characteristics and advantages of modern distance medical education. In view of the diversity of teaching objects, we should give full play to the advantages of various teaching models and reasonably apply these teaching modes. To strengthen the construction of practical training courses, improve all kinds of teaching resources and the safeguard, train a contingent of qualified teachers is the important factor to improve teaching quality. Key words: Distance medical education; The model of teaching; Adult education

Instability in Linear Cooperative Systems of Ordinary Differential Equations

It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ is in no relation to the sign of the real parts of the eigenvalues of $A(t)$. In particular, the real parts of all eigenvalues can be negative and bounded away from zero, nonetheless there is a solution of magnitude growing to infinity. In this paper we present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems). We illustrate those examples both with interactive animations and analytically. The paper is written in such a way that it can be accessible to students with diverse mathematical backgrounds/skills.

On some difference equations with exponential nonlinearity

The problem of the existence of complex $\ell_1$ solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and $z-$ transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.

On a fractional degenerate Kirchhoff-type problem

In this paper, we study a highly nonlocal parametric problem involving a fractional-type operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Our approach is of variational nature; by working in suitable fractional Sobolev spaces which encode Dirichlet homogeneous boundary conditions, and exploiting an abstract critical point theorem for smooth functionals, we derive the existence of at least three weak solutions to our problem for suitable values of the parameters. Finally, we provide a concrete estimate of the range of these parameters in the autonomous case, by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. The methods adopted here can be exploited to study different classes of elliptic problems in presence of a degenerate nonlocal term.

Ground state solutions for a fractional Schrödinger equation with critical growth

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u+V(x)u= f(u) \mbox{ in } \mathbb{R}^{N}, \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ is a bounded potential satisfying suitable assumptions, and $f\in C^{1, \beta}(\mathbb{R}, \mathbb{R})$ has critical growth. We first analyze the case $V$ constant, and then we develop a Jeanjean-Tanaka argument \cite{JT} to deal with the non autonomous case. As far as we know, all results presented here are new.

An autonomous Kirchhoff-type equation with general nonlinearity in [formula omitted

We consider the following autonomous Kirchhoff-type equation −(a+b∫RN|∇u|2)Δu=f(u),u∈H1(RN), where a≥0,b>0 are constants and N≥1. Under general Berestycki–Lions type assumptions on the nonlinearity f, we establish the existence results of a ground state and multiple radial solutions for N≥2, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for N=1. The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and a new description of the critical values in association with the level sets argument.

Global attractors for $p$-Laplacian differential inclusions in unbounded domains

In this work we consider a differential inclusion governed by a p-Laplacianoperator with a diffusion coefficient depending on a parameter in which thespace variable belongs to an unbounded domain. We prove the existence of aglobal attractor and show that the family of attractors behaves uppersemicontinuously with respect to the diffusion parameter. Both autonomous andnonautonomous cases are studied.

Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps

In this paper, we study an averaging principle for stochastic FitzHugh-Nagumo system with different time scales driven by cylindrical Wiener processes and Poisson jumps, where the slow equation is non-autonomous and the fast equation is autonomous case. Under suitable assumptions, we show that the slow component mean-square strongly converges to the solution of the corresponding averaging equation, and the rate of the convergence as a by-product is also affirmed. Finally, we give some open problems which are derived from this paper.

Phase Difference in Lateral Synchronization of Pedestrian Floors Using a Modified Hybrid Van der pol/Rayleigh Oscillator

The modified hybrid Van der Pol/Rayleigh (MHVR) oscillator was originally proposed by the authors to model the lateral oscillations of a pedestrian walking on a rigid floor and it was shown that for the autonomous case, the MHVR oscillator can correctly fit the experimental data. The case of a pedestrian walking on a laterally moving floor is modeled by a nonautonomous oscillator. The case of a floor subjected to a harmonic lateral motion has been then studied by the authors, with focus on the amplitude and stability of the entrained response, i.e. the response having the same frequency as that of the given periodic excitation. For the nonautonomous (moving floor) case, the main focus of this paper is on the analysis of the phase difference between the oscillator entrained response and the external excitation. Both analytical and numerical calculations have been performed. The approximate analytical method is the harmonic balance method. Then, the model is used to represent the experimental results for the pedestrian lateral oscillations during walking. Comparison is made for the examples along with discussions.

Entorno ecológico del caso índice de Leishmaniasis cutánea, Unquillo, Córdoba

ABSTRACT INTRODUCTION AND OBJECTIVES: Cordoba hills are located to the WEST of the province and are arranged in three mountain ranges, in one of these, on the EASTERN slope, we find the town of Unquillo, place of the first autonomous case of skin leishmaniasis in the province of Cordoba, Argentina, which was reported and published by us (1).We consider important to gather information about environmental and demographic factors that may have favored the settlement of Lutzomya spp. MATERIALS AND METHODS: On-site inspection, observational examination of the city and surrounding areas, bibliographic analysis of statistics, census, geomorphology and phytogeography were used to characterize the pattern which functions as the ecological niche for the vector.RESULTS: The geography of Unquillo presents slopes, faults and ravines, with soil rich in organic nutrients, good levels of rainfalls and human activity patches in the wild environment resulting in the ideal setting for the settlement and reproduction of the transmitting insect. CONCLUSIONS: The study shows the so-called draining corridors of Lutzomya spp., convergent from endemic provinces with the zoonosis towards these mountain tows; Unquillo among them. The analysis adds extra value and a threat for public health if we consider the proximity of this town to the large city of Cordoba

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

The dynamics of combat module with nonlinear elastic support characteristics

The angular combat module oscillations are investigated. The combat module is situated in the combat vehicle platform using a nonlinear elastic characteristic support. The non autonomous and autonomous cases, the influence of resistance force are considered. The analytical methods for constructing solutions of appropriate models of combat module dynamics are presented by the using the Van der Pol and Ateb-periodic functions. The obtained results is the basis for evaluation of influence of angular oscillations on shooting accuracy on the move on rough terrain forming design solutions to reduce these vibrations.

Transit times and mean ages for nonautonomous and autonomous compartmental systems.

We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick–von Förster equation, we show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time and show that our nonautonomous theory generalises the autonomous case. We apply these results to study a nine-dimensional nonautonomous compartmental system modeling the terrestrial carbon cycle, which is a modification of the Carnegie–Ames–Stanford approach model, and we demonstrate that the nonautonomous versions of transit time and mean age differ significantly from the autonomous quantities when calculated for that model.

Extinction, periodicity and multistability in a Ricker model of stage-structured populations

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage (adult-juvenile) biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case yielding general conditions for extinction in the biological context. We also study the dynamics of an autonomous special case of the equation that generates multistable periodic and non-periodic orbits in the positive quadrant of the plane.