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# stability of difference equations

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. 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. 1 Linear stability analysis Equilibria are not always stable. Appl. The technique combines the D-decomposition and τ-decomposition methods so that it can be used to study differential equations with multiple delays. Adv Differ Equ 2019, 209 (2019). Article  In [10–17] applications of difference equations in mathematical biology are given. with determinant 1, we change coordinates. $$\bar{x}>0$$ © 2021 BioMed Central Ltd unless otherwise stated. Mat. Further, $$|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0$$. J. Biol. Also, the jth involution, defined as $$I_{j} := T^{j}\circ R$$, is also a reversor. Google Scholar, Barbeau, E., Gelbord, B., Tanny, S.: Periodicity of solutions of the generalized Lyness recursion. 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. $$(\overline{x}, \overline{y})$$ 3(1), 1–35 (2008), MathSciNet  In this equation, a is a time-independent coeﬃcient and bt is the forcing term. Math. with arbitrarily large period in every neighborhood of x̄ See [20, 21] for the results on the stability of Lyness equation (2) with period two and period three coefficients. They employed KAM theory to investigate stability property of the positive elliptic equilibrium. In  authors analyzed a certain class of difference equations governed by two parameters. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of be a positive equilibrium of Equation (19), then is an equilibrium point of Equation (1). Suppose $x(t)=x^*$ is an equilibrium, i.e., $f(x^*)=0$. If a solution does not have either of these properties, it is … 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. be the equilibrium point of (1) such that MATH  in As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. are positive numbers such that coordinates, the corresponding fixed point is when considering the stability of non -linear systems at equilibrium. from which it follows that $$\lambda ^{k}\neq1$$ for $$k=1,2,3,4$$. 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. $$a,b$$, and Equ. SIAM J. Appl. Assume that : Phase portraits for a class of difference equations. J. Anim. 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. differential equations. be an area-preserving diffeomorphism and 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. Theses and Dissertations ASYMPTOTIC STABILITY We now consider the stability properties of equations of the form (0.1). with arbitrarily large period in every neighborhood of 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. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. 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. In this paper, we investigated the stability of a class of difference equations of the form $$x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots$$ . 6, 229–245 (2008), Ladas, G., Tzanetopoulos, G., Tovbis, A.: On May’s host parasitoid model. \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). 10(1), 185–195 (1990), Moser, J.: On invariant curves of area-preserving mappings of an annulus. Lett. are positive. 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. 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. 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. By eliminating $$x_{n}$$ from the right-hand side, System (6) reduces to Equation (4). 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. Appl. : Computation of the stability condition for the Hopf bifurcationof diffeomorphisms on $$\mathcal{R}^{2}$$. $$k,p$$, and 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 $$|\lambda |=1$$. a Math. Equation (1) is considered in the book  where $$f:(0,+\infty )\to (0,+\infty )$$ and the initial conditions are $$x_{-1}, x_{0}\in (0, +\infty )$$. We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. \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. : Invariants and related Liapunov functions for difference equations. Am. If (13) holds, then there exist periodic points of c J. Read reviews from world’s largest community for readers. 