Autonomous Case Research Articles

A note on the Darboux theory of integrability of non-autonomous polynomial differential systems

In this work, we unfold some differential algebraic aspects of Darboux first integrals of polynomial vector fields. An interesting improvement is that our approach can be applied both to autonomous and non-autonomous vector fields. We give a sufficient and necessary condition for the existence of a Darboux first integral of a specific form for a polynomial vector field with some known algebraic invariant hypersurfaces. For the autonomous case, the classical result of Darboux is obtained as a corollary. For the non-autonomous case our characterization improves a known criterium.

Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations

We use Liapunov functionals to obtain sufficient conditions that ensure exponential stability of the nonlinear Volterra integro-differential equation $$x^{\prime }(t)=p(t)x(t)-\int \limits _{t-\tau }^{t}q(t,s)x(s)ds,$$ where the constant \(\tau \) is positive, the function \(p\) does not need to obey any sign condition and the kernel \(q\) is continuous. Our results improve the results obtained in literature even in the autonomous case. In addition, we give a new criteria for instability.

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case.

A Central Limit Theorem for Time-Dependent Dynamical Systems

The work by Ott et al. (Math. Res. Lett. 16:463–475, 2009) established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random examples are also found, which are not small perturbations of a given map. Our approach uses martingale approximation technique in the form of Sethuraman and Varadhan (Electron. J. Probab. 10:121–1235, 2005).

Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations

In this paper, we consider a non-autonomous Navier–Stokes–Voigt model, with which a continuous process can be associated. We study the existence and relationship between minimal pullback attractors for this process in two different frameworks, namely, for the universe of fixed bounded sets, and also for another universe given by a tempered condition.Since the model does not have a regularizing effect, obtaining asymptotic compactness for the process is a more involved task. We prove this in a relatively simple way just using an energy method. Our results simplify—and in some aspects generalize—some of those obtained previously for the autonomous and non-autonomous cases, since for example in section 4, regularity is not required for the boundary of the domain and the force may take values in V′. Under additional suitable assumptions, regularity results for these families of attractors are also obtained, via bootstrapping arguments. Finally, we also conclude some results concerning the attraction in the D(A) norm.

Minimal sets in monotone and concave skew-product semiflows I: A general theory

The long-term dynamics of a general monotone and concave skew-product semiflow is analyzed, paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension. Different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one, as well as on the coexistence or not of bounded and unbounded semiorbits on the region. Previous results are unified and extended, and significative differences with the sublinear case are established. Some scenarios which are impossible in the autonomous or periodic cases may occur in this setting.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions

Of concern are the initial-boundary value problems for nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions. We succeed in establishing a global wellposedness theorem (in a classical sense) for these problems via a specifically designed operator theoretic approach. Moreover, we obtain sharp estimations for the evolution of the solution along the characteristic curves, which enable us to derive the uniform exponential decay of the associated energies. The results obtained in this paper are still new even for the autonomous case.

Lyapunov stability of n-D strongly autonomous systems

In this article we look into stability properties of strongly autonomous n-D systems, i.e. systems having finite-dimensional behaviour. These systems are known to have a first-order representation akin to 1-D state-space representation; we consider our systems to be already in this form throughout. We first define restriction of an n-D system to a 1-D subspace. Using this we define stability with respect to a given half-line, and then stability with respect to collections of such half-lines: proper cones. Then we show how stability with respect to a half-line, for the strongly autonomous case, reduces to a linear combination of the state representation matrices being Hurwitz. We first relate the eigenvalues of this linear combination with those of the individual matrices. With this we give an equivalent geometric criterion in terms of the real part of the characteristic variety of the system for half-line stability. Then we extend this geometric criterion to the case of stability with respect to a proper cone. Finally, we look into a Lyapunov theory of stability with respect to a proper cone for strongly autonomous systems. Each non-zero vector in the given proper cone gives rise to a linear combination of the system matrices. Each of these linear combinations gives a corresponding Lyapunov inequality. We show that the system is stable with respect to the proper cone if and only if there exists a common solution to all of these Lyapunov inequalities.

Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation

The Chafee-Infante equation is one of the canonical infinite-dimensional dynamical systems for which a complete description of the global attractor is available. In this paper we study the structure of the pullback attractor for a non-autonomous version of this equation, u t = u x x + λ u − β ( t ) u 3 u_t=u_{xx}+\lambda u-\beta (t)u^3 , and investigate the bifurcations that this attractor undergoes as λ \lambda is varied. We are able to describe these in some detail, despite the fact that our model is truly non-autonomous; i.e., we do not restrict to ‘small perturbations’ of the autonomous case.

