Periodicity of a class of nonautonomous maxtype difference. In this article, we investigate the existence of positive periodic solutions for a class of nonautonomous dierence equations. In the second part of the manuscript is present the dynamics of some concrete models that may be used in ecology, biology and. The longtime behavior of autonomous difference equations or discrete semidynamical systems exhibits various features, which occur in their. We address the stability properties in a nonautonomous difference equation xn. The normal form theory has applications in bifurcation theory see e. Nonautonomous difference equations 87 for k,l e zz2. This paper focuses on stability and boundedness of certain nonlinear nonautonomous secondorder stochastic differential equations.
Forward attraction in nonautonomous difference equations. Lyapunovs second method is employed by constructing a suitable complete lyapunov function and is used to obtain criteria, on the nonlinear functions, that guarantee stability and boundedness of solutions. Nonautonomous difference equations 1061 the second equation and vice versa for h l. Strictly increasing solutions of nonautonomous difference. Sell 23 have shown that applications of topological dynamics are possible when treating nonautonomous differential equations that are either periodic, or almost periodic, in t. Asymptotic theory for a class of nonautonomous delay.
Smooth linearization of nonautonomous difference equations. Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the. For instance, this is the case of nonautonomous mechanics. Nonautonomous terms dont have to be periodic, but in applications, they often are. Global behavior of certain nonautonomous linearizable. We also obtain some global results about the behavior of solutions of the nonautonomous nonhomogeneous difference equation where g i, i 0, 1, 2 are. In this section we will consider the simplest cases. Reducing the linear part of the nonautonomous system, defined by a sequence of invertible linear operators on \\mathbb rd\, to a bounded linear operator on a banach space, we discuss the spectrum and its spectral gaps. Open problems and conjectures find, read and cite all the research you need on researchgate. The paper provides conditions sufficient for the existence of strictly increasing solutions of the secondorder nonautonomous difference equation, where is a parameter and is lipschitz continuous and has three real zeros.
On a nonautonomous difference equation with bounded. The general form of linear homogeneous nonautonomous difference equation of order is given by and the associated nonhomogeneous equation is given by where and are realvalued functions defined for. Nonautonomous riccati difference equation with real kperiodic k. Sell 6, 7 has thatshown there is a way of viewing the solutions of nonautonomous di. Autonomous equations stability of equilibrium solutions. Normal forms for nonautonomous difference equations. Under these conditions the solutions of the differential equation over. Periodicity of a class of nonautonomous maxtype difference equations. Reducing the linear part to a bounded linear operator on a banach space, we discuss the spectrum and its spectral gaps. On constrained volterra cubic stochastic operators.
On the local stability of nonautonomous difference equations in n 1. Then we obtain a gap condition for \c1\ linearization of such a nonautonomous difference equation. These equations usually describe the evolution of certain phenomena over the course of time. Qualitative analysis of nonautonomous firstorder ode. This process is experimental and the keywords may be updated as the learning algorithm improves. Open problems and conjectures find, read and cite all the. A nonautonomous system is a dynamic equation on a smooth fiber bundle over. Because of this fact, the theory of topological dynamics has not developed into a powerful technique in applications to nonautonomous equations. The problem is motivated by some models arising in. Our results are based on the explicit formulas for. Oct 18, 2017 in this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The slope depends on both the dependent and independent variables. The longtime behavior of autonomous difference equations or discrete semidynamical systems exhibits various features, which occur in their continuous counterpart of evolutionary differential equations only for higher or even infinitedimensional state spaces. The state space x is no longer a proper phase space for nonautonomous differential equations because the behavior at a given point in the.