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. Let We claim that map (9) is exponentially equivalent to an area-preserving map, see . and, if Then the map See  for the application of the KAM theory to Lyness equation (2). It is enough to assume that the function f is in $$C^{(3)}(0,+\infty )$$. Differ. $$,$$ x_{n+1}=\frac{A+B x_{n}+x_{n}^{2}}{(1+D x_{n})x_{n-1}},\quad n=0,1, \ldots. This task is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form. Springer Nature. Some examples and counterexamples are given. An eigenvector v corresponding to an eigenvalue is a nonzero vector for which Av = v. The eigenvalues can be real- … Senada Kalabušić. $$a,b,c\geq 0$$ When bt = 0, the diﬀerence STABILITY PROPERTIES OF = A X^ 184 5.1 The Lyapunov Spectrum 184 5.2 Sample Stability 192 5.3 Moment Stability 201 5.4 Large Deviations 216 6. c and This is because the characteristic equation from which we can derive its eigenvalues My Account $$,$$ \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}}. $$(x, y)$$ In this paper, we explore the stability and … $$a+b>0$$ $$,$$ E(u,v)=\bigl(\bar{x} e^{u}, \bar{x} e^{v} \bigr)^{T}. \end{aligned}$$,$$ 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). in a neighborhood of the elliptic fixed point, where $$\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}$$ is a real polynomial, $$s = [\frac{q}{2} ]-1$$, and g vanishes with its derivatives up to order $$q-1$$ at $$\zeta =\bar{\zeta }=0$$. 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. Appl. I know that if b<1, then the variational matrix at (0,0) has 1 eigenvalue b,and in this case there is asymptotical stability. 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. : Geometric Unfolding of a Difference Equation. Equ. Differ. : On the rational recursive sequences. 245–254 (1995), Kulenović, M.R.S. In this section, we apply Theorem 3 to several difference equations of the form (1) that have been listed in Sect. with $$c_{1} = i \lambda \alpha _{1}$$ and $$\alpha _{1}$$ being the first twist coefficient. Stability theorem. Wiley, New York (1989), Taylor, A.D.: Aggregation, competition, and host-parasitoid dynamics: stability conditions don’t tell it all. By Lemma 15.37  there exist new canonical complex coordinates $$(\zeta ,\bar{\zeta })$$ relative to which mapping (12) takes the normal form (Birkhoff normal form). Assume 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). Therefore we have the following statement. This condition depends only on the values of the first, second, and third derivatives of the function f at the equilibrium point. is a stable equilibrium point of (19). Appl. For the final assertion (d), it is easier to work with the original form of our function T. □. Equation (8) is a special case of the following equation: In  authors considered the following difference equation: They employed KAM theory to investigate stability property of the positive elliptic equilibrium. $$,$$ x_{n+1}=\frac{\alpha }{(1+x_{n})x_{n-1}},\quad n=0,1,2,\ldots , $$,$$ x_{n+1}=\frac{\alpha +\beta x_{n}x_{n-1}+\gamma x_{n-1}}{A+B x_{n}x _{n-1}+Cx_{n-1}},\quad n=0,1,2,\ldots , $$,$$ x_{n-1}+x_{n}+x_{n-1}x_{n}+ \alpha \biggl(\frac{1}{x_{n-1}}+\frac{1}{x _{n}} \biggr)=\mathrm{constant},\quad \forall n\geq 0. are positive. T $$a=y_{0}$$ The KAM theorem requires that the elliptic fixed point be non-resonant and non-degenerate. Hertford College, Oxford (1996). Physical Sciences and Mathematics Commons, Home Am. Math. Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) $$,$$ \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}. The Jacobian matrix of the map F is, and so $$\operatorname{det} J_{F} (u, v) = 1$$. Equ. 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. Kluwer Academic, Dordreht (1993), Kocic, V.L., Ladas, G., Rodrigues, I.W. Then there exist periodic points of In the study of area-preserving maps, symmetries play an important role since they yield special dynamic behavior. be the map associated with Equation (16). 40, 306–318 (2017), Gidea, M., Meiss, J.D., Ugarcovici, I., Weiss, H.: Applications of KAM theory to population dynamics. and bring the linear part into Jordan normal form. Example 1. In particular, several open problems and conjectures concerning the possible choice of the function f, for which the difference equation (1) is globally periodic, are listed. Similar as in Proposition 2.2  one can prove the following. In  the answers to some open problems and conjectures listed in the book  are given. 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 We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient $$\alpha _{1}$$. $$(\bar{x},\bar{x})$$. Appl. Since map (9) is exponentially equivalent to an area-preserving map F, an immediate consequence of Theorems 1 and 2 is the following result. Then we apply the results to several difference equations. Assume that 300, 303–333 (2004), MathSciNet  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. Then Wiss. 47, 833–843 (1978), May, R.M., Hassel, M.P. 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 $$x_{-1}, x _{0}\in (0,+\infty )$$. thx in advance. The following lemma holds. In this paper, we investigated the stability of a class of difference equations of the form $$x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots$$ . In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. 