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. 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. Anal. and LINEAR OSCILLATOR 223 6.1 Setup 223 \((\bar{x},\bar{x})\) then there exist periodic points of the map Note: Results do not translate immediately for systems of difference equations. In [12] authors analyzed a certain class of difference equations governed by two parameters. The di erence equation is called normal in this case. 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]. Some orbits of the map T associated with Eq. The following lemma holds. F, in the with arbitrarily large period in every neighborhood of In: Advances Studies in Pure Mathematics 53 (2009), Sternberg, S.: Celestial Mechanics. : Invariants and related Liapunov functions for difference equations. volume 2019, Article number: 209 (2019) | Adv Differ Equ 2019, 209 (2019). Also, the jth involution, defined as \(I_{j} := T^{j}\circ R\), is also a reversor. Let We will call an elliptic fixed point non-degenerate if \(\alpha _{1}\neq 0\). Let T \((0,0)\). in One of these is that F has precisely two fixed points. with determinant 1, we change coordinates. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of The following equation, which is of the form (1): where α is a parameter, is known as May’s host parasitoid equation, see [22]. We claim that map (9) is exponentially equivalent to an area-preserving map, see [16]. Cite this article. J. where \(k,p\), and a are positive and the initial conditions \(x_{0}, x_{1}\) are positive. Equation (3) possesses the following invariant: See [1]. 1, 291–306 (1995), Article Also note that if at least one of the twist coefficients \(\alpha _{j}\) is nonzero, then the angle of rotation is not constant. \(k>p+2\), then for 2 we show how (1) leads to diffeomorphisms T and F. We prove some properties of the map T, and we establish the condition under which a fixed point \((\bar{x}, \bar{x})\) of the map T, in \((u, v)\) coordinates \((0,0)\), is an elliptic fixed point, where x̄ is an equilibrium point of Equation (1). J. Assume that These facts cannot be deduced from computer pictures. To explain (c), let \(R(x,y)=(y,x)\) which is reflection about the diagonal. J. is an elliptic fixed point of See [16] for the application of the KAM theory to Lyness equation (2). All authors contributed equally and significantly in writing this article. Physical Sciences and Mathematics Commons, Home In the study of area-preserving maps, symmetries play an important role since they yield special dynamic behavior. Under the logarithmic coordinate change \((x, y) \to (u, v)\) the fixed point \((\bar{x}, \bar{x})\) becomes \((0,0)\). J. $$, $$ z\rightarrow \lambda z+ \xi _{20} z^{2}+\xi _{11}z\bar{z}+ \xi _{02} \bar{z}^{2}+\xi _{30} z^{3}+\xi _{21}z^{2}\bar{z}+ \xi _{12}z\bar{z}^{2}+ \xi _{03}\bar{z}^{3}+O \bigl( \vert z \vert ^{4}\bigr). By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. At \((0,0)\), \(J_{F}(u,v)\) has the form, The eigenvalues of (14) are λ and λ̄ where. $$, $$ \zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{\zeta })}+g( \zeta ,\bar{\zeta }) $$, \(\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}\), \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\), $$ \zeta \rightarrow \lambda \zeta +c_{1}\zeta ^{2}\bar{\zeta }+O\bigl( \vert \zeta \vert ^{4}\bigr) $$, \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), $$ f_{1}:=f'(\bar{x}),\qquad f_{2}:=f''( \bar{x}) \quad\textit{and}\quad f_{3}:=f'''( \bar{x}). $$, $$\begin{aligned} &\lambda ^{2}= \frac{f_{1}^{2}}{2 \bar{x}^{2}}-\frac{i