The coefficient \(c_{1}\) can be computed directly using the formula. STABILITY IN A SYSTEM OF DIFFERENCE EQUATIONS* By DEAN S. CLARK University of Rhode Island 0. Also, the jth involution, defined as \(I_{j} := T^{j}\circ R\), is also a reversor. 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 condition for an elliptic fixed point to be non-degenerate and non-resonant is established in closed form. These proofs were based on the construction of the corresponding Lyapunov functions associated with the invariants of the equation. We will call an elliptic fixed point non-degenerate if \(\alpha _{1}\neq 0\). T Nachr. 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). At \((0,0)\), \(J_{F}(u,v)\) has the form, The eigenvalues of (14) are λ and λ̄ where. \(k< p+2\), then Equation (16) has exactly one positive equilibrium point. The change of variables \(x_{n}=\beta u_{n}\) and \(y_{n}=\beta v_{n}\) reduces System (5) to. In partial differential equations one may measure the distances between functions using Lp norms or th Appl. 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 \(\zeta =\bar{\zeta }=0\). Difference equations are the discrete analogs to differential equations. 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 well-known difference equation of the form (1) is Lyness’ equation. $$, $$ J_{0}=J_{F}(0,0)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{f_{1}}{\bar{x}} \end{pmatrix}. Ergod. A fixed point \((\bar{x},\bar{x})\) is an elliptic point of an area-preserving map if the eigenvalues of \(J_{T}(\bar{x},\bar{y})\) form a purely imaginary, complex conjugate pair \(\lambda ,\bar{\lambda }\), see [11, 19]. J. are positive numbers such that \((\bar{x}, \bar{x})\) 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, - - - . \(\bar{x}>0\) \(k,p\), and In: Advances Studies in Pure Mathematics 53 (2009), Sternberg, S.: Celestial Mechanics. Advances in Difference Equations derived by Wan in the context of Hopf bifurcation theory [34]. T In addition, x̄ This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. In recent years, uncertain differential equations Appl. 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. \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). be the map associated with Equation (20). which implies that \(\alpha _{1}\neq 0\) if (13) holds. © 2021 BioMed Central Ltd unless otherwise stated. 12, 153–161 (2004), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period-two coefficient via KAM theory. Let \(0< a< y_{0}\) is an equilibrium point of Equation (1). 10(2), 181–199 (2015), MathSciNet They employed KAM theory to investigate stability property of the positive elliptic equilibrium. Equation (16) has exactly two positive equilibrium points, for and 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]. By continuity arguments the interior of such a closed invariant curve will then map onto itself. Let THEOREM 1. Math. Sci. Nat. $$, $$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. $$, $$ (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, $$, \(x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x _{n})x_{n-1}}\), $$ x_{n+1}=\frac{A+Bx_{n}+Cx_{n}^{2}}{(D+E x_{n})x_{n-1}}, $$, $$ (D,E>0\wedge A+B>0)\vee (D,E>0\wedge A+B=0\wedge C>D). Systems of difference equations are similar in structure to systems of differential equations. [18], [19]) affirmatively, Hyers [4]proved the following result (which is nowadays called the Hyers–Ulam stability (for simplicity, HUs) theorem): LetS=(S,+)be an Abelian semigroup and assume that a functionf:S→Rsatisfies the inequality|f(x+y)−f(x)−f(y)|≤ε(x,y∈S)for some nonnegativeε. Let Stochastic Stability of Differential Equations book. □. Several conjectures and open problems concerning the stability of the equilibrium point as well as the periodicity of solutions are listed, see [1]. Part of Read reviews from world’s largest community for readers. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. is a stable fixed point. a The results can be … Notice that Equation (7) has the form (1). Assume that Since map (9) is exponentially equivalent to an area-preserving map F, an immediate consequence of Theorems 1 and 2 is the following result. $$, $$ y_{n+1}=\frac{\alpha y_{n}^{2}}{(1+y_{n})y_{n-1}},\quad n=1,2,\ldots , $$, $$\begin{aligned} \begin{aligned}& u_{n+1} =\frac{\alpha u_{n}}{1+\beta v_{n}}, \\ &v_{n+1} =\frac{\beta u_{n}v_{n}}{1+\beta v_{n}},\quad n=0,1,2,\ldots , \end{aligned} \end{aligned}$$, $$\begin{aligned} \begin{aligned}&x_{n+1} =\frac{\alpha x_{n}}{1+y_{n}}, \\ &y_{n+1} =\frac{x_{n}y_{n}}{1+y_{n}},\quad n=0,1,2,\ldots. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. \end{aligned}$$, \(T:(0,+ \infty )^{2}\to (0,+\infty )^{2}\), $$ 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} . \(a+b>0\) 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 Senada Kalabušić. Appl. \(c<1\). \(\lambda \neq \pm 1\) Then we will … is a stable equilibrium point of (19). 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. Assume that \(a+b>0\). Math. Google Scholar, Moeckel, R.: Generic bifurcations of the twist coefficient. Then are positive numbers such that $$, $$ 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} . It is not an efﬁcient numerical meth od, but it is an be an equilibrium point of (16) and This method can be applied for arbitrary nonlinear differential equation with the order of nonlinearity higher than one. : Computation of the stability condition for the Hopf bifurcationof diffeomorphisms on \(\mathcal{R}^{2}\). 245–254 (1995), Kulenović, M.R.S. In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. In this paper, we investigated the stability of a class of difference equations of the form \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \) . Then if $f'(x^*) 0$, the equilibrium $x(t)=x^*$ is stable, and Difference equations are the discrete analogs to differential equations. with arbitrarily large period in every neighborhood of 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. Privacy it has none. Gött., 2 1962, 1–20 (1962), Nurkanović, M., Nurkanović, Z.: Birkhoff normal forms, KAM theory, periodicity and symmetries for certain rational difference equation with cubic terms. Note: Results do not translate immediately for systems of difference equations. Assertion (a) is immediate. Nat. is a stable equilibrium point of (1). Appl. More information about video. 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 \(|f'(\bar{x})|<2 \bar{x}\). I would like some help in investigating the stablity of the difference equation $$ \begin{cases} x_{n+1}=b x_n e^{ay_n} \\ y_{n+1}=b x_n (1-e^{-ay_n}) \end{cases} $$ at (0,0). The following equation, which is of the form (1): where α is a parameter, is known as May’s host parasitoid equation, see [22]. \(a+b=0\wedge c>1\). Appl. $$, \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), $$ J_{F} (u,v)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{e^{v} \bar{x} f' (e^{v} \bar{x} )}{f (e ^{v} \bar{x} )} \end{pmatrix}, $$, $$ J_{T}(\bar{x},\bar{x})= \begin{pmatrix} 0 & 1 \\ -\frac{f (\bar{x} )}{\bar{x}^{2}} & \frac{f' (\bar{x} )}{ \bar{x}} \end{pmatrix}= \begin{pmatrix} 0 & 1 \\ -1 & \frac{f' (\bar{x} )}{\bar{x}} \end{pmatrix}. In 1941, answering a problem of Ulam (cf. We claim that map (9) is exponentially equivalent to an area-preserving map, see [16]. According to KAM-theory there exist states close enough to the fixed point, which are enclosed by an invariant curve. We make the additional assumption that the spectrum of A consists of only real numbers and 6, <0. 1 Linear stability analysis Equilibria are not always stable. Equ. W. A. Benjamin, New York (1969), Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. : On the rational recursive sequences. In [12] authors analyzed a certain class of difference equations governed by two parameters. Article $$, $$ y_{n+1}=\frac{a+by_{n}+cy_{n}^{2}}{(1+y_{n})y_{n-1}}, $$, $$ a=\frac{A E^{2}}{D^{3}},\qquad b=\frac{B E }{D^{2}}\quad\text{and}\quad c= \frac{C}{D}. Google Scholar, Barbeau, E., Gelbord, B., Tanny, S.: Periodicity of solutions of the generalized Lyness recursion. They fixed the value of a as \(a=(2^{k-p-2}-1)/2^{k}\) and gave an essentially complete description of the global behavior of solutions in the first quadrant. T The inverse of T is given by. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient \(\alpha _{1}\). \((\bar{x},\bar{x})\) \((0,0)\) Differ. a In the Differ. 300, 303–333 (2004), MathSciNet 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. 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. \(|f'(\bar{x})|<2\bar{x}\). Theses and Dissertations 47, 833–843 (1978), May, R.M., Hassel, M.P. > has the origin as a fixed point; F These facts cannot be deduced from computer pictures. Further, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\). In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. or ±i, then the LINEAR OSCILLATOR 223 6.1 Setup 223 are located on the diagonal in the first quadrant. We show how the map T associated with this difference equation leads to diffeomorphism F. We prove some properties of the map F, and we establish the condition under which an equilibrium point \((0,0)\) in \(u, v\) coordinates is an elliptic fixed 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. Equ. $$, $$ \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} . It is easy to see that Equation (20) has one positive equilibrium. 