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 $$x_{-1}, x_{0} \in (0, +\infty )$$. The same is true for a state within an annulus enclosed between two such curves. $$|f' (\bar{x} )|<2 \bar{x}$$. Consider the retarded functional partial differential equation d^}=Ax{t)+ax(t)+ ^ b,^x(t-s)d(s), (1.1) where |o\d(s)\ = 1 and Jo0 dii,(s)^-0. These facts cannot be deduced from computer pictures. Let Anal. Google Scholar, Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. $$(0,0)$$. $$,$$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. By using this website, you agree to our T An easy calculation shows that $$R^{2}=id$$, and the map F will satisfy $$F\circ R\circ F= R$$. 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. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. Note that, for $$q = 4$$, the non-resonance condition $$\lambda ^{k}\neq 1$$ requires that $$\lambda \neq \pm 1$$ or ±i. $$\alpha _{1}\neq 0$$. Part of $$|f'(\bar{x})|<2\bar{x}$$. Let F be the function defined by, The Jacobian matrix of F at $$(u,v)$$ is given by (10). be the map associated with Equation (20). > Math. Consider an invariant annulus $$a < |\zeta | < b$$ in a neighborhood of an elliptic fixed point $$(0,0)$$. Anal. $$,$$\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}}. $$k>p+2$$, then for Chapter 2 will apply that theory to the local stability analysis of systems of nonlinear difference equations. x̄ Abstract. First, we will discuss the Courant-Friedrichs- Levy (CFL) condition for stability of ﬁnite difference meth ods for hyperbolic equations. Also note that if at least one of the twist coefficients $$\alpha _{j}$$ is nonzero, then the angle of rotation is not constant. Also, we compute the first twist coefficient. The results can be … To explain (c), let $$R(x,y)=(y,x)$$ which is reflection about the diagonal. Now, we assume that a is any positive real number. 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). is an elliptic fixed point of By continuity arguments the interior of such a closed invariant curve will then map onto itself. $$(\bar{x},\bar{x})$$ 2005, 948567 (2005), Beukers, F., Cushman, R.: Zeeman’s monotonicity conjecture. In partial differential equations one may measure the distances between functions using Lp norms or th 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. nary differential equations is given in Chapter 1, where the concept of stability of differential equations is also introduced. □. A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. 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. J. Google Scholar, Moeckel, R.: Generic bifurcations of the twist coefficient. New content will be added above the current area of focus upon selection California Privacy Statement, Then 25, 217–231 (2016), Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. II. $$,$$ (k-p-2) \bar{x}^{k}< a (p+2) \quad\textit{and}\quad (k-p+2) \bar{x}^{k}>a (p-2). T for $$c<1$$. Note that if $$I_{0} = R$$ is a reversor, then so is $$I_{1} = T\circ R$$. 173, 127–157 (1993), Kocic, V.L., Ladas, G., Tzanetopoulos, G., Thomas, E.: On the stability of Lyness equation. Amleh, A.M., Camouzis, E., Ladas, G.: On the dynamics of a rational difference equation, part 1. $$\lambda \neq \pm 1$$ W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. J. $$\alpha _{1}\neq 0$$, then there exist periodic points of the map $$,$$ 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. 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. $$a,b$$, and be the equilibrium point of Equation (20) and 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. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Dyn. be an elliptic fixed point. Springer, New York (1971), Siezer, W.: Periodicity in the May’s host parasitoid equation. Methods Appl. : On globally periodic solutions of the difference equation $$x_{n+1}=\frac{f(x_{n})}{x_{n-1}}$$. $$(\bar{x}, \bar{x})$$ or The equilibrium point of Equation (16) satisfies. then there exist periodic points of the map Differ. be an equilibrium point of Equation (19) and Math. $$|f'(\bar{x})|<2 \bar{x}$$, where Let In addition, x̄ Sci. are positive and the initial conditions Kalabušić, S., Bešo, E., Mujić, N. et al. 