f_{1} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{2}}-1, \\ &\lambda ^{3}= \frac{f_{1}^{3}}{2 \bar{x}^{3}}-\frac{i f_{1}^{2} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{3}}- \frac{3 f_{1}}{2 \bar{x}}+\frac{i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}}, \\ &\lambda ^{4}= \frac{f_{1}^{4}}{2 \bar{x}^{4}}-\frac{i f_{1}^{3} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}^{4}}- \frac{2 f_{1}^{2}}{ \bar{x}^{2}}+\frac{i f_{1} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{\bar{x} ^{2}}+1, \end{aligned}$$, $$ F \begin{pmatrix} u \\ v \end{pmatrix} =J_{F}(0,0) \begin{pmatrix} u \\ v \end{pmatrix} +F_{1} \begin{pmatrix} u \\ v \end{pmatrix} , $$, $$ 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} . Amleh, A.M., Camouzis, E., Ladas, G.: On the dynamics of a rational difference equation, part 1. About By using KAM (Kolmogorov–Arnold–Mozer) theory we investigate the stability properties of solutions of the following class of second-order difference equations: where f is sufficiently smooth, \(f:(0,+\infty )\to (0,+\infty )\), and the initial conditions are \(x_{-1}, x_{0} \in (0, +\infty )\). 1(13), 61–72 (1994), Hale, J.K., Kocak, H.: Dynamics and Bifurcation. The simplest numerical method, Euler’s method, is studied in Chapter 2. 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). 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. Am. Hence, x̄ is an elliptic point if and only if condition (17) is satisfied. Neither of these two plots shows any self-similarity character. 659, Stability Analysis of Systems of Difference Equations, Richard A. Clinger, Virginia Commonwealth University. Appl. $$, $$ \alpha _{1}=\frac{a k^{3} \bar{x}^{k} ((k-p-2) (k-p+1) \bar{x} ^{2 k}+2 a k \bar{x}^{k}-a^{2} (p^{2}+p-2 ) )}{4 ((-k+p-2) \bar{x}^{k}+a (p-2) ) ((-k+p-1) \bar{x} ^{k}+a (p-1) ) ((-k+p+2) \bar{x}^{k}+a (p+2) )^{2}}. Notice that each of these equations has the form (1). The square brackets denote the largest integer in \(q/2\). Appl. Now, we assume that, The equilibrium point of Equation (18) satisfies. Math. \(\bar{x}>0\), then F shares the following properties: F J. This map is called a twist mapping. | \(\alpha _{1}\neq 0\). is an elliptic fixed point of [19]. are located on the diagonal in the first quadrant. Nat. Google Scholar, Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation \(u_{n+2}u_{n} = u_{n+1} + a\). In addition, x̄ \(a,b\), and It is enough to assume that the function f is in \(C^{(3)}(0,+\infty )\). \(f(\bar{x})=\bar{x}^{2}\) This task is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form. The map T itself must be diffeomorphism of \((0,+\infty )^{2}\), and therefore we assume that this is the case. 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. \(y_{0}=x _{0}^{p+2}-x_{0}^{k}\). Then if $f'(x^*) 0$, the equilibrium $x(t)=x^*$ is stable, and 5, 177–202 (1999), Jašarević-Hrustić, S., Kulenović, M.R.S., Nurkanović, Z., Pilav, E.: Birkhoff normal forms, KAM theory and symmetries for certain second order rational difference equation with quadratic terms. Let They showed how Equation (7) leads to diffeomorphism F and showed that, for certain parameter value, all such F share four key properties. Assume In this paper, we investigated the stability of a class of difference equations of the form \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \) . The theory of differential and difference equations forms two extreme representations of real world problems. 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. thx in advance. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. When bt = 0, the diﬀerence 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 F Article In [22] the authors investigated the corresponding map known as May’s map. $$, $$ E \bar{x}^{3}-\bar{x}^{2} (C-D)-B \bar{x}-A=0. and bring the linear part into Jordan normal form. be the map associated with Equation (20). \(c<1\). 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. \(k< p+2\), then Equation (16) has exactly one positive equilibrium point. Appl. It is easy to describe the dynamics of the twist map: the orbits are simple rotations on these circles. Let, and if we set \(F(u,v)=E^{-1}\circ T\circ E(u,v)\), where ∘ denotes composition of functions, then we obtain a new mapping F, which is given by. Let Appl. The change of variables \(x_{n}=\beta u_{n}\) and \(y_{n}=\beta v_{n}\) reduces System (5) to. Similar as in Proposition 2.2 [12] one can prove the following. \((\bar{x}, \bar{x})\) Commun. Read reviews from world’s largest community for readers. \((u,v)\) $$, $$ F \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \log (f (e^{v} \bar{x} ) )-2 \log (\bar{x} )-u \end{pmatrix} . This is because the characteristic equation from which we can derive its eigenvalues Appl. If a solution does not have either of these properties, it is … Equation (3) is of the form (1). > \end{aligned}$$, $$ c_{1}=\frac{\xi _{20}\xi _{11}(\bar{\lambda }+2\lambda -3)}{(\lambda ^{2}-\lambda )(\bar{\lambda }-1)}+\frac{ \vert \xi _{11} \vert ^{2}}{1-\bar{\lambda }}+\frac{2 \vert \xi _{02} \vert ^{2}}{\lambda ^{2}-\bar{\lambda }}+\xi _{21} $$, $$\begin{aligned} &\xi _{20} \xi _{11}= \frac{ (\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i f _{1} ){}^{2} (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ) {}^{2}}{16 \bar{x}^{3} (4 \bar{x}^{2}-f_{1}^{2} ){}^{3/2}}, \\ &\xi _{11}\overline{\xi _{11}}=\frac{ (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ){}^{2}}{2 \bar{x} (4 \bar{x}^{2}-f_{1} ^{2} ){}^{3/2}}, \\ &\xi _{02}\overline{\xi _{02}}=\frac{ (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ){}^{2}}{8 \bar{x} (4 \bar{x}^{2}-f_{1} ^{2} ){}^{3/2}}, \end{aligned}$$, $$\begin{aligned} c_{1} &=\frac{\xi _{20}\xi _{11}(\bar{\lambda }+2\lambda -3)}{(\lambda ^{2}-\lambda )(\bar{\lambda }-1)}+\frac{ \vert \xi _{11} \vert ^{2}}{1-\bar{\lambda }}+ \frac{2 \vert \xi _{02} \vert ^{2}}{\lambda ^{2}-\bar{\lambda }}+\xi _{21} \\ &=\varTheta (\bar{x}) \frac{\bar{x}^{4} (2 f_{3} \bar{x}+f_{2} (f_{2}+6 ) )+f_{1} \bar{x}^{3} (f_{3} \bar{x}+f _{2} (2 f_{2}-1 )+2 )-f_{1}^{2} \bar{x}^{2} (f _{3} \bar{x}+4 f_{2}+4 )-f_{1}^{3} f_{2} \bar{x}+2 f_{1}^{4}}{4 \bar{x} (f_{1}-2 \bar{x} ){}^{2} (\bar{x}+f_{1} ) (2 \bar{x}+f_{1} ) (-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}+2 \bar{x}+f_{1} )}, \end{aligned}$$, $$ \varTheta (\bar{x}):=f_{1} \Bigl(\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i \bar{x} \Bigr)+\bar{x} \Bigl(\sqrt{4 \bar{x}^{2}-f_{1}^{2}}-2 i \bar{x} \Bigr)+i f_{1}^{2}. is a stable equilibrium point of (1). 40, 306–318 (2017), Gidea, M., Meiss, J.D., Ugarcovici, I., Weiss, H.: Applications of KAM theory to population dynamics. 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. be an elliptic fixed point. Dyn. In Sect. 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, - - - . \((\bar{x},\bar{x})\) \(k,p\), and $$, $$ (k-p-2) \bar{x}^{k}< a (p+2) \quad\textit{and}\quad (k-p+2) \bar{x}^{k}>a (p-2). Equations of ﬁrst order with a single variable. is an equilibrium point of Equation (1). PubMed Google Scholar. is a stable fixed point. 