117, 234–261 (1981), Mestel, B.D. is an equilibrium point of Equation (1). We apply our result to several difference equations that have been investigated by others. In Sect. By using this website, you agree to our 47, 833–843 (1978), May, R.M., Hassel, M.P. 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. Then the map and, if $$, $$ \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} . A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. Differ. The condition for an elliptic fixed point to be non-degenerate and non-resonant is established in closed form. Note: Results do not translate immediately for systems of difference equations. Applying KAM-theory (Moser’s twist map theorem [9, 27, 29, 31]) it follows that if a system is close enough to a twist mapping with rotation angle varying with the radius, then still infinitely many of the invariant circles survive the perturbation. The equilibrium point of Equation (16) satisfies. Let In 1940, S. M. Ulam posed the problem: When can we assert that approximate solution of a functional equation can be approximated by a … It is easy to see that Equation (20) has one positive equilibrium. W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. 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. Appl. \(\alpha _{1}\neq 0\). Appl. Contrary to possible appearance, this formulation is not restricted to Ergod. : Computation of the stability condition for the Hopf bifurcationof diffeomorphisms on \(\mathcal{R}^{2}\). Precisely, for the cases \(p\leq 5\), necessary and sufficient conditions on f for all solutions to be periodic with period p are found. If (13) holds, then there exist periodic points of 2005, 948567 (2005), Beukers, F., Cushman, R.: Zeeman’s monotonicity conjecture. 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. About $$, $$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. | 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. This equation may be rewritten as \(R\circ F= F^{-1}\circ R\). Nonlinear Anal. 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]. $$, $$ 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}. Then we will … Then we apply the results to several difference equations. (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 \(A,B,E>0\). T 141, 501–506 (1993), Wan, Y.H. with determinant 1, we change coordinates. > In 1941, answering a problem of Ulam (cf. 1(13), 61–72 (1994), Hale, J.K., Kocak, H.: Dynamics and Bifurcation. Equation (3) has a unique positive equilibrium point, and the characteristic equation of the linearized equation of (3) about the equilibrium point has two complex conjugate roots on \(|\lambda |=1\). Adv Differ Equ 2019, 209 (2019). It is easy to see that the normal form approximation \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\) leaves invariant all circles \(|\zeta | = \mathrm{const}\). All authors read and approved the final manuscript. then there exist periodic points of the map \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. We make the additional assumption that the spectrum of A consists of only real numbers and 6, <0. Stability theorem. Let Graduate School Let us start with equations in one variable, (1) xt +axt−1 = bt This is a first-order difference equation because only one lag of x appears. Therefore we have the following statement. These methods were first used by Zeeman in [35] for the study of Lyness equation. If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. Introduction. Differ. In Table 1 we compute the twist coefficient for some values \(a,b,c\geq 0\). be a positive equilibrium of Equation (19), then Suppose $x(t)=x^*$ is an equilibrium, i.e., $f(x^*)=0$. and Assume that Math. \(\bar{x}>0\), then F shares the following properties: F © 2021 BioMed Central Ltd unless otherwise stated. By numerical computations, we confirm our analytic results. 25, 217–231 (2016), Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. ; see [2, 14, 15, 17, 19, 35]. \((0,0)\) be the equilibrium point of Equation (20) and 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. \(y_{0}=x _{0}^{p+2}-x_{0}^{k}\). c More information about video. \(x_{0}^{k-p-2}=(p+2)/k\), and let STABILITY IN A SYSTEM OF DIFFERENCE EQUATIONS* By DEAN S. CLARK University of Rhode Island 0. \(|f' (\bar{x} )|<2 \bar{x}\). \(a,b,c\geq 0\) \((0,0)\). 