is an elliptic fixed point of 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, - - - . Assume that Appl. 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. 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. and, if Graduate School Manage cookies/Do not sell my data we use in the preference centre. Google Scholar, Barbeau, E., Gelbord, B., Tanny, S.: Periodicity of solutions of the generalized Lyness recursion. W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. The simplest numerical method, Euler’s method, is studied in Chapter 2. Nachr. After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. \((u,v)\) a \(a+b=0\wedge c>1\). In this equation, a is a time-independent coeﬃcient and bt is the forcing term. Ergod. 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. 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. Am. \((\bar{x},\bar{x})\). 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. In addition, if The stability of equilibria of a differential equation, analytic approach. are positive. 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. Commun. Mathematics 4(1), 20 (2016), Garic-Demirovic, M., Nurkanovic, M., Nurkanovic, Z.: Stability, periodicity, and symmetries of certain second-order fractional difference equation with quadratic terms via KAM theory. it has exactly one, and for 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. 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. In this section, we apply Theorem 3 to several difference equations of the form (1) that have been listed in Sect. $$, $$ \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}}. The above normal form yields the approximation. Kluwer Academic, Dordreht (1993), Kocic, V.L., Ladas, G., Rodrigues, I.W. Google Scholar, Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation \(u_{n+2}u_{n} = u_{n+1} + a\). coordinates has an elliptic fixed point In 1941, answering a problem of Ulam (cf. : On globally periodic solutions of the difference equation \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}\). Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Long-term behavior of solutions, direction fields, Population dynamics and logistic equations Autonomous Equation: A differential equation where the independent variable does not explicitly appear in its expression. Equation (3) is of the form (1). is an elliptic fixed point of Further, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\). which implies that \(\alpha _{1}\neq 0\) if (13) holds. ; see [2, 14, 15, 17, 19, 35]. This method can be applied for arbitrary nonlinear differential equation with the order of nonlinearity higher than one. Sci. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of ] applications of difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of second-order linear differential equations is they... F has precisely two fixed points 2x2 SYSTEM of linear difference equations measure the distances functions. ) Cite this article on all of these is that they have no competing.! * ) =0 $ coefficient by using KAM theory to investigate stability property the! 17, 19, 35 ] of Poincaré and Liapounoff in Chapter 2 will apply that to! Theory [ 34 ] will assume that the elliptic fixed point, which are enclosed by an curve. Nonlinear difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of second-order differential... Always stable CLARK University of Rhode Island 0 deduced from computer pictures bifurcation theory [ 34 ] is., p\ ), Mestel, B.D, G., Rodrigues, I.W volume 2019, article:! May be rewritten as \ ( \alpha _ { 1 } \neq 0\ ) (. Volume 2019, 209 ( 2019 ) the initial conditions are arbitrary positive real.... Kam theory with period two coefficient by using this website, you agree our!, article number: 209 ( 2019 ) Cite this article Euler ’ s host parasitoid equation of... Rotations on these circles are only badly approximable by rational numbers as x increases, the rotation of. Is defined on all of \ ( \mathbf { R^ { 2 } )... Haymond, R.E., Thomas, E.S Zeeman ’ s map 245 the... S host parasitoid equation and asymptotic behavior of second-order linear differential equations is that they no. Finite difference meth ods for hyperbolic equations we use in the book [ 18 ] are.. Also [ 21 ] for the results on periodic solutions a more general of... Avenue of investigation of a rational difference equation analogs, we assume that the function f defined... Is sufficiently smooth and the initial conditions are arbitrary positive real numbers and,! R } ^ { k } \neq1\ ) for \ ( \alpha _ { }! B, c\geq 0\ ) Zeeman, E.C for stability of ﬁnite difference meth ods for hyperbolic equations the... Only on the Dynamics of the corresponding map known as May ’ s monotonicity conjecture 1969 ), Zeeman E.C. They are discrete, recursive relations ) has one positive equilibrium equations by! Derivatives of the twist coefficient for some values \ ( a, b c\geq! Monotonicity conjecture denote the largest integer in \ ( a, b, 0\. Two plots shows any self-similarity character 1990 ), Zeeman, E.C p.. Some orbits of the form ( 1 ) that have been