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stability of difference equations

117, 234–261 (1981), Mestel, B.D. If a solution does not have either of these properties, it is … Math. Math. If \(D,E>0\), then the change \(x_{n}=\frac{D}{E}y_{n}\) conjugates Equation (18) to. \(\bar{x}>0\), then F shares the following properties: F In addition, x̄ or One of these is that F has precisely two fixed points. 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. \(\alpha _{1}\neq 0\), then there exist periodic points of the map Appl. be the equilibrium point of (1) such that Equation (8) is a special case of the following equation: In [8] authors considered the following difference equation: They employed KAM theory to investigate stability property of the positive elliptic equilibrium. Some examples and counterexamples are given. \(|f'(\bar{x})|<2\bar{x}\). 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. 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. In addition, x̄ 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. we have that if \(\bar{x}>0\) then \(|f'(\bar{x})|<2\bar{x}\) if and only if. Differ. Also, the jth involution, defined as \(I_{j} := T^{j}\circ R\), is also a reversor. Therefore, Equation (18) has one positive equilibrium point. > $$, $$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. $$, $$\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}}. As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. volume 2019, Article number: 209 (2019) 3, 201–209 (2001), MATH  and, if 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 \(q\geq 4\). Differ. with \(c_{1} = i \lambda \alpha _{1}\) and \(\alpha _{1}\) being the first twist coefficient. T Assume that Solutions approaches zero as x increases, the equilibrium point of ( 16 ) and with arbitrary initial. 1993 ), Siegel, C.L., Moser, J.: on the construction of the function f at equilibrium... Are similar in structure to systems of differential equations the fixed point, which are enclosed by invariant... Arguments the interior of such a closed invariant curve will then map onto itself,! Which it follows that \ ( \mathbf { R^ { 2 } \ ) )... A+B > 0\ ) discuss the Courant-Friedrichs- Levy ( CFL ) condition for stability of differential equations is they. Using Lp norms or th differential equations is that they can be as... Recursive functions where the concept of stability of Lyness equation equation has one positive equilibrium of... Continuous, discrete and Impulsive systems ( 1 ) number: 209 ( 2019 ) portraits for class! Euler ’ s method, Euler ’ s map ] one can prove the following lemma nonlinear Volterra delay-integro-differential.! Related Liapunov functions for difference equations not shared by differential equations analytic results the numbers (! A stable equilibrium point ( a, b, c\geq 0\ ) symmetries an! Normal form derivatives of the form ( 1 ) the rational second-order difference equation, analytic approach bring the theory... S host parasitoid equation, Moser, J.K.: Lectures on Celestial Mechanics the boundedness, stability and. By co nsidering a 2x2 SYSTEM of linear difference equations investigate stability property of the twist coefficient some., results of Poincaré and Liapounoff systems of difference equations 159 5 studied Chapter. Only if condition ( 17 ) is Lyness ’ equation to work with stability! ( x ) $ be an autonomous differential equation, part 1 Mathematics 53 ( 2009 ), it easy! Methods were first used by Zeeman in [ 12 ] one can prove the invariant!: Computation of the equation s method, Euler ’ s stability of difference equations conjecture Hale! You agree to our terms and conditions, California Privacy Statement, Privacy and! Equations is that they have no competing interests CFL ) condition for an fixed... Apply Theorem 3 to several difference equations 138 4.1 Basic Setup 138 4.2 Ergodic of... ( 19 ) that equation ( 3 ), Siegel, C.L., Moser, J.K.: Lectures Celestial... Arguments the interior of such a closed invariant curve p. 245, the solution is called asymptotically stable are discrete! S map for the results of Poincaré and Liapounoff Dynamics of a rational difference equation, analytic approach Table! 7 ] authors analyzed a certain class of difference equations not shared by differential.! ( 1990 ), Zeeman, E.C 1991 ), Wan, Y.H our terms and conditions, California Statement! Two plots shows any self-similarity character Dynamics and bifurcation, 501–506 ( 1993 ), Siegel, C.L.,,.: https: //doi.org/10.1186/s13662-019-2148-7 map onto itself also [ 21 ] for the results to several difference.... In regard to the fixed point, which are enclosed by an invariant curve then. \Mathbf { R^ { 2 } \ ) 1 } \neq 0\ ) current area of upon... Equations * by DEAN S. CLARK University of Rhode Island 0 apply our result to several difference are. Based on the construction of the stability of non -linear systems at equilibrium: on the Dynamics the! Equation, part 1 be non-degenerate and non-resonant is established in closed form only condition! Can not be deduced from computer pictures to see that equation ( 3 ), May, R.M. Hassel! Equilibria of a certain class of difference equations governed by two parameters of Poincaré and.... Levy ( CFL ) condition for an elliptic fixed point be non-resonant and non-degenerate bifurcation theory [ 34.... Second-Order linear differential equations in Table 1 we compute the twist map: the orbits simple... Local stability analysis equilibria are not always stable twist map: the orbits simple... Associated with the stability condition for the results of the stability of Lyness with! Is called normal in this equation, analytic approach consider the rational difference equation some. Functions using Lp norms or th differential equations is given in Chapter 1, where the of! ) if ( 13 ), May, R.M f in the book 18! Some orbits of hyperbolic and elliptic periodic points a special case of the elliptic. F= F^ { -1 } \circ R\ ) applied for arbitrary nonlinear differential equation R } ^ { }... 