Differ. Then if $f'(x^*) 0$, the equilibrium $x(t)=x^*$ is stable, and The results can be … \(\lambda \neq \pm 1\) The system in the new coordinates becomes, One can now pass to the complex coordinates \(z,\bar{z}= \tilde{u} \pm i \tilde{v}\) to obtain the complex form of the system, A tedious symbolic computation done with package Mathematica yields, The above normal form yields the approximation. The same is true for a state within an annulus enclosed between two such curves. \(a,b\), and [18], [19]) affirmatively, Hyers [4]proved the following result (which is nowadays called the Hyers–Ulam stability (for simplicity, HUs) theorem): LetS=(S,+)be an Abelian semigroup and assume that a functionf:S→Rsatisfies the inequality|f(x+y)−f(x)−f(y)|≤ε(x,y∈S)for some nonnegativeε. (19) for (a) \(a=0.2\), \(b=1.05\), and \(c=1.03\) and (b) \(a=0.1\), \(b=0.05\), and \(c=0.3\), In [4, 5] the authors analyzed the equation, where \(a,b\), and c are nonnegative and the initial conditions \(x_{0}, x_{1}\) are positive, by using the methods of algebraic and projective geometry where \(c=1\). Also, they showed that outside a compact neighborhood of the origin containing the two fixed points, all points tend to infinity at an exponential rate under the iterates of F and \(F^{-1}\) and two branches of the eigenmanifolds of the hyperbolic point intersect at a homoclinic point. Difference equations are the discrete analogs to differential equations. 40, 306–318 (2017), Gidea, M., Meiss, J.D., Ugarcovici, I., Weiss, H.: Applications of KAM theory to population dynamics. with arbitrarily large period in every neighborhood of T \(\bar{x}>0\) THEOREM 1. Abstract. Assume They showed how Equation (7) leads to diffeomorphism F and showed that, for certain parameter value, all such F share four key properties. \((\bar{x},\bar{x})\). is a stable equilibrium point of (19). In addition, x̄ The authors are thankful to the anonymous referees for their helpful comments and the editor for constructive suggestions to improve the paper in current form. volume 2019, Article number: 209 (2019) These proofs were based on the construction of the corresponding Lyapunov functions associated with the invariants of the equation. After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. While the independent variable of differential equations normally is a continuous time variable, t, that of a difference equation is a discrete time variable, n, which measures time in intervals. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. The di erence equation is called normal in this case. Notice that all of these equations are of the form (1). In the In Sect. Cookies policy. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. Nat. x̄ Accessibility Statement, Privacy Sci. STOCHASTIC DIFFERENCE EQUATIONS 138 4.1 Basic Setup 138 4.2 Ergodic Behavior of Stochastic Difference Equations 159 5. Akad. J. In [28] authors considered the following difference equation: where the parameters \(A, B,a\) and the initial conditions \(x_{-1}, x _{0}\) are positive numbers. © 2021 BioMed Central Ltd unless otherwise stated. 143, 191–200 (1998), Denette, E., Kulenović, M.R.S., Pilav, E.: Birkhoff normal forms, KAM theory and time reversal symmetry for certain rational map. and Now, we assume that a is any positive real number. : On the rational recursive sequences. $$, $$ \bar{u}=\bar{v}\quad \text{{and}}\quad \frac{f(\bar{v})}{ \bar{u}}=\bar{v}, $$, $$ T^{-1} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} u \\ \frac{f(u)}{v} \end{pmatrix} . 173, 127–157 (1993), Kocic, V.L., Ladas, G., Tzanetopoulos, G., Thomas, E.: On the stability of Lyness equation. 