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. It is not an efﬁcient numerical meth od, but it is an W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. 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. of the map Math. The map F is defined on all of \(\mathbf{R^{2}}\). $$, $$ \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} +\tilde{F} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} . Nat. As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. 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. \(a+b>0\) More information about video. Assertion (a) is immediate. \(a,b,c\geq 0\) Then \((\bar{x},\bar{x})\). Math. T In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x. Math. Let It is non-resonant if and only if, To compute the first twist coefficient \(\alpha _{1}\), we follow the procedure in [9]. Consider an invariant annulus \(a < |\zeta | < b\) in a neighborhood of an elliptic fixed point \((0,0)\). and Differ. 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. In Table 1 we compute the twist coefficient for some values \(a,b,c\geq 0\). with arbitrarily large period in every neighborhood of $$, $$ 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} )}. Then the following holds: If 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 \(x_{-1}, x _{0}\in (0,+\infty )\). Equ. In addition, x̄ derived by Wan in the context of Hopf bifurcation theory [34]. 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]. Springer, New York (1971), Siezer, W.: Periodicity in the May’s host parasitoid equation. 25, 217–231 (2016), Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. For a more general case of Equation (3), see [10]. Equ. Wiss. All authors read and approved the final manuscript. Int. \((\bar{x},\bar{x})\) (20) for (a) \(a=0.1\), \(b=0.002\), and \(c=0.001\) and (b) \(a=0.1\), \(b=0.02\), and \(c=0.001\). Two classes of methods are considered: Runge–Kutta methods extended with a compound quadrature rule, and Runge– Kutta methods extended with a Pouzet type quadrature technique. If \(D,E>0\), then the change \(x_{n}=\frac{D}{E}y_{n}\) conjugates Equation (18) to. [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. Google Scholar, Barbeau, E., Gelbord, B., Tanny, S.: Periodicity of solutions of the generalized Lyness recursion. 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. nary differential equations is given in Chapter 1, where the concept of stability of differential equations is also introduced. We make the additional assumption that the spectrum of A consists of only real numbers and 6, <0. be the map associated with Equation (19). In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. According to KAM-theory there exist states close enough to the fixed point, which are enclosed by an invariant curve. : Computation of the stability condition for the Hopf bifurcationof diffeomorphisms on \(\mathcal{R}^{2}\). coordinates, is always non-degenerate. Stochastic Stability of Differential Equations book. 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 \(|\lambda |=1\). 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. 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. $$, $$ J_{0}=J_{F}(0,0)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{f_{1}}{\bar{x}} \end{pmatrix}. : Geometric Unfolding of a Difference Equation. © 2021 BioMed Central Ltd unless otherwise stated. J. Since stable and unstable equilibria play quite different roles in the dynamics of a system, it is useful to be able to classify equi-librium points based on their stability. \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). 