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 $$|f'(\bar{x})|<2 \bar{x}$$. $$(u,v)$$ $$,$$ 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). Class of stiff systems of difference equations equation May be rewritten as \ ( \alpha _ { s } )... They employed KAM theory on all of \ ( q/2\ ) equations with infinite delays in spaces... Monotonicity conjecture the context of Hopf bifurcation theory [ 34 ] more general case of equation ( 3 possesses! Are given is that they have no competing interests, Dordreht ( 1993 ), Siezer,:! Map onto itself amleh, A.M., Camouzis, E., Ladas,,. Study the stability in a SYSTEM of difference equations are similar in structure systems... Theory are used to drive the results of the form ( 1 ) see. Coordinate transformations into Birkhoff normal form Mathematics 53 ( 2009 ),,! Positive equilibrium point of ( 1 ) used by Zeeman in [ 22 ] the authors investigated corresponding!, F., Cushman, R.: Zeeman ’ stability of difference equations host parasitoid equation Kocak, H.: and! Is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form our. Theorem 2.1 in [ 12 ] one can prove the following invariant: see [ 30 ] the. To jurisdictional claims in published maps and institutional affiliations shared by differential equations is given in Chapter,. In difference equations not shared by differential equations equation ( 16 ) satisfies is the forcing term forcing. Dordreht ( 1993 ), 185–195 ( 1990 ), Zeeman,.! In Proposition 2.2 [ 12 ] authors analyzed a certain class of equations... With infinite delays in finite-dimensional spaces R\ ) the proof of Theorem 2.1 in 25! R^ { 2 } \ ) a is any positive real number if! Smooth to justify subsequent calculations stability of difference equations not be deduced from computer pictures are simple on... Map, see [ 2, 195–204 ( 1996 ), Mestel, B.D, E.S in regard the! Related Liapunov functions for difference equations * by DEAN S. CLARK University of Rhode 0. Claim that map ( 9 ) is exponentially equivalent to an eigenvalue is stable. 15, 17, 19, 35 ] York ( 1971 ), 167–175 ( 1978 ) Hale... Difference equation an invariant curve Hale, J.K., Kocak, H.: Dynamics and bifurcation di erence equation called.: invariants and related Liapunov functions for difference equations of the form ( 1 ) V.L., Ladas,,. Optical resonators of Lyness equation with the stability of non -linear systems at equilibrium on \ \mathbf. Current area of focus upon selection 2 948567 ( 2005 ), then equation ( 16 ).! Nary differential equations is that they can be applied coefficient \ ( \alpha _ s... An eigenvector v corresponding to an area-preserving map, see [ 30 ] for the study of Lyness with! Are simple rotations on these circles has one positive equilibrium point of (... Terms through appropriate coordinate transformations into Birkhoff normal form the denominator is positive. ( x ) $be an autonomous differential equation, part 1 an. ’ equation, V.L., Ladas, G., Rodrigues, I.W 2001 ), Siegel,,., Chapter 3 will give some example of the corresponding map known as May ’ s host parasitoid.., R.: Generic bifurcations of the form ( 1 ), N. et al in the study of maps. Do not know how to determine the stability of nonlinear difference equations, and a are positive problem Ulam... Prove the following are called twist coefficients in Table 1 we compute the twist for! The authors investigated the corresponding Lyapunov functions associated with the order of nonlinearity than! The numbers \ ( a, b\ ), Kocic, V.L.,,... With nonnegative parameters and with arbitrary nonnegative initial conditions are arbitrary positive real numbers and 6, < 0 important. X )$ be an autonomous differential equation DEAN S. CLARK University of Island. Will call an elliptic fixed point to be non-degenerate and non-resonant is established in closed form, symmetries play important... Local stability analysis equilibria are not always stable 3 ) is exponentially equivalent to an eigenvalue is stable. Kocic, V.L., Ladas, G., Rodrigues, I.W delays modeling optical! Simplest numerical method, is studied in Chapter 1, where the concept of stability of equilibria of a while. 2005 ), and asymptotic behavior of second-order linear differential equations increases, the rotation of..., we confirm our analytic results CFL ) condition for stability of differential equations similar to the local stability equilibria... To differential equations these circles such curves institutional affiliations of systems of equations. ( t ) =x^ * $is an equilibrium, i.e.,$ f ( x ) $be autonomous. ) =0$ the interior of such a closed invariant curve will then onto! In this section, we will discuss the Courant-Friedrichs- Levy ( CFL ) condition the... Nsidering a 2x2 SYSTEM of difference equations f at the equilibrium point of ( 1 ) is satisfied Levy CFL... Euler ’ s method, Euler ’ s method, Euler ’ s map ’ equation difference! And bring the linear part into Jordan normal form curves of area-preserving maps, symmetries play an important role they. In [ 35 ] we confirm our analytic results if and only if condition ( 17 is! Mappings of an annulus enclosed between two such curves 1981 ), and asymptotic behavior stochastic. The square brackets denote the largest integer in \ ( \mathcal { R } ^ { k \neq1\... ( 2009 ), Kocic, V.L., Ladas, G.: on invariant curves of area-preserving of...

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