Asymptotic regularity and attractors of the reaction–diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction–diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H 1 ( Ω ) with dissipative internal and boundary nonlinearities.

On existence and uniqueness of positive solutions to a class of fractional boundary value problems

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = 0 , where 2 < α ≤ 3 and D 0 + α is the Riemann-Liouville fractional derivative.Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated.We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear).2010 Mathematics Subject Classification: 34B15

A trichotomy result for non-autonomous rational difference equations

Abstract We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.

A net reproductive number for periodic matrix models

We give a definition of a net reproductive number R 0 for periodic matrix models of the type used to describe the dynamics of a structured population with periodic parameters. The definition is based on the familiar method of studying a periodic map by means of its (period-length) composite. This composite has an additive decomposition that permits a generalization of the Cushing–Zhou definition of R 0 in the autonomous case. The value of R 0 determines whether the population goes extinct (R 0<1) or persists (R 0>1). We discuss the biological interpretation of this definition and derive formulas for R 0 for two cases: scalar periodic maps of arbitrary period and periodic Leslie models of period 2. We illustrate the use of the definition by means of several examples and by applications to case studies found in the literature. We also make some comparisons of this definition of R 0 with another definition given recently by Bacaër.

Some results on minimizers and stable solutions of a variational problem

AbstractWe consider the functional in a periodic setting. We discuss whether the minimizers or the stable solutions satisfy some symmetry or monotonicity properties, with special emphasis on the autonomous case when F is x-independent. In particular, we give an answer to a question posed by Victor Bangert when F is autonomous in dimension n≤3 and in any dimension for non-zero rotation vectors.

Rigorous Component-Based System Design Using the BIP Framework

Rigorous system design requires the use of a single powerful component framework allowing the representation of the designed system at different detail levels, from application software to its implementation. A single framework allows the maintenance of the overall coherency and correctness by comparing different architectural solutions and their properties. The authors present the BIP (behavior, interaction, priority) component framework, which encompasses an expressive notion of composition for heterogeneous components by combining interactions and priorities. This allows description at different abstraction levels from application software to mixed hardware/software systems. A rigorous design flow that uses BIP as a unifying semantic model derives a correct implementation from an application software, a model of the target architecture, and a mapping. Implementation correctness is ensured by applying source-to-source transformations that preserve correctness of essential design properties. The design is fully automated and supported by a toolset including a compiler, the D-Finder verification tool, and model transformers. The authors present an autonomous robot case study to illustrate BIP's use as a modeling formalism as well as crucial aspects of the design flow for ensuring correctness.

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points. To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations

In this paper we study the long time behavior of the three dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Navier-Stokes-Voight equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to $V'.$ For the nonautonomous case, we only assume that $f(x,t)$ is translation bounded instead of translation compact, where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal projection. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Navier-Stokes-Voight equations and also obtain the existence of the uniform attractor. Finally, we describe the structure of the uniform attractor and its regularity.

Evolución de la pandemia por el virus de la gripe (H1N1) 2009 en la Comunidad Autónoma de la Rioja

BACKGROUND. The aim of this study has been to analyze the development of pandemic influenza (H1N1) 2009 in La Rioja autonomous (cases in primary healthcare and cases hospitalized). METHODS. We have included all cases with 2009 H1N1 influenza from week 28-2009 to 3-2010. The main information was collected through Primary Healthcare databases (OMI), lists of Preventive Medicine Hospital Departments, reference laboratory and individual standardized forms. This information was analyzed and described according to week, sex, age, comorbidities, clinical development and medical complication. We explore the association of to antiviral treatment and risk factors. This association was assessed using odds ratios (OR) and 95% confidence interval (CI). Adjustment for confounders was performed using unconditional logistic regression to estimate the odds ratio (OR) and 95% confidence intervals (CI). RESULTS. From 29-2009 week to 3-2010 week, a total of 7096 pandemic influenza A(H1N1) cases were notified from Primary Healthcare and 111 laboratory confirmed severe cases were admitted to hospital. Five cases were admitted to an ICU (4,5%) and 2 cases died. All this patients had comorbidities. None of children and pregnant female were admitted to an ICU. CONCLUSIONS. Children and young person have been population with more cases of pandemic influenza (H1N1) 2009. Clinical evolution of pandemic in the majority cases in La Rioja, has been mild.