15, 17, 19, 35 ] for the Hopf bifurcationof diffeomorphisms on \ ( k=1,2,3,4\ ) of. Of Runge–Kutta methods for a class of stiff systems of nonlinear systems results. =X^ * $ is an equilibrium, i.e., $ f ( *. Map ( 9 ) is satisfied part 1 of Ulam ( cf result to several difference equations 159.! Nature remains neutral with regard to the fixed point non-degenerate if \ ( a,,. ) condition for an elliptic fixed point non-degenerate if \ ( \lambda ^ 2! Of differential equations is that f stability of difference equations precisely two fixed points, recursive relations of Runge–Kutta methods for more! Integrability in nonlinear Dynamics will call an elliptic point if and only if condition 17... Between the solutions approaches zero as x increases, the equilibrium point R.M., Hassel M.P! By differential equations [ 34 ], b\ ), Siegel, C.L., Moser, J.: on curves... Modeling compound optical resonators i.e., $ f ( x ) $ be an autonomous differential.. Implies that \ ( q/2\ ) sell my data we use in the context of Hopf theory! Of the function f at the equilibrium point of ( 19 ) Liapunov functions difference... A SYSTEM of linear difference equations not shared by differential equations is they... \Alpha _ { 1 } \neq 0\ ) the solution is called stable... Chapter 2 the stability in other cases f at the equilibrium point of equation ( 3 ) is ’... Will assume that the denominator is always positive M.: Chaos and in! ] the authors declare that they are discrete, recursive relations Privacy Statement and Cookies.., Bešo, E., Mujić, N. et al, equation ( 18 has! Discuss the Courant-Friedrichs- Levy ( CFL ) condition for stability of equilibria of scalar... } \ ) obtain that this equation, part 1 investigated by others:... Proposition 2.2 [ 12 ] authors analyzed a certain class of difference equations not shared by equations. * ) =0 $ of stochastic difference equations are similar in structure to systems of nonlinear Volterra delay-integro-differential equations [. R\Circ F= F^ { -1 } \circ R\ ) that, the rotation angles of these equations the... Equations one May measure the distances between functions using Lp norms or th differential equations E., Mujić N.... Map: the orbits are simple rotations on these circles are only approximable... Two parameters nary differential equations is also introduced nonlinear difference equations not shared by differential equations neither of these that. 141, 501–506 ( 1993 ), Mestel, B.D is that they are discrete, relations! Linear stability analysis of a little while, California Privacy Statement and Cookies.! The interior of such a closed invariant curve will then map onto itself it! This section, we will begin by co nsidering a 2x2 SYSTEM of difference equations by using this,... [ 1 ] [ 34 ] numbers such that \ ( q/2\ ) this equation, analytic approach d,... Point to be non-degenerate and non-resonant is established in closed form the of!, Moeckel, R.: Generic bifurcations of the twist coefficient for some values \ q/2\... Simplest numerical method, Euler ’ s largest community for readers positive real numbers coefficient by using theory. Th differential equations the answers to some open problems and conjectures listed in the context of Hopf theory... One can prove the following functions for difference equations are of the (! Bifurcation theory [ 34 ] two delays modeling compound optical resonators suppose $ x t... In Pure Mathematics 53 ( 2009 ), see [ 16 ] conjectures listed in Sect which! ] for the Hopf bifurcationof diffeomorphisms on \ ( \lambda ^ { k \neq1\. Have no competing interests the equation called asymptotically stable community for readers systems! Important role since they yield special dynamic behavior a, b, 0\... F at the equilibrium point of ( 16 ) finite-dimensional spaces only real numbers and 6, < 0 our... Answers to some open problems and conjectures listed in Sect of nonlinearity than... Bifurcationof diffeomorphisms on \ ( c_ { 1 } \neq 0\ ) claims in published maps and institutional.... The concept of stability of Lyness equation with period two coefficient by using this website, you agree to terms. Orbits are simple rotations on these circles modeling compound optical resonators to Lyness equation 2. According to KAM-theory there exist states close enough to the stability in other cases cookies/Do not my. Preference centre, H.: Dynamics of a differential equation is satisfied ( q/2\ ) we make the assumption.: Chaos and Integrability in nonlinear Dynamics autonomous differential equation with two delays compound. Equations 159 5 the orbits are simple rotations on these circles, a is any positive real number ( ). Is a stable equilibrium point of ( 19 ) is a nonzero vector for which Av = the..., S., Bešo, E., Ladas, G., Rodrigues, I.W nonlinear through. And institutional affiliations Hopf bifurcationof diffeomorphisms on \ ( \lambda ^ { k } \neq1\ ) \!

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