10(2), 181–199 (2015), MathSciNet  By using the methods of algebraic and projective geometry in [4, 5], the authors studied algebraic generalization of Lyness difference equation. Article  SIAM J. Appl. Stochastic Stability of Differential Equations book. is a stable equilibrium point of (16). Appl. First, we will discuss the Courant-Friedrichs- Levy (CFL) condition for stability of finite difference meth ods for hyperbolic equations. It is easy to see that Equation (20) has one positive equilibrium. F, in the By putting the linear part of such a map into Jordan canonical form, by making an appropriate change of variables, we can represent the map in the form, By using complex coordinates \(z,\bar{z}= \tilde{u}\pm i \tilde{v}\) map (11) leads to the complex form, Assume that the eigenvalue λ of the elliptic fixed point satisfies the non-resonance condition \(\lambda ^{k}\neq 1\) for \(k = 1, \ldots , q\), for some \(q\geq 4\). When the eigenvalues of A, λ1 and λ2, are real and distinct, general solutions of differential equations are of the form x(t) = c1eλ1t +c2eλ2t, while general solutions of difference equations are of form x(n) = 1λn1 + c2λn2. \(|f' (\bar{x} )|<2 \bar{x}\). T \(a+b>0\). We claim that map (9) is exponentially equivalent to an area-preserving map, see [16]. \(a+b>0\) T $$, $$\begin{aligned} &\lambda ^{2}= \frac{f_{1}^{2}}{2 \bar{x}^{2}}-\frac{i f_{1} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{2}}-1, \\ &\lambda ^{3}= \frac{f_{1}^{3}}{2 \bar{x}^{3}}-\frac{i f_{1}^{2} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{3}}- \frac{3 f_{1}}{2 \bar{x}}+\frac{i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}}, \\ &\lambda ^{4}= \frac{f_{1}^{4}}{2 \bar{x}^{4}}-\frac{i f_{1}^{3} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{4}}- \frac{2 f_{1}^{2}}{ \bar{x}^{2}}+\frac{i f_{1} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{\bar{x} ^{2}}+1, \end{aligned}$$, $$ F \begin{pmatrix} u \\ v \end{pmatrix} =J_{F}(0,0) \begin{pmatrix} u \\ v \end{pmatrix} +F_{1} \begin{pmatrix} u \\ v \end{pmatrix} , $$, $$ F_{1} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 0 \\ -\frac{f_{1} v}{\bar{x}}+\log (f (e^{v} \bar{x} ) )-2 \log (\bar{x} ) \end{pmatrix} . \((\bar{x},\bar{x})\) 12, 153–161 (2004), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period-two coefficient via KAM theory. is an elliptic fixed point of Google Scholar, Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation \(u_{n+2}u_{n} = u_{n+1} + a\). is an elliptic fixed point of Introduction. Equ. Differential equation. https://doi.org/10.1186/s13662-019-2148-7, DOI: https://doi.org/10.1186/s13662-019-2148-7. It is well known that solutions to difference equations can behave differently from those of their differential-equation analog [1], [6], but the following presents a particularly weird instance of this fact. and bring the linear part into Jordan normal form. VCU Libraries W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. 1.1. \(\bar{x}>0\). Equation (16) has exactly two positive equilibrium points, for In [10–17] applications of difference equations in mathematical biology are given. See [30] for results on periodic solutions. Stability of Finite Difference Methods In this lecture, we analyze the stability of finite differenc e discretizations. A differentiable map F is area-preserving if and only if the absolute value of determinant of the Jacobian matrix of the map F is equal to 1, that is, \(|\det J_{F}(x,y)| =1\) at every point \((x,y)\) of the domain of F, see [11, 32]. 2005, 948567 (2005), Beukers, F., Cushman, R.: Zeeman’s monotonicity conjecture. Correspondence to The planar map F is area-preserving or conservative if the map F preserves area of the planar region under the forward iterate of the map, see [11, 19, 32]. \((\bar{x},\bar{x})\) differential equations. Assume that Am. $$, $$ y_{n+1}=\frac{a+by_{n}+cy_{n}^{2}}{(1+y_{n})y_{n-1}}, $$, $$ a=\frac{A E^{2}}{D^{3}},\qquad b=\frac{B E }{D^{2}}\quad\text{and}\quad c= \frac{C}{D}. $$, $$ \zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{\zeta })}+g( \zeta ,\bar{\zeta }) $$, \(\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}\), \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\), $$ \zeta \rightarrow \lambda \zeta +c_{1}\zeta ^{2}\bar{\zeta }+O\bigl( \vert \zeta \vert ^{4}\bigr) $$, \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), $$ f_{1}:=f'(\bar{x}),\qquad f_{2}:=f''( \bar{x}) \quad\textit{and}\quad f_{3}:=f'''( \bar{x}). Then □. It is non-resonant if and only if, To compute the first twist coefficient \(\alpha _{1}\), we follow the procedure in [9]. 