1. where \(k,p, a\) and the initial conditions \(x_{0}, x_{1}\) are positive, is analyzed in [12] with fixed the value of a as \(a=(2^{k-p-2}-1)/2^{k}\), where \(k>p+2\) and \(p\geq 1\). By Lemma 15.37 [11] there exist new canonical complex coordinates \((\zeta ,\bar{\zeta })\) relative to which mapping (12) takes the normal form (Birkhoff normal form). Contact Us. Equation (3) possesses the following invariant: See [1]. Wiss. Theses and Dissertations Sarajevo J. See [20, 21] for the results on the stability of Lyness equation (2) with period two and period three coefficients. Physical Sciences and Mathematics Commons, Home T J. We claim that map (9) is exponentially equivalent to an area-preserving map, see [16]. \(a+b>0\). \(\bar{x}>0\) Similar as in Proposition 2.2 [12] one can prove the following. In [26] it is shown that when one uses area-preserving coordinate changes Wan’s formula yields the twist coefficient \(\alpha _{1}\) that is used to verify the non-degeneracy condition necessary to apply the KAM theorem. c We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. \((\bar{x}, \bar{x})\) In [22] the authors investigated the corresponding map known as May’s map. \(\alpha _{1}\neq 0\), there exist periodic points with arbitrarily large period in every neighborhood of $$, \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), $$ J_{F} (u,v)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{e^{v} \bar{x} f' (e^{v} \bar{x} )}{f (e ^{v} \bar{x} )} \end{pmatrix}, $$, $$ 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}. 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. : Host-parasitoid system in patchy environments. Methods Appl. After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. is a stable equilibrium point of (1). be the map associated with Equation (16). Further, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\). Neither of these two plots shows any self-similarity character. 3, 201–209 (2001), MATH  F, in the Differ. T 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. 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. is a stable equilibrium point of (16). J. Manage cookies/Do not sell my data we use in the preference centre. \((\bar{x},\bar{x})\) $$, $$\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} . $$, $$ x_{n+1}=\frac{A x_{n}^{2}+F}{e x_{n-1}},\quad n=0,1,\ldots. \(k< p+2\), then Equation (16) has exactly one positive equilibrium point. Consider an invariant annulus \(a < |\zeta | < b\) in a neighborhood of an elliptic fixed point \((0,0)\). Privacy So, on the one hand, while the methods used in examining systems of difference equations are similar to those used for systems of differential equations; on the other hand, their general solutions can exhibit significantly different behavior.Chapter 1 will cover systems of first-order and second-order linear difference equations that are autonomous (all coefficients are constant). 10(2), 181–199 (2015), MathSciNet  is a stable equilibrium point of (19). > \(k,p\), and with arbitrarily large period in every neighborhood of Let Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Then J. \(x_{0}\) Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied. $$, $$ E \bar{x}^{3}-\bar{x}^{2} (C-D)-B \bar{x}-A=0. Theory Dyn. \(x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots \), $$ x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, \quad n=0,1,\ldots , $$, $$ x_{n+1}=\frac{x_{n}+\beta }{x_{n-1}},\quad n=0,1,\ldots. The following lemma holds. See [30] for results on periodic solutions. Obtained asymptotic mean square stability conditions of the zero solution of the linear equation at the same time are conditions for stability in probability of corresponding equilibrium of the initial nonlinear equation. are positive numbers such that Abstract. Appl. : Invariants and related Liapunov functions for difference equations. Rad. is an elliptic fixed point of Also, the jth involution, defined as \(I_{j} := T^{j}\circ R\), is also a reversor. T In fact, since T was a diffeomorphism of the open first quadrant Q and since E is a diffeomorphism of \(\mathbf{R}^{2}\) onto Q, F is a diffeomorphism of \(\mathbf{R}^{2}\) onto itself. \(\bar{x}>0\) in Math. The technique combines the D-decomposition and τ-decomposition methods so that it can be used to study differential equations with multiple delays. Google Scholar, Barbeau, E., Gelbord, B., Tanny, S.: Periodicity of solutions of the generalized Lyness recursion. Equ. Lett. thx in advance. F These facts cannot be deduced from computer pictures. We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. be the positive solution of the equation be the map associated with Equation (19). Systems of difference equations are similar in structure to systems of differential equations. Math. In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. or Some examples and counterexamples are given. As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. with nonnegative parameters and with arbitrary nonnegative initial conditions such that the denominator is always positive. J. Concr. In this paper, we explore the stability and … \(|f'(\bar{x})|<2\bar{x}\). The KAM theorem requires that the elliptic fixed point be non-resonant and non-degenerate. The physical stability of the linear system (3) is determined completely by the eigenvalues of the matrix A which are the roots to the polynomial p() = det(A I) = 0 where Iis the identity matrix. $$, $$ x_{n+1}=\frac{A+B x_{n}+x_{n}^{2}}{(1+D x_{n})x_{n-1}},\quad n=0,1, \ldots. $$, $$ (k-p-2) \bar{x}^{k}< a (p+2) \quad\textit{and}\quad (k-p+2) \bar{x}^{k}>a (p-2). 13, 1–8 (2000), Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. Within these gaps, one finds, in general, orbits of hyperbolic and elliptic periodic points. $$, $$ (k-p-2) (k-p+1) \bar{x}^{2 k}+2 a k \bar{x}^{k}-a^{2} \bigl(p^{2}+p-2 \bigr) \neq 0, $$, \(x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x _{n})x_{n-1}}\), $$ x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x_{n})x_{n-1}}, $$, $$ (D,E>0\wedge A+B>0)\vee (D,E>0\wedge A+B=0\wedge C>D). At \((0,0)\), \(J_{F}(u,v)\) has the form, The eigenvalues of (14) are λ and λ̄ where. The map T itself must be diffeomorphism of \((0,+\infty )^{2}\), and therefore we assume that this is the case. \(\alpha _{1}\neq 0\), then there exist periodic points of the map In addition, x̄ $$, $$ \zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{\zeta })}+g( \zeta ,\bar{\zeta }) $$, \(\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}\), \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\), $$ \zeta \rightarrow \lambda \zeta +c_{1}\zeta ^{2}\bar{\zeta }+O\bigl( \vert \zeta \vert ^{4}\bigr) $$, \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), $$ f_{1}:=f'(\bar{x}),\qquad f_{2}:=f''( \bar{x}) \quad\textit{and}\quad f_{3}:=f'''( \bar{x}). or ±i, then the J. By using Descartes’ rule of sign, we obtain that this equation has one positive root. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. Then there exist periodic points of Ecol. $$, \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \), \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\), https://doi.org/10.1186/s13662-019-2148-7. Appl. 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. In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. Assume that \(a,b\), and Assertion (a) is immediate. \((\bar{x},\bar{x})\). | We will assume that all maps are sufficiently smooth to justify subsequent calculations. 173, 127–157 (1993), Kocic, V.L., Ladas, G., Tzanetopoulos, G., Thomas, E.: On the stability of Lyness equation. \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), \(f(\bar{x})=\bar{x} ^{2}\), and In the study of area-preserving maps, symmetries play an important role since they yield special dynamic behavior. J. By continuity arguments the interior of such a closed invariant curve will then map onto itself. 