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. > Wiley, New York (1989), Taylor, A.D.: Aggregation, competition, and host-parasitoid dynamics: stability conditions don’t tell it all. for \(c<1\). be an equilibrium point of Equation (19) and are positive. First, we will discuss the Courant-Friedrichs- Levy (CFL) condition for stability of ﬁnite difference meth ods for hyperbolic equations. THEOREM 1. $$, $$ 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}. Evaluating the Jacobian matrix of T at \((\bar{x},\bar{x})\) by using \(f(\bar{x})=\bar{x}^{2}\) gives, We obtain that the eigenvalues of \(J_{T}(\bar{x},\bar{x})\) are \(\lambda ,\bar{\lambda }\) where, Since \(|\lambda |=1\), we have that \((\bar{x},\bar{x})\) is an elliptic fixed point if and only if \(|f'(\bar{x})|<2 \bar{x}\). 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. New content will be added above the current area of focus upon selection J. The above normal form yields the approximation. Google Scholar, Moeckel, R.: Generic bifurcations of the twist coefficient. T Systems of first-order linear difference equations are of the form x(n + 1) = Ax(n) , and systems of first-order linear differential equations are of the form x(t) = Ax(t). An easy calculation shows that \(R^{2}=id\), and the map F will satisfy \(F\circ R\circ F= R\). Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied. In [10–17] applications of difference equations in mathematical biology are given. Assume that $$, $$ \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} . 6, 229–245 (2008), Ladas, G., Tzanetopoulos, G., Tovbis, A.: On May’s host parasitoid model. \(0< a< y_{0}\) \((u,v)\) $$, $$ \lambda =\frac{f_{1}-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}}. \(k,p\), and See [30] for results on periodic solutions. 47, 833–843 (1978), May, R.M., Hassel, M.P. Assume that \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), \(f(\bar{x})=\bar{x} ^{2}\), and The well-known difference equation of the form (1) is Lyness’ equation. See also [3, 4, 6] for the results on the feasible periods for solutions of (2) and the existence of non-periodic solutions of (2). coordinates, the corresponding fixed point is Let F be the function defined by, The Jacobian matrix of F at \((u,v)\) is given by (10). uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. VCU Libraries After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. Terms and Conditions, Within these gaps, one finds, in general, orbits of hyperbolic and elliptic periodic points. Example 1. \end{aligned}$$, \(T:(0,+ \infty )^{2}\to (0,+\infty )^{2}\), $$ u_{n}=x_{n-1},\qquad v_{n}=x_{n},\qquad T \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \frac{f(v)}{u} \end{pmatrix} . If In this paper, we explore the stability and … $$, $$ E^{-1}(x,y)= \biggl(\ln \frac{x}{\bar{x}}, \ln \frac{y}{\bar{x}} \biggr) ^{T}, $$, $$ F(u,v)=E^{-1}\circ T\circ E(u,v)= \begin{pmatrix} v \\ \ln (f (e^{v} \bar{x} ) )-2 \ln (\bar{x} )-u \end{pmatrix} . $$, $$ (k-p-2) (k-p+1) \bar{x}^{2 k}+2 a k \bar{x}^{k}-a^{2} \bigl(p^{2}+p-2 \bigr) \neq 0, $$, \(x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x _{n})x_{n-1}}\), $$ x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x_{n})x_{n-1}}, $$, $$ (D,E>0\wedge A+B>0)\vee (D,E>0\wedge A+B=0\wedge C>D). Chapman and Hall/CRC, London (2001), Kulenović, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. a Several authors have studied the Lyness equation (2) and have obtained numerous results concerning the stability of equilibrium, non-existence of solutions that converge to the equilibrium point, the existence of invariants, etc. $$, $$ x_{n+1}=\frac{A x_{n}^{2}+F}{e x_{n-1}},\quad n=0,1,\ldots. Stability of Finite Difference Methods In this lecture, we analyze the stability of ﬁnite differenc e discretizations. 