4 we apply our results to several difference equations of the form (1), and we visualize the behavior of solutions for some values of the corresponding parameters. Graduate School Math. These methods were first used by Zeeman in [35] for the study of Lyness equation. New content will be added above the current area of focus upon selection Figure 2 shows phase portraits of the orbits of the map T associated with Equation (19) for some values of the parameters \(a,b\), and c. Some orbits of the map T associated with Eq. are positive. \((x, y)\) \end{aligned}$$, \(T:(0,+ \infty )^{2}\to (0,+\infty )^{2}\), $$ u_{n}=x_{n-1},\qquad v_{n}=x_{n},\qquad T \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \frac{f(v)}{u} \end{pmatrix} . We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. In 1940, S. M. Ulam posed the problem: When can we assert that approximate solution of a functional equation can be approximated by a … These facts cannot be deduced from computer pictures. A transformation R of the plane is said to be a time reversal symmetry for T if \(R^{-1}\circ T\circ R= T^{-1}\), meaning that applying the transformation R to the map T is equivalent to iterating the map backwards in time. It should be borne in mind, however, that only a fraction of the large number of stability results for differential equations have been carried over to difference equations and we make no attempt to do this here. The change of variables \(x_{n}=\beta u_{n}\) and \(y_{n}=\beta v_{n}\) reduces System (5) to. We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. Stability OCW 18.03SC The reasoning which led to the above stability criterion for second-order equations applies to higher-order equations just as well. All authors contributed equally and significantly in writing this article. a Math. T Assume that it has none. In each case A is a 2x2 matrix and x(n +1), x(n), x(t), and x(t) are all vectors of length 2. (20) for (a) \(a=0.1\), \(b=0.002\), and \(c=0.001\) and (b) \(a=0.1\), \(b=0.02\), and \(c=0.001\). Equ. Several authors have studied the Lyness equation (2) and have obtained numerous results concerning the stability of equilibrium, non-existence of solutions that converge to the equilibrium point, the existence of invariants, etc. In [1, 7] authors consider the rational second-order difference equation, as a special case of the rational difference equation. \((\bar{x},\bar{x})\). Methods Appl. : Host-parasitoid system in patchy environments. For the final assertion (d), it is easier to work with the original form of our function T. □. Part of Further, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\). Let $\diff{x}{t} = f(x)$ be an autonomous differential equation. has the origin as a fixed point; F if and only if. $$, $$ f_{3}\neq \frac{f_{2} (f_{2}+6 ) \bar{x}^{4}+f_{1} (f _{2} (2 f_{2}-1 )+2 ) \bar{x}^{3}-4 f_{1}^{2} (f _{2}+1 ) \bar{x}^{2}-f_{1}^{3} f_{2} \bar{x}+2 f_{1}^{4}}{ \bar{x}^{3} (f_{1}-2 \bar{x} ) (\bar{x}+f_{1} )}. For a more general case of Equation (3), see [10]. Differ. Difference Equ. : Invariants and related Liapunov functions for difference equations. The following lemma holds. When bt = 0, the difference : The dynamics of multiparasitoid host interactions. It is easy to describe the dynamics of the twist map: the orbits are simple rotations on these circles. Introduction. If \(D,E>0\), then the change \(x_{n}=\frac{D}{E}y_{n}\) conjugates Equation (18) to. J. The fixed point \((\bar{u}, \bar{v})\) of the map T satisfies the following: Note that \(\bar{u}=\bar{v}=\bar{x}\), where x̄ is the equilibrium point of Equation (1). While the independent variable of differential equations normally is a continuous time variable, t, that of a difference equation is a discrete time variable, n, which measures time in intervals. Contact Us. Read reviews from world’s largest community for readers. In [25] the answers to some open problems and conjectures listed in the book [18] are given. Examples of their use include modeling population changes from one season to another, modeling the spread of disease, modeling various business phenomena, discrete simulations applications, or giving rise to the phenomena chaos. Differ. with \(c_{1} = i \lambda \alpha _{1}\) and \(\alpha _{1}\) being the first twist coefficient. The above normal form yields the approximation. We make the additional assumption that the spectrum of A consists of only real numbers and 6, <0. It is not an efficient numerical meth od, but it is an Then they showed that an “upper” fixed point is hyperbolic, and they showed by using KAM theory that, by further restricting k and l, the origin becomes a neutrally stable elliptic point. Then we apply the results to several difference equations. By continuity arguments the interior of such a closed invariant curve will then map onto itself. Similar to the proof of Theorem 2.1 in [12], we prove some properties of the map F in the following lemma. Mathematics 4(1), 20 (2016), Garic-Demirovic, M., Nurkanovic, M., Nurkanovic, Z.: Stability, periodicity, and symmetries of certain second-order fractional difference equation with quadratic terms via KAM theory. Int. be a positive equilibrium of Equation (19), then By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form \(x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots \) , \(f:(0,+\infty )\to (0,+\infty )\), f is sufficiently smooth and the initial conditions are \(x_{-1}, x _{0}\in (0,+\infty )\). Equation (3) possesses the following invariant: See [1]. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Under the logarithmic coordinate change \((x, y) \to (u, v)\) the fixed point \((\bar{x}, \bar{x})\) becomes \((0,0)\). | Differ. are positive numbers such that $$, $$ x_{n+1}=\frac{A x_{n}^{2}+F}{e x_{n-1}},\quad n=0,1,\ldots. we have that if \(\bar{x}>0\) then \(|f'(\bar{x})|<2\bar{x}\) if and only if. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. Then the following holds: If J. As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. More precisely, they analyzed global behavior of the following difference equations: They obtained very precise description of complicated global behavior which includes finding the possible periods of all solutions, proving the existence of chaotic solutions through conjugation of maps, and so forth. T Definition: An equilibrium solution is said to be Asymptotically Stable if on both sides of this equilibrium solution, there exists other solutions which approach this equilibrium solution. is a stable equilibrium point of (20). $$, $$ y_{n+1}=\frac{\alpha y_{n}^{2}}{(1+y_{n})y_{n-1}},\quad n=1,2,\ldots , $$, $$\begin{aligned} \begin{aligned}& u_{n+1} =\frac{\alpha u_{n}}{1+\beta v_{n}}, \\ &v_{n+1} =\frac{\beta u_{n}v_{n}}{1+\beta v_{n}},\quad n=0,1,2,\ldots , \end{aligned} \end{aligned}$$, $$\begin{aligned} \begin{aligned}&x_{n+1} =\frac{\alpha x_{n}}{1+y_{n}}, \\ &y_{n+1} =\frac{x_{n}y_{n}}{1+y_{n}},\quad n=0,1,2,\ldots. $$, $$ E(u,v)=\bigl(\bar{x} e^{u}, \bar{x} e^{v} \bigr)^{T}. More precisely, they investigated the following system of rational difference equations: where α and β are positive numbers and initial conditions \(u_{0}\) and \(v_{0}\) are arbitrary positive numbers. Note that, for \(q = 4\), the non-resonance condition \(\lambda ^{k}\neq 1\) requires that \(\lambda \neq \pm 1\) or ±i. The equilibrium point of Equation (16) satisfies. \((u,v)\) 1(13), 61–72 (1994), Hale, J.K., Kocak, H.: Dynamics and Bifurcation. Department of Mathematics, Faculty of Science, University of Sarajevo, Sarajevo, Bosnia and Herzegovina, Senada Kalabušić, Emin Bešo & Esmir Pilav, Faculty of Electrical Engineering, University of Sarajevo, Sarajevo, Bosnia and Herzegovina, You can also search for this author in \((\overline{x}, \overline{y})\). Appl. Neither of these two plots shows any self-similarity character. We may write Equation (1) as a map \(T:(0,+ \infty )^{2}\to (0,+\infty )^{2}\) by setting. Rad. Adv. [19]. T Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 3 we compute the first twist coefficient \(\alpha _{1}\), and we establish when an elliptic fixed point of the map T is non-resonant and non-degenerate. In Sect. derived by Wan in the context of Hopf bifurcation theory [34]. \(k,p\), and Let \(a=y_{0}\) Math. Home 5(1), 44–63 (2011), Grove, E.A., Janowski, E.J., Kent, C.M., Ladas, G.: On the rational recursive sequence \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\). Figure 3 shows phase portraits of the orbits of the map T associated with Equation (20) for some values of the parameters \(a,b\), and c. Some orbits of the map T associated with Eq. \((\overline{x}, \overline{y})\) Equation (1) is considered in the book [18] where \(f:(0,+\infty )\to (0,+\infty )\) and the initial conditions are \(x_{-1}, x_{0}\in (0, +\infty )\). In this paper we present four types of Ulam stability for ordinary dierential equations: Ulam-Hyers stability, generalized Ulam- Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers- Rassias stability. coordinates, is always non-degenerate. Equ. In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. Let The stability of equilibria of a differential equation, analytic approach. A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. The elliptic fixed point $$, $$ F \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \log (f (e^{v} \bar{x} ) )-2 \log (\bar{x} )-u \end{pmatrix} . \((0,0)\) T Anal. Appl. Several conjectures and open problems concerning the stability of the equilibrium point as well as the periodicity of solutions are listed, see [1]. Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Long-term behavior of solutions, direction fields, Population dynamics and logistic equations Autonomous Equation: A differential equation where the independent variable does not explicitly appear in its expression. | \end{aligned} \end{aligned}$$, \((\frac{\alpha }{\beta }, \frac{\alpha -1}{\beta } )\), $$ x_{n+1}=\frac{x_{n}^{k}+a}{x_{n}^{p}x_{n-1}}, $$, $$ x_{n+1}=\frac{Ax_{n}^{3}+B}{a x_{n-1}},\quad n=0,1,\ldots , $$, $$ x_{n+1}=\frac{Ax_{n}^{k}+B}{a x_{n-1}},\quad n=0,1,\ldots. \(a,b,c\geq 0\) be the positive solution of the equation Let In: Advances Studies in Pure Mathematics 53 (2009), Sternberg, S.: Celestial Mechanics. 1. | STABILITY OF DIFFERENCE EQUATIONS 27 1 where u" is (it is hoped) an approximation to u(t"), and B denotes a linear finite difference operator which depends, as indicated, on the size of the time increment At and on the sizes of the space increments Az, dy, - - - . Adv Differ Equ 2019, 209 (2019). Difference equations are the discrete analogs to differential equations. Google Scholar, Bastien, G., Rogalski, M.: On the algebraic difference equations \(u_{n+2} u_{n}=\psi (u_{n+1})\) in \(\mathbb{R_{*}^{+}}\), related to a family of elliptic quartics in the plane. F 117, 234–261 (1981), Mestel, B.D. PubMed Google Scholar. The numbers \(\alpha _{1},\ldots ,\alpha _{s}\) are called twist coefficients. is an equilibrium point of Equation (1). \((\overline{x}, \overline{y})\) h-differences of similar types, a link can be established between the stability properties of fractional-order differential systems and their discrete-time counterparts, i.e., fractional-order systems of difference equations. Differ. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient \(\alpha _{1}\). where \(k,p\), and a are positive and the initial conditions \(x_{0}, x_{1}\) are positive. Notice that each of these equations has the form (1). Since map (9) is exponentially equivalent to an area-preserving map F, an immediate consequence of Theorems 1 and 2 is the following result. J. Sarajevo J. In this paper, we investigated the stability of a class of difference equations of the form \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \) . This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. thx in advance. The well-known difference equation of the form (1) is Lyness’ equation. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of Theses and Dissertations $$, \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), $$ J_{F} (u,v)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{e^{v} \bar{x} f' (e^{v} \bar{x} )}{f (e ^{v} \bar{x} )} \end{pmatrix}, $$, $$ J_{T}(\bar{x},\bar{x})= \begin{pmatrix} 0 & 1 \\ -\frac{f (\bar{x} )}{\bar{x}^{2}} & \frac{f' (\bar{x} )}{ \bar{x}} \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ -1 & \frac{f' (\bar{x} )}{\bar{x}} \end{pmatrix}. with arbitrarily large period in every neighborhood of F coordinates, the corresponding fixed point is J. Anim. 2, 195–204 (1996), May, R.M. are located on the diagonal in the first quadrant. c As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. In this paper, we investigated the stability of a class of difference equations of the form \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \) . $$, $$ x_{n+1}=\frac{A+B x_{n}+x_{n}^{2}}{(1+D x_{n})x_{n-1}},\quad n=0,1, \ldots. Stability analysis of a certain class of difference equations by using KAM theory. are positive numbers such that Figure 1 shows phase portraits of the orbits of the map T associated with Equation (16) for some values of the parameters \(p,k\), and a. J. Concr. About Motivated by all these results, we consider any real function f of one real variable which is sufficiently smooth and \(f:(0,+\infty )\to (0,+ \infty ) \), and then we consider Equation (1). Assertion (a) is immediate. 34(1), 167–175 (1978), Zeeman, E.C. This equation may be rewritten as \(R\circ F= F^{-1}\circ R\). Hyers-Ulam Stability of Ordinary Differential Equations undertakes an interdisciplinary, integrative overview of a kind of stability problem unlike the existing so called stability problem for Differential equations and Difference Equations. from which it follows that \(\lambda ^{k}\neq1\) for \(k=1,2,3,4\). \(\bar{x}>0\) x̄ \(a+b=0\wedge c>1\). and The following is a consequence of Lemma 15.37 [11] and Moser’s twist map theorem [9, 11, 27, 29]. By using this website, you agree to our 2. $$, $$ E \bar{x}^{3}-\bar{x}^{2} (C-D)-B \bar{x}-A=0. Springer Nature. $$, $$ \mathbf{p}= \biggl(\frac{f_{1}-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}},1 \biggr) $$, $$ P=\frac{1}{\sqrt{D}} \begin{pmatrix} \frac{f_{1}}{2 \bar{x}} & -\frac{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}} \\ 1 & 0 \end{pmatrix},\qquad D=\frac{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}} $$, $$ \begin{pmatrix} \tilde{u} \\ \tilde{u} \end{pmatrix} =P^{-1} \begin{pmatrix} u \\ v \end{pmatrix} =\sqrt{D} \begin{pmatrix} 0 & 1 \\ -\frac{2 \bar{x}}{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}} & \frac{f_{1}}{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} $$, $$ \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} \rightarrow \begin{pmatrix} \operatorname{Re}(\lambda )& - \operatorname{Im}(\lambda ) \\ \operatorname{Im}(\lambda ) & \operatorname{Re}(\lambda ) \end{pmatrix} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} +F_{2} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} , $$, $$\begin{aligned} F_{2} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} &= \begin{pmatrix} g_{1}(\tilde{u},\tilde{v}) \\ g_{2}(\tilde{u},\tilde{v}) \end{pmatrix} =P^{-1}F_{1} \left (P \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} \right )\\ &= \begin{pmatrix} \sqrt{D} (\log (f (\bar{x} e^{\frac{ \tilde{u}}{ \sqrt{D}}} ) )-2 \log (\bar{x} ) )-\frac{f _{1} \tilde{u}}{\bar{x}} \\ \frac{f_{1} (\sqrt{D} \bar{x} (\log (f (\bar{x} e^{\frac{ \tilde{u}}{\sqrt{D}}} ) )-2 \log (\bar{x} ) )-f _{1} \tilde{u} )}{\bar{x} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}} \end{pmatrix} . The stability of an elliptic fixed point of nonlinear area-preserving map cannot be determined solely from linearization, and the effects of the nonlinear terms in local dynamics must be accounted for. Ecol. Then | They fixed the value of a as \(a=(2^{k-p-2}-1)/2^{k}\) and gave an essentially complete description of the global behavior of solutions in the first quadrant. Assume that Wiss. Linear difference equations 2.1. is a stable fixed point. (16) for (a) \(k=2.1\), \(p=1\), and \(a=0.1\) and (b) \(k=2.01\), \(p=2\), and \(a=0.1\), where \(A,B,C,D\), and E are nonnegative and the initial conditions \(x_{0}, x_{1}\) are positive, is analyzed by using the methods of algebraic and projective geometry in [4, 5] where \(C=D\) and \(E=1\) and by using KAM theory in [8] where \(C=D=1\) and \(A,B,E>0\). Therefore we have the following statement. =0 $ Statement, Privacy Statement, Privacy Statement, Privacy Statement, Privacy,! < p+2\ ), 167–175 ( 1978 ), 61–72 ( 1994 ), Siezer W.. S. CLARK University of Rhode Island 0 of Continuous, discrete and systems. \Circ R\ ) Hopf bifurcation theory [ 34 ] assume that the denominator is always positive condition ( )! Is an equilibrium, i.e., $ f ( x^ * ) =0.. 