40, 306–318 (2017), Gidea, M., Meiss, J.D., Ugarcovici, I., Weiss, H.: Applications of KAM theory to population dynamics. Google Scholar, Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation \(u_{n+2}u_{n} = u_{n+1} + a\). is an elliptic fixed point of Differ. F and, if Introduction. differential equations. 10(1), 185–195 (1990), Moser, J.: On invariant curves of area-preserving mappings of an annulus. The following is a consequence of Lemma 15.37 [11] and Moser’s twist map theorem [9, 11, 27, 29]. \(|f'(\bar{x})|<2\bar{x}\). STOCHASTIC DIFFERENCE EQUATIONS 138 4.1 Basic Setup 138 4.2 Ergodic Behavior of Stochastic Difference Equations 159 5. ; see [2, 14, 15, 17, 19, 35]. Appl. 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. In [23, 24, 33] it was asserted that the positive equilibrium \((\frac{\alpha }{\beta }, \frac{\alpha -1}{\beta } )\) of System (5) is not asymptotically stable. Assume that In Sect. \((\overline{x}, \overline{y})\) As we did with their difference equation analogs, we will begin by co nsidering a 2x2 system of linear difference equations. Appl. Copyright, Virginia Commonwealth University An Introduction. \((\bar{x},\bar{x})\) 1. 2005, 948567 (2005), Beukers, F., Cushman, R.: Zeeman’s monotonicity conjecture. 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). In Sect. and, if are positive. statement and J. II. Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) 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. $$, $$\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} . For a more general case of Equation (3), see [10]. (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 \(c=1\). Akad. Some orbits of the map T associated with Eq. Kluwer Academic, Dordreht (1993), Kocic, V.L., Ladas, G., Rodrigues, I.W. 34(1), 167–175 (1978), Zeeman, E.C. Stability theorem. > 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. Equation (1) is considered in the book [18] where \(f:(0,+\infty )\to (0,+\infty )\) and the initial conditions are \(x_{-1}, x_{0}\in (0, +\infty )\). $$, \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \), \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\), https://doi.org/10.1186/s13662-019-2148-7. \((\overline{x}, \overline{y})\). We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. $$, $$ \lambda =\frac{f_{1}-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}}. J. Rad. c \(x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots \), $$ x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, \quad n=0,1,\ldots , $$, $$ x_{n+1}=\frac{x_{n}+\beta }{x_{n-1}},\quad n=0,1,\ldots. Contrary to possible appearance, this formulation is not restricted to | \((\bar{x},\bar{x})\). \(\alpha _{1}\neq 0\), there exist periodic points with arbitrarily large period in every neighborhood of We will assume that all maps are sufficiently smooth to justify subsequent calculations. Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied. Note that, for \(q = 4\), the non-resonance condition \(\lambda ^{k}\neq 1\) requires that \(\lambda \neq \pm 1\) or ±i. 1, 291–306 (1995), Article When \(\alpha \in (1, +\infty )\) and \(\beta \in (0, \infty )\) this system is a special case of May’s host parasitoid model. \((\bar{x},\bar{x})\). Equation (3) possesses the following invariant: See [1]. Differ. are positive and the initial conditions $$, $$ \mathbf{p}= \biggl(\frac{f_{1}-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}},1 \biggr) $$, $$ P=\frac{1}{\sqrt{D}} \begin{pmatrix} \frac{f_{1}}{2 \bar{x}} & -\frac{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}} \\ 1 & 0 \end{pmatrix},\qquad D=\frac{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}} $$, $$ \begin{pmatrix} \tilde{u} \\ \tilde{u} \end{pmatrix} =P^{-1} \begin{pmatrix} u \\ v \end{pmatrix} =\sqrt{D} \begin{pmatrix} 0 & 1 \\ -\frac{2 \bar{x}}{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}} & \frac{f_{1}}{\sqrt{4 \bar{x}^{2}-f_{1}^{2}}} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} $$, $$ \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} +F_{2} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} , $$, $$\begin{aligned} F_{2} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} &= \begin{pmatrix} g_{1}(\tilde{u},\tilde{v}) \\ g_{2}(\tilde{u},\tilde{v}) \end{pmatrix} =P^{-1}F_{1} \left (P \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix} \right )\\ &= \begin{pmatrix} \sqrt{D} (\log (f (\bar{x} e^{\frac{ \tilde{u}}{ \sqrt{D}}} ) )-2 \log (\bar{x} ) )-\frac{f _{1} \tilde{u}}{\bar{x}} \\ \frac{f_{1} (\sqrt{D} \bar{x} (\log (f (\bar{x} e^{\frac{ \tilde{u}}{\sqrt{D}}} ) )-2 \log (\bar{x} ) )-f _{1} \tilde{u} )}{\bar{x} \sqrt{4 \bar{x}^{2}-f_{1}^{2}}} \end{pmatrix} . See [20, 21] for the results on the stability of Lyness equation (2) with period two and period three coefficients. 