In this paper we extend this result to periodic nonautonomous difference equations via the concept of skewproduct dynamical systems. Finally, the last section is devoted to prove the existence and uniqueness of almost automorphic solution of the nonautonomous semilinear di erence equation 1. Under certain constraints, stable periodic solutions can be guaranteed to exist, and this is used to compare the analogous behavior of a nonautonomous periodic hyperbolic difference equation to that of the nonautonomous periodic pearlverhulst logistic differential equation. Jul 17, 2010 difference equation kutta method global attractor discrete interval forward solution these keywords were added by machine and not by the authors. Bounded uniform attractors and repellors are the natural nonautonomous analogues of autonomous stable and unstable equilibria. An autonomous differential equation is an equation of the form. Difference equation kutta method global attractor discrete interval forward solution these keywords were added by machine and not by the authors. In simple cases, a di erence equation gives rise to an associated auxiliary equation rst explained in 7. This paper is devoted to the systematic study of some qualitative properties of solutions of a nonautonomous nonlinear delay equation, which can be utilized to model single population growths. Approximation of solutions to nonautonomous difference. These rules, in general, can be described by discrete mathematical models.
Nonautonomous difference equations and discrete dynamical. Aug 10, 2017 hence, one of the mathematical concepts of a dynamical system is based on the simple fact that there are certain rules that governs our natural laws. For example, the new equation will not have any bounded motions, nor any periodic motions, nor any almost periodic motions. Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. A study of the skewproduct dynamical system, periodicity, stability, centre manifold and bifurcation is present. Invariant manifolds with asymptotic phase for nonautonomous. In the discrete time case, such nonautonomous systems are generated by difference equations, i. Given a planar system of nonautonomous ordinary differential equations, conditions are given for the existence of an associative commutative unital algebra with unit and a function on an open set such that and the maps and are lorch differentiable with respect to for all, where and represent variables in. Introduction to autonomous differential equations math. An r order differential equation on a fiber bundle q r \displaystyle q\to \mathbb r is represented by a closed subbundle of a jet bundle j r q \displaystyle jrq of q r \displaystyle q. Pdf on the local stability of nonautonomous difference. That being the case, there is some additional structure to the equations we can exploit. Pdf simple nonautonomous differential equations with many limit. For every point, a solution to the equation through consists of an differentiable function defined in a neighborhood of, with and derivative with respect to satisfies for all.
This paper is devoted to the systematic study of some qualitative properties of solutions of a nonautonomous nonlinear delay equation, which can be. Qualitative analysis of a nonautonomous nonlinear delay differential equation yang kuang, binggen zhang and tao zhao received june 27, 1990, revised march 22, 1991 abstract. On the other hand, the autonomous and nonautonomous evolution equations and related topics were studied in, for example, 6, 7, 1020, and the nonlocal cauchy problem was considered in, for example, 2, 5, 18, 2126. Nonautonomous differential equations arno berger department of mathematics and statistics university of canterbury christchurch, new zealand abstract.
Finally, equation 2 is said to possess an exponential ttichotomy if there exist real numbers. We investigate the nonautonomous difference equation with real initial conditions and coefficients g i, i 0, 1 which are in general functions of n andor the state variables x n, x n. Periodic solutions to nonautonomous difference equations. Now we can see that the limiting velocity is just the equilibrium solution of the motion equation which is an autonomous equation.
In particular we prove that for each sufficiently small there exists a solution such that is increasing, and. The linearization of flows arising from autonomous ordinary differential equations and autonomous difference equations has a long history start. Jump to content jump to main navigation jump to main navigation. On the asymptotic behavior of a nonautonomous difference. A spectral theory for nonautonomous difference equations.