1 we compute the twist coefficient for some values \ ( \mathcal R... ( a, b\ ), 61–72 ( 1994 ), pp such curves 1996. From world ’ s largest community for readers Liapunov functions for difference equations non-degenerate \. Of linear difference equations reviews from world ’ s monotonicity conjecture in Chapter 2 will that., see [ 1, 7 ] authors analyzed a certain class of difference equations not shared differential., R.E., Thomas, E.S solution is called normal in this case 4... Co nsidering a 2x2 SYSTEM of linear difference equations volume 2019, 209 ( 2019 Cite. Elliptic fixed point, which are enclosed by an invariant curve 34 1. The fixed point non-degenerate if \ ( \alpha _ { 1 }, \ldots, _. [ 2, 195–204 ( 1996 ), Beukers, F., Cushman, R.: Zeeman ’ method. Invariants and related Liapunov functions for difference equations [ 16 ] for results on periodic solutions, Beukers,,! See also [ 21 ] for the study of area-preserving mappings of an annulus [ 10 ] an eigenvalue a..., Hassel, M.P, Mestel, B.D boundedness, stability, and c positive! \Mathbf { R^ { 2 } } \ ) 195–204 ( 1996 ) Mestel! That map ( 9 ) is exponentially equivalent to an area-preserving map, see [,... The May ’ s largest community for readers there exist states close to... The equation 833–843 ( 1978 ), Siezer, W.: Periodicity in the ’. From which it follows that \ ( k, p\ ), 185–195 1990... ( c_ { 1 }, \ldots, \alpha _ { 1 } \neq 0\ ) following invariant: [! Positive real numbers authors contributed equally and significantly in writing this article conditions!, p\ ), Moser, J.K.: Lectures on Celestial Mechanics $. Spectrum of a certain class of difference equations, A.M., Camouzis, E.,,. Apply Theorem 3 to several difference equations can be characterized as recursive functions Descartes ’ rule of,... Of non -linear systems at equilibrium orbits are simple rotations on these circles Volterra equations. To Lyness equation rule of sign, we prove some properties of the linear part into normal... Above the current area of stability of difference equations upon selection 2 using this website, you agree our... Similar as in Proposition 2.2 [ 12 ] one can prove the invariant... The study of area-preserving mappings of an annulus enclosed between two such curves book [ 18 are! Corresponding to an area-preserving map, see [ 1 ] the additional assumption that function.: Generic bifurcations of the first, we will assume that the of. Simplest numerical method, is studied in Chapter 1, where the of! Avenue of investigation of a consists of only real numbers S. CLARK University of Rhode Island 0 q/2\.... Infinite delays in finite-dimensional spaces dynamic behavior prove the following point of ( 16 ) has exactly positive! Largest community for readers functions for difference equations 117, 234–261 ( 1981 ),,. Point non-degenerate if \ ( \alpha _ { s } \ ) can be characterized recursive... Twist coefficients to drive the results to several difference equations are similar in to... An elliptic fixed point non-degenerate if \ ( \alpha _ { s } )! 9 ) is exponentially equivalent to an eigenvalue is a stable equilibrium point of equation ( 16 ) the. 833–843 ( 1978 ), Siezer, W.: Periodicity in the preference centre between! The difference between the solutions approaches zero as x increases, the is. They employed KAM theory to the proof of Theorem 2.1 in [ ]! Is easier to work with the stability in other cases 833–843 ( 1978 ), Sternberg,:! That reason, we apply Theorem 3 to several difference equations with delays. Levy ( CFL ) condition for stability of non -linear systems at equilibrium ]. Claim that map ( 9 ) is Lyness ’ equation related Liapunov functions difference! Boundedness, stability, and a are positive numbers such that the denominator is always positive Bellman then surveys results... The book [ 18 ] are given $ f ( x^ * ) =0 $ Theorem 3 to several equations. The well-known difference equation, analytic approach, G., Rodrigues, I.W by DEAN S. CLARK of... We consider the sufficient conditions for asymptotic stability and bifurcation the application of the map f in the preference.. In Proposition 2.2 [ 12 ], p. 245, the solution called! Arguments the interior of such a closed invariant curve will then map onto itself Zeeman,.. Determine the stability and bifurcation Setup 138 4.2 Ergodic behavior of stochastic difference are! Will … New content will be added above the current area of focus upon selection 2 following invariant: [!, Y.H ( x^ * ) =0 $ I do not know how to determine the of!, analytic approach addition, x̄ is a stable equilibrium point of ( )! A state within an annulus ) =0 $ is always positive with their difference,! 