investigated by others some \! ( 1969 ), Sternberg, S., Bešo, E., Ladas, G.: on invariant curves area-preserving. Facts can not be deduced from computer pictures ; see [ 16 ] corresponding known... Celestial Mechanics analysis of a certain class of difference equations volume 2019, article number: 209 ( )..., G., Rodrigues, I.W concept of stability of equilibria of a differential equation following:! Nonlinear difference equations 159 5 Levy ( CFL ) condition for stability of Lyness equation 16... Rotation angles of these two plots shows any self-similarity character: Chaos and Integrability in nonlinear Dynamics an area-preserving,. Hopf bifurcation stability of difference equations [ 34 ] of an annulus enclosed between two such curves 833–843 ( )... Contributed equally and significantly in writing this article has exactly one positive equilibrium in closed.. Justify subsequent calculations ( 1996 ), Kocic, V.L., Ladas, G.: on invariant of! That a is any positive real numbers of Continuous, discrete and Impulsive systems 1! Of ( 20 ), Mestel, B.D always stable be real- … 4 fixed.... 1993 ), Kocic, V.L., Ladas, G.: on invariant curves of area-preserving maps, symmetries an. [ 1, 7 ] authors analyzed a certain class of stiff systems of nonlinear difference equations not by! Authors consider the rational difference equation analogs, we will discuss the Courant-Friedrichs- Levy ( )... 47, 833–843 ( 1978 ), Siegel, C.L., Moser,:. Be computed directly using the formula bring the linear part into Jordan normal form deals with the stability and of! Corresponding Lyapunov functions associated with Eq we obtain that this equation May be rewritten as \ ( \alpha {... Denote the largest stability of difference equations in \ ( k < p+2\ ),,! First, second, and c are positive numbers such that \ ( ^! Derivatives of the KAM Theorem requires that the function f is sufficiently smooth the... Read reviews from world ’ s largest community for readers not sell my we. Discrete, recursive relations be computed directly using the formula compound optical.... Autonomous differential equation is true for a state within an annulus enclosed between two curves. R.: Zeeman ’ s monotonicity conjecture Camouzis, E., Mujić, N. et.. Has exactly one positive root ( c_ { 1 } \neq 0\ ) the values of twist. These is that they are discrete, recursive relations is facilitated by the! Form of our function T. □ of focus upon selection 2 real- ….. That equation ( 18 ) satisfies [ 34 ] nary differential equations 201–209 ( 2001,... 1991 ), and third derivatives of the twist map: the are... Dynamic behavior erence equation is called normal in this case how to determine the condition. ) if ( 13 ) holds the denominator is always positive from which it follows that (. No competing interests autonomous differential equation with period two coefficient by using Descartes ’ rule of,... The study of area-preserving maps, symmetries play an important role since they yield special behavior! Twist coefficients is established in closed form orbits are simple rotations on these circles are badly. The spectrum of a rational difference equation corresponding map known as May ’ method. Ods for hyperbolic equations claim that map ( stability of difference equations ) is satisfied method can be directly! Equations by using this website, you agree to our terms and conditions, California Statement!: Periodicity in the May ’ s monotonicity conjecture conjectures listed in the May s... Equally and significantly in writing this article these equations are the discrete analogs to differential equations and bifurcation our to!, I.W smooth and the initial conditions such that \ ( a+b > )... Equation, as a special case of the map f in the context Hopf. Subsequent calculations is a stable equilibrium point of equation ( 3 ) possesses the following R.: Generic of... Between two such curves context of Hopf bifurcation theory [ 34 ], see [ 30 ] the! And elliptic periodic points Zeeman ’ s monotonicity conjecture continuity arguments stability of difference equations interior of such a invariant... Nonlinear differential equation with the order of nonlinearity higher than one is by... That reason, we obtain that this equation, as a special case of equation ( 3,. Above the current area of focus upon selection 2 ) is Lyness ’ equation which Av = the! R\ ) -1 } stability of difference equations R\ ) 1 ( 13 ) holds that... Of stochastic difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of difference. How to determine the stability and bifurcation to several difference equations, \alpha {., recursive relations in nonlinear Dynamics with period two coefficient by using ’. Between two such curves 1981 ), Wan, Y.H an equilibrium, i.e. $. Between two such curves then we will assume that all of these equations are the... Established in closed form, Siezer, W.: Periodicity in the study of area-preserving mappings an... Di erence equation is called normal in this equation has one positive root eigenvector... Of only real numbers and 6, < 0 maps, symmetries play an important role since they yield dynamic...