For instance, this is the case of nonautonomous mechanics an rorder differential equation on a fiber bundle is represented by a closed subbundle of a jet bundle of. The differential equation gives a formula for the slope. Autonomous di erential equations and equilibrium analysis an autonomous rst order ordinary di erential equation is any equation of the form. Volume 25, 2019 vol 24, 2018 vol 23, 2017 vol 22, 2016 vol 21, 2015 vol 20, 2014 vol 19, 20 vol 18, 2012 vol 17, 2011 vol 16, 2010 vol 15, 2009 vol 14, 2008 vol, 2007 vol 12, 2006 vol 11, 2005 vol 10. In summary, our system of differential equations has three critical points, 0,0, 0,1 and 3,2. In mathematics, an autonomous system is a dynamic equation on a smooth manifold. Home about us subjects contacts about us subjects contacts. In both cases, the component sets are constructed using a pullback. Nonlinear delay differential equation yang kuang, binggen zhang and tao zhao received june 27, 1990, revised march 22, 1991 abstract. We show that for a kperiodic difference equation, if a. The present paper deals with the existence and uniqueness of solutions of fractional difference equations. Being a quadratic, the auxiliary equation signi es that the di erence equation is of second order. No other choices for x, y will satisfy algebraic system 43. Various examples are given in the form of corollaries with a highly flexible integrand.
This manuscript contains a framework of the theory of nonautonomous periodic difference equations. On a class of nonautonomous maxtype difference equations wanping liu,1 xiaofan yang,1 and stevo stevic. Stability of an equilibrium solution the stability of an equilibrium solution is classified according to the. An ordinary differential equation or ode is an equation involving derivatives of an unknown quantity with respect to a single variable. Elaydi and others published nonautonomous difference equations. Nonautonomous definition of nonautonomous by merriam. Introduction to autonomous differential equations math insight. The linear autonomous difference equation of order has following general form. We present here several conjectures and open problems pertaining to the properties of omegalimitsets see kempf7andthequestion ofliftingproperties fromthe limiting equation to the original equation. Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. Then, we obtain the conditions for the existence of an. A difference equation is a relation between the differences of a function at one or more general values of the independent variable. In our opinion, the real difference is that, while sedaghats result relies on a contraction argument, the halanaytype approach is based on monotonicity arguments. More precisely, if ak is discrete almost automorphic and a nonsingular matrix and.
In this paper, we consider the following nonlocal cauchy problem for nonautonomous fractional evolution equations. The article is devoted to the study of almost periodic solutions of dierence bevertonholt equation. Pdf stability and boundedness of solutions to a certain. Then we obtain a gap condition for c1 linearization of such a nonautonomous di. On the asymptotic behavior of a nonautonomous difference equation a. Global behavior of certain nonautonomous linearizable three. Using the kras noselskii fixed point theorem, we establish sucient criteria that are easily verifiable and that generalize and improve related studies in the literature. Nonlocal cauchy problem for nonautonomous fractional. A technique is presented for determining when periodic solutions to nonautonomous periodic difference equations exist. Any two fundamental matrix solutions are related as. The equation is called a differential equation, because it is an equation involving the derivative. Nonautonomous definition of nonautonomous by the free. Nonlinear autonomous systems of differential equations.
Request pdf on a nonautonomous difference equation with bounded coefficient in this paper we investigate the boundedness, the persistence and the attractivity of the positive solutions of the. Autonomous di erential equations and equilibrium analysis. Qualitative analysis of nonautonomous firstorder ode consider the nonautonomous ode yt2y. Stability of nonautonomous differential equations lecture.
A nonautonomous differential equation over an algebra is denoted by where is a function defined in an open set. A dynamic equation on is a differential equation which. Topological equivalence of nonautonomous difference equations. On a class of nonautonomous maxtype difference equations. Mar 16, 2010 the paper provides conditions sufficient for the existence of strictly increasing solutions of the secondorder nonautonomous difference equation, where is a parameter and is lipschitz continuous and has three real zeros. Numerical simulations are presented which support our theoretical results for some concrete models. Normal forms for nonautonomous differential equations core. We also obtain some global results about the behavior of solutions of the nonautonomous nonhomogeneous difference equation where g i, i 0, 1, 2 are functions of. Algebrization of nonautonomous differential equations. The zero on the righthand side signi es that this is a homogeneous di erence equation. We prove that such equation admits an invariant continuous section an invariant manifold.
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