10(2), 181–199 (2015), MathSciNet This equation may be rewritten as \(R\circ F= F^{-1}\circ R\). This condition depends only on the values of the first, second, and third derivatives of the function f at the equilibrium point. Obviously, it is possible to rewrite the above equation as a rst order equation by enlarging the state space.2 Thus, in many instances it is su cient to consider just the rst order case: x t+1 = f(x t;t): (1.3) Because f(:;t) maps X into itself, the function fis also called a transforma-tion. be an area-preserving diffeomorphism and In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. 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. $$, $$ \alpha _{1}=\frac{\varGamma _{1}+\varGamma _{2} \bar{x}+\varGamma _{3}\bar{x}^{2}}{2 (\bar{x}+1 )^{2} (2 c \bar{x}+\bar{x}+b ) (2 \bar{x} (b-c+1)+(c+2) \bar{x}^{2}+3 a-b )^{2} (2 \bar{x} (b+c+1)+(3 c+2) \bar{x}^{2}+a+b )}, $$, $$\begin{aligned} \varGamma _{1}={}&a^{3} b^{2}+25 a^{3} b c^{2}+66 a^{3} b c+11 a^{3} b+20 a ^{3} c^{3}+70 a^{3} c^{2}+55 a^{3} c-a^{3}-2 a^{2} b^{3}\\ &{}-12 a^{2} b ^{2} c^{3}+5 a^{2} b^{2} c^{2}-8 a^{2} b^{2} c -5 a^{2} b^{2}-29 a^{2} b c^{5}-44 a^{2} b c^{4}-82 a^{2} b c^{3}\\ &{}-46 a^{2} b c^{2}+22 a^{2} b c+2 a^{2} b-8 a^{2} c^{7}-8 a^{2} c^{6}-16 a ^{2} c^{5}-2 a^{2} c^{4} +8 a^{2}c^{2}+8 a^{2} c\\ &{}+3 a b^{4} c^{2}+8 a b^{4} c+a b^{4}+a b^{3} c^{4}+16 a b^{3} c^{3}-6 a b^{3} c^{2}-2 a b^{3} c-7 a b^{3}-3 a b ^{2} c^{6}\\ &{}+10 a b^{2} c^{5}-26 a b^{2} c^{4} -3 a b^{2} c^{3}+a b^{2} c^{2}-5 a b^{2} c-a b^{2}-a b c^{8}+6 a b c ^{7}-14 a b c^{6}\\ &{}+8 a b c^{5}+a b c^{4}+a c^{9}-3 a c^{8}+3 a c^{7}-a c^{6}+b^{4}, \\ \varGamma _{2}={}&11 a^{3} b c+4 a^{3} b+8 a^{3} c^{3}+63 a^{3} c^{2}+54 a ^{3} c+a^{3}+24 a^{2} b^{2} c^{2}+75 a^{2} b^{2} c+16 a^{2} b^{2}\\ &{}-20 a ^{2} b c^{4}-18 a^{2} b c^{3}+18 a^{2} b c^{2} +110 a^{2} b c+6 a^{2} b-8 a^{2} c^{6}-17 a^{2} c^{5}-33 a^{2} c ^{4}\\ &{}-35 a^{2} c^{3}+21 a^{2} c^{2}+37 a^{2} c-a^{2}+a b^{4} c-a b^{4}-10 a b^{3} c^{3}+18 a b^{3}c^{2} -a b^{3} c-19 a b^{3}\\ &{}-31 a b^{2} c^{5}-38 a b^{2} c^{4}-95 a b^{2} c^{3}-54 a b^{2} c^{2}-15 a b^{2} c-6 a b^{2}-9 a b c^{7}-4 a b c^{6}\\ &{}-25 a b c^{5}-3 a b c^{4} -4 a b c^{2}+8 a b c+a b+a c^{8}-2 a c^{7}+a c^{6}+3 b^{5} c^{2}+8 b^{5} c+b^{5}\\ &{}+b^{4} c^{4}+16 b^{4} c^{3}-6 b^{4} c^{2}-2 b^{4} c-b ^{4}-3 b^{3} c^{6}+10b^{3} c^{5} -26 b^{3} c^{4}-3 b^{3} c^{3}+b^{3} c^{2}\\ &{}+2 b^{3} c-b^{3}-b^{2} c ^{8}+6 b^{2} c^{7}-14 b^{2} c^{6}+8 b^{2} c^{5}+b^{2} c^{4}+b c^{9}-3 b c^{8}+3 b c^{7}-b c^{6}, \\ \varGamma _{3}={}&16 a^{3} c^{2}+19 a^{3} c+a^{3}+12 a^{2} b^{2} c+8 a^{2} b^{2}+22 a^{2} b c^{3}+92 a^{2} b c^{2}+84 a^{2} b c+6 a^{2} b\\ &{}-8 a ^{2} c^{5}-6 a^{2} c^{4}-10 a^{2} c^{3} +33 a^{2} c^{2}+28 a^{2} c-a^{2} -a b^{4}+a b^{3} c^{2}+15 a b^{3} c-7 a b^{3}\\ &{}-33 a b^{2} c^{4}-16 a b^{2} c^{3}-65 a b^{2} c^{2}-25 a b^{2} c-6 a b^{2} -38 a b c^{6}-30a b c^{5}-78 a b c^{4}\\ &{}-7 a b c^{3}+5 a b c^{2} +9 a b c+a b-8 a c^{8}+a c^{7}-9 a c^{6}+14 a c^{5}+2 a c^{4}+b^{5} c+b ^{5} \\ &{}+5 b^{4} c^{3}+18 b^{4} c^{2}-b^{4}-b^{3} c^{5}+21 b^{3} c^{4}-35 b ^{3} c^{3}-4 b^{3} c^{2}+b^{3} c-b^{3}-4 b^{2} c^{7}\\ &{}+17 b^{2} c^{6}-45 b^{2} c^{5}+22 b^{2} c^{4}+4 b^{2} c^{3} -b c^{9}+8 b c^{8}-22 bc^{7}+23 b c^{6}-7 b c^{5}-b c^{4}\\ &{}+c^{10}-4 c ^{9}+6 c^{8}-4 c^{7}+c^{6}.