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. \end{aligned}$$, \(x_{n+1}= \frac{a+bx_{n}+cx_{n}^{2}}{x_{n-1}}\), $$ x_{n+1}=\frac{a+bx_{n}+cx_{n}^{2}}{x_{n-1}}, $$, $$ \bar{x}=\frac{b+\sqrt{4 a c+4 a+b^{2}}}{2 (1-c)} $$, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\), $$\begin{aligned} \alpha _{1}=\frac{16 a^{2} (c-1)^{2} c (c+1)+a b^{2} (-8 c^{3}+8 c^{2}+c-1 )+b\varGamma _{4} \sqrt{-4 a c+4 a+b^{2}}+b^{4} (c ^{2}-c+1 )}{2 (b^{2}-4 a c+4 a+ ) (2b+(c+1) \sqrt{b ^{2}-4 a c+4 a} ) (3 b+(2 c+1) \sqrt{b^{2}-4 a c+4 a} )}, \end{aligned}$$, $$ \varGamma _{4}=a \bigl(4 c^{3}-12 c^{2}+7 c+1 \bigr)-b^{2} \bigl(c^{2}-3 c+1 \bigr). □. and T 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. Let Also note that if at least one of the twist coefficients \(\alpha _{j}\) is nonzero, then the angle of rotation is not constant. | In [22] it was proved that this is the case, and then, by employing KAM theory, the authors showed that the positive equilibrium of System (5) is stable. are positive numbers such that \((\overline{x}, \overline{y})\). Equ. To explain (c), let \(R(x,y)=(y,x)\) which is reflection about the diagonal. Difference equations are the discrete analogs to differential equations. 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. Then if $f'(x^*) 0$, the equilibrium $x(t)=x^*$ is stable, and My Account 245–254 (1995), Kulenović, M.R.S. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. 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. : The dynamics of multiparasitoid host interactions. T Accessibility Statement, Privacy Mat. 6, 229–245 (2008), Ladas, G., Tzanetopoulos, G., Tovbis, A.: On May’s host parasitoid model. Differ. 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 \(q\geq 4\). VCU Libraries In addition, x̄ Let $\diff{x}{t} = f(x)$ be an autonomous differential equation. $$, $$\begin{aligned} \xi _{20}&=\frac{1}{8} \bigl\{ (g_{1})_{\tilde{u} \tilde{u}}-(g_{1})_{ \tilde{v} \tilde{v}}+2(g_{2})_{\tilde{u} \tilde{v}}+i \bigl[(g_{2})_{ \tilde{u} \tilde{u}}-(g_{2})_{ \tilde{v} \tilde{v}}-2(g_{1})_{ \tilde{u} \tilde{v}} \bigr] \bigr\} \\ &=\frac{ (\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i f_{1} ) (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} )}{4 \sqrt{2} \bar{x}^{3/2} (4 \bar{x}^{2}-f_{1}^{2} ){}^{3/4}}, \\ \xi _{11} &=\frac{1}{4} \bigl\{ (g_{1})_{\tilde{u} \tilde{u}}+(g_{1})_{ \tilde{v} \tilde{v}}+i \bigl[(g_{2})_{\tilde{u} \tilde{u}}+(g_{2})_{ \tilde{v} \tilde{v}} \bigr] \bigr\} =\frac{ (\sqrt{4 \bar{x} ^{2}-f_{1}^{2}}+i f_{1} ) (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} )}{2 \sqrt{2} \bar{x}^{3/2} (4 \bar{x}^{2}-f_{1}^{2} ){}^{3/4}}, \\ \xi _{02} &=\frac{1}{8} \bigl\{ (g_{1})_{\tilde{u} \tilde{u}}-(g_{1})_{ \tilde{v} \tilde{v}}-2(g_{2})_{ \tilde{u} \tilde{v}}+i \bigl[(g_{2})_{ \tilde{u} \tilde{u}}-(g_{2})_{ \tilde{v} \tilde{v}}+2(g_{1})_{ \tilde{u} \tilde{v}} \bigr] \bigr\} \\ &=\frac{ (\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i f_{1} ) (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} )}{4 \sqrt{2} \bar{x}^{3/2} (4 \bar{x}^{2}-f_{1}^{2} ){}^{3/4}}, \\ \xi _{21} &=\frac{1}{16} \bigl\{ (g_{1})_{\tilde{u} \tilde{u} \tilde{u}}+(g _{1})_{\tilde{u} \tilde{v} \tilde{v}}+(g_{2})_{\tilde{u} \tilde{u} \tilde{v}}+(g_{2})_{ \tilde{v} \tilde{v} \tilde{v}}+i \bigl[(g_{2})_{ \tilde{u} \tilde{u} \tilde{u}}+(g_{2})_{\tilde{u} \tilde{v} \tilde{v}}-(g _{1})_{\tilde{u} \tilde{u} \tilde{v}}-(g_{1})_{ \tilde{v} \tilde{v} \tilde{v}} \bigr] \bigr\} \\ &=\frac{ (\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i f_{1} ) (\bar{x}^{3} (f_{3} \bar{x}+3 f_{2} )+f_{1} (1-3 f_{2} ) \bar{x}^{2}-3 f_{1}^{2} \bar{x}+2 f_{1}^{3} )}{32 \bar{x}^{4}-8 f_{1}^{2} \bar{x}^{2}}. This map is called a twist mapping. Appl. is an elliptic fixed point of J. Anim. 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]. be the equilibrium point of (1) such that Nachr. \((\bar{x},\bar{x})\) of the map In Sect. are located on the diagonal in the first quadrant. 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, - - - . 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. Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) \(\bar{x}>0\). By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. Note that if \(I_{0} = R\) is a reversor, then so is \(I_{1} = T\circ R\). Sci. coordinates has an elliptic fixed point (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 \(c=1\). Studies in Pure Mathematics 53 ( 2009 ), May, R.M of sign, we study the stability Runge–Kutta! Asymptotic behavior of second-order linear differential equations is that they can be characterized as recursive functions