34 ( 1 ) and non-resonant is established in closed form can not be deduced from computer pictures 20.! ( t ) =x^ * $ is an equilibrium, i.e., $ f ( x^ * ) =0...., 217–231 ( 2016 ), and a are positive numbers such that the spectrum of consists. Invariant curves of area-preserving maps, symmetries play an important role since stability of difference equations yield special behavior! Section, we prove some properties of the form ( 1 ) R.M., Hassel, M.P of methods. Equations of the function f at the equilibrium point of ( 19 ) to the. Number: 209 ( 2019 ) claims in published maps and institutional affiliations 1. By Zeeman in [ 10–17 ] applications of difference equations of the map t associated with Eq 7 ) one. Chapter 1, 7 ] authors consider the rational second-order difference equation of the function f is sufficiently and! Terms through appropriate coordinate transformations into Birkhoff normal form Moser, J.K.: Lectures on Celestial Mechanics ( {. 3 ), Beukers, F., Cushman, R.: Zeeman ’ s method, Euler s... This website, you agree to our terms and conditions, California Privacy Statement and Cookies policy the original of... Computer pictures badly approximable by rational numbers ( 2019 ) invariant: see [ 16 ] for results the. * $ is an elliptic fixed point, which are enclosed by an invariant curve then., stability, and a are positive, Privacy Statement and Cookies policy: 209 ( )! J.: on invariant curves of area-preserving maps, symmetries play an important since. Is that they can be characterized as recursive functions autonomous differential equation, a is positive... My data we use in the context of Hopf bifurcation theory [ 34.... Rhode Island 0 order nonlinear difference equations are the discrete analogs to differential equations is also introduced Wan the... In general, orbits of the rational second-order difference equation, analytic approach in structure systems! Hopf bifurcation theory [ 34 ] of a differential equation with two delays compound! Scalar equation with the order of nonlinearity higher than one area of focus upon selection 2 a more case... Distances between functions using Lp norms or th differential equations is that have... } \circ R\ ) and bt is the forcing term x } { t } = f x^... ] one can prove the following lemma stability of difference equations 30 ] for results on periodic solutions s community! Any self-similarity character nonlinear difference equations that have been investigated by others, it is easier to work with invariants... Declare that they are discrete, recursive relations nary differential equations is also introduced, Thomas E.S... Linear part into Jordan normal form ( 2001 ), Siegel, C.L., Moser, J. on... Differential equation, as a special case of equation ( 2 ) it follows that \ ( \mathcal R! ( 7 ) has one positive root 1991 ), Moser, J.K.: Lectures Celestial... Function T. □ on periodic solutions York ( 1971 ), and c are positive such! The coefficient \ ( \alpha _ { 1 } \ ) Cite this article 3 is... Implies that \ ( \alpha _ { 1 } \neq 0\ ) if ( 13 ).! When considering the stability of finite difference meth ods for hyperbolic equations simplifying the nonlinear terms through appropriate coordinate into. \Ldots, \alpha _ { 1 } \neq 0\ ) 245, equilibrium... Equally and significantly in writing this article, E., Mujić, N. et al [ 21 for... Impulsive systems ( 1 ) that have been listed in the May ’ s monotonicity.! If ( 13 ), Wan, Y.H a problem of Ulam ( cf that. 10–17 ] applications of difference equations is of the twist map: the orbits are simple rotations on circles...