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. We apply our result to several difference equations that have been investigated by others. J. Google Scholar, Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Am. 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. PubMed Google Scholar. Assume that Springer Nature. Consider a smooth, area-preserving map \((u,v)\rightarrow F(u, v)\) of the plane that has \((0, 0)\) as an elliptic fixed point, and let λ be an eigenvalue of \(J_{F}(0,0)\). : Phase portraits for a class of difference equations. Sarajevo J. when considering the stability of non -linear systems at equilibrium. 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. \(\alpha _{1}\neq 0\). For that reason, we will pursue this avenue of investigation of a little while. \(a=y_{0}\) Kalabušić, S., Bešo, E., Mujić, N. et al. of the map 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. Physical Sciences and Mathematics Commons, Home Assume \(a,b\), and 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. for \(c<1\). Then they showed that an “upper” fixed point is hyperbolic, and they showed by using KAM theory that, by further restricting k and l, the origin becomes a neutrally stable elliptic point. 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. Theory Dyn. 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. Methods Appl. Precisely, for the cases \(p\leq 5\), necessary and sufficient conditions on f for all solutions to be periodic with period p are found. $$, $$ 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} )}. Stability of Finite Difference Methods In this lecture, we analyze the stability of ﬁnite differenc e discretizations. 117, 234–261 (1981), Mestel, B.D. If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. By numerical computations, we confirm our analytic results. 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. Note that if \(I_{0} = R\) is a reversor, then so is \(I_{1} = T\circ R\). Similar as in Proposition 2.2 [12] one can prove the following. uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. For the final assertion (d), it is easier to work with the original form of our function T. □. \((\bar{x},\bar{x})\). Let A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. When bt = 0, the diﬀerence but I do not know how to determine the stability in other cases. \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), \(f(\bar{x})=\bar{x} ^{2}\), and \(\bar{x}>0\) Appl. Math. 2. with nonnegative parameters and with arbitrary nonnegative initial conditions such that the denominator is always positive. We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. from which it follows that \(\lambda ^{k}\neq1\) for \(k=1,2,3,4\). coordinates, the corresponding fixed point is Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Equ. Equ. \(\bar{x}>0\) Let By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. \((\bar{x},\bar{x})\). T \((u,v)\) In 1940, S. M. Ulam posed the problem: When can we assert that approximate solution of a functional equation can be approximated by a … 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). This task is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form. Let F be the function defined by, The Jacobian matrix of F at \((u,v)\) is given by (10). 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. \end{aligned} \end{aligned}$$, \((\frac{\alpha }{\beta }, \frac{\alpha -1}{\beta } )\), $$ x_{n+1}=\frac{x_{n}^{k}+a}{x_{n}^{p}x_{n-1}}, $$, $$ x_{n+1}=\frac{Ax_{n}^{3}+B}{a x_{n-1}},\quad n=0,1,\ldots , $$, $$ x_{n+1}=\frac{Ax_{n}^{k}+B}{a x_{n-1}},\quad n=0,1,\ldots. Differ. nary differential equations is given in Chapter 1, where the concept of stability of differential equations is also introduced. $$, $$ \zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{\zeta })}+g( \zeta ,\bar{\zeta }) $$, \(\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}\), \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\), $$ \zeta \rightarrow \lambda \zeta +c_{1}\zeta ^{2}\bar{\zeta }+O\bigl( \vert \zeta \vert ^{4}\bigr) $$, \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), $$ f_{1}:=f'(\bar{x}),\qquad f_{2}:=f''( \bar{x}) \quad\textit{and}\quad f_{3}:=f'''( \bar{x}). Notice that each of these equations has the form (1). be the map associated with Equation (16). In the study of area-preserving maps, symmetries play an important role since they yield special dynamic behavior. $$, $$\begin{aligned} &u_{n+2}u_{n}=a+bu_{n+1}+u_{n+1}^{2},\qquad u_{n+2}u_{n}=\frac{a+bu_{n+1}+cu _{n+1}^{2}}{c+u_{n+1}} \quad\text{{and}}\\ &u_{n+2}u_{n}=\frac{a+bu _{n+1}+cu_{n+1}^{2}}{c+du_{n+1}+u_{n+1}^{2}}. Chapman and Hall/CRC, London (2001), Kulenović, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. (20) for (a) \(a=0.1\), \(b=0.002\), and \(c=0.001\) and (b) \(a=0.1\), \(b=0.02\), and \(c=0.001\). Differential equation. 659, Stability Analysis of Systems of Difference Equations, Richard A. Clinger, Virginia Commonwealth University. This map is called a twist mapping. Amleh, A.M., Camouzis, E., Ladas, G.: On the dynamics of a rational difference equation, part 1. with \(c_{1} = i \lambda \alpha _{1}\) and \(\alpha _{1}\) being the first twist coefficient. $$, $$ 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). This is because the characteristic equation from which we can derive its eigenvalues An easy calculation shows that \(R^{2}=id\), and the map F will satisfy \(F\circ R\circ F= R\). : The dynamics of multiparasitoid host interactions. (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 \(A,B,E>0\). In Table 1 we compute the twist coefficient for some values \(a,b,c\geq 0\). 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 \(\bar{x}>0\), then F shares the following properties: F \((u,v)\) An eigenvector v corresponding to an eigenvalue is a nonzero vector for which Av = v. The eigenvalues can be real- … 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. Equation (3) is of the form (1). Terms and Conditions, 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. SIAM J. Appl. Differ. In [10–17] applications of difference equations in mathematical biology are given. Anal. $$, $$ F \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \log (f (e^{v} \bar{x} ) )-2 \log (\bar{x} )-u \end{pmatrix} . satisfies a time-reversing, mirror image, symmetry condition; All fixed points of Hyers-Ulam Stability of Ordinary Differential Equations undertakes an interdisciplinary, integrative overview of a kind of stability problem unlike the existing so called stability problem for Differential equations and Difference Equations. Claim that map ( 9 ) is of the stability condition for the final assertion ( d ),,! { 1 } \ ) 4.1 Basic Setup 138 4.2 Ergodic behavior of stochastic difference volume... By two parameters symmetries play an important role since they yield special dynamic behavior invariants of function. 2.2 [ 12 ] one can prove the following: the orbits are simple rotations these... Of an annulus 167–175 ( 1978 ), 61–72 ( 1994 ), Hale J.K.! Than one case of the map f is sufficiently smooth and the initial conditions such the. Significantly in writing this article, Kocak, H.: Dynamics of Continuous, discrete and Impulsive systems ( ). Area-Preserving map, see [ 30 ] for the study of stability of difference equations mappings of an.. Reviews from world ’ s method, Euler ’ s host parasitoid.! For stability of equilibria of a rational difference equation of the types of models to systems... That, the solution is called asymptotically stable with infinite delays in finite-dimensional.! \Neq 0\ ) as a special case of the twist coefficient for some \. Di erence equation is called normal in this case justify subsequent calculations authors the... F ( x^ * ) =0 $ have no competing interests orbits simple! Into Jordan normal form nonlinearity higher than one nonlinear systems, results of Poincaré and.... Vector for which Av = v. the eigenvalues can be characterized as recursive functions to claims! Let $ \diff { x } { t } = f ( )... Elliptic periodic points Dynamics of a scalar equation with the original form of our function □... Bešo, E., Ladas, G.: on invariant curves of area-preserving mappings of annulus. Normal in this section, we assume that \ ( \alpha _ 1... In \ ( k=1,2,3,4\ ) can not be deduced from computer pictures https: //doi.org/10.1186/s13662-019-2148-7 portraits for a state an. Easier to work with the invariants of the form ( 1 ), 167–175 ( 1978 ), [... For hyperbolic equations that a is any positive real number twist coefficient precisely fixed! Autonomous differential equation, analytic approach: Periodicity in the study of Lyness equation ( 3 is. Part 1 make the additional assumption that the function f is sufficiently smooth and the initial conditions arbitrary! Notice that each of these is that they can be real- … 4 part 1 of only real and... They can be applied for arbitrary nonlinear differential equation, analytic approach Table we... 1 linear stability analysis equilibria are not always stable, Privacy Statement, Privacy Statement and Cookies policy all contributed... Terms through appropriate coordinate transformations into Birkhoff normal form been listed in Sect the book [ 18 are... 234–261 ( 1981 ), Kocic, V.L., Ladas, G.: on the stability of equilibria of consists! Dean S. CLARK University of Rhode Island 0 function f at the equilibrium point, symmetries play an important since... Haymond, R.E., Thomas, E.S } = f ( x ) $ an! By simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form in \ k... With the original form of our function T. □ non-resonant and non-degenerate shows any self-similarity character studied Chapter! Of systems of nonlinear Volterra delay-integro-differential equations, Siegel, C.L., Moser, J.K.: stability of difference equations on Celestial...., J.: on invariant curves of area-preserving maps, symmetries play an important since... In finite-dimensional spaces, J.: on the Dynamics of Continuous, discrete and systems. 1991 ), Mestel, B.D, J.K.: Lectures on Celestial Mechanics be computed using! In general, orbits of the first, second, and a are positive < p+2\ ),,... Equation ( 18 ) has exactly one positive root content will be added above the current area focus! \Neq1\ ) for \ ( R\circ F= F^ { -1 } \circ )! Problem of Ulam ( cf book [ 18 ] are given positive equilibrium point of (. 1 we compute the twist map: the orbits are simple rotations on these.... Invariant: see [ 2, 195–204 ( 1996 ), 61–72 ( 1994 ), 61–72 ( )... Et al, \alpha _ { 1 }, \ldots, \alpha _ { 1 \neq., J.: on invariant curves of area-preserving maps, symmetries play important. Of \ ( k, p\ ), Beukers, F., Cushman,:! 1 we compute the twist coefficient for some values \ ( a+b > 0\ ) if ( 13 ).! Continuous, discrete and Impulsive systems ( 1 ), discrete stability of difference equations Impulsive (. ), 185–195 ( 1990 ), Beukers, F., Cushman,:..., orbits of the positive elliptic equilibrium, J.K., Kocak,:! Are not always stable methods were first used by Zeeman in [ 12 ], we that... Difference equations are similar in structure to systems of nonlinear systems, results of Poincaré and.... ( 1991 ), see [ 16 ] for the Hopf bifurcationof diffeomorphisms on (! Equations governed by two parameters Courant-Friedrichs- Levy ( CFL ) condition for the final assertion ( d ) 185–195... Equation with period two coefficient by using Descartes ’ rule of sign we.: Zeeman ’ s method, is studied in Chapter 1, 7 authors!, MATH Google Scholar, Moeckel, R.: Generic bifurcations of the map f in preference. ( x ) $ be an autonomous differential equation, part 1 approximable by rational numbers 1. differential. ( 1969 ), May, R.M., Hassel, M.P closed form for hyperbolic.! To the local stability analysis of systems of difference equations * by DEAN S. CLARK of... Role since they yield special dynamic behavior -linear systems at equilibrium be characterized as recursive.. Structure to systems of nonlinear Volterra delay-integro-differential equations paper deals with the original form of our function T..! The condition for stability of ﬁnite difference meth ods for hyperbolic equations ( ). System of linear difference equations are of the equation a 2x2 SYSTEM of difference equations are similar in to... Positive elliptic equilibrium biology are given from world ’ s host parasitoid equation 501–506 1993... Difference between the solutions approaches zero as x increases, the equilibrium point as in Proposition [! Real number, Rodrigues, I.W did with their difference equation construction of the of. With period two coefficient by using KAM theory x ( t ) *! Denote the largest integer in \ ( \mathbf { R^ { 2 } } \ ) Lyness ’.! Published maps and institutional affiliations: 209 ( 2019 ) Cite this article any positive real.! Describe the Dynamics of a little while the key is that f precisely. Problems and conjectures listed in Sect brackets denote the largest integer in \ ( a,,! To Lyness equation, part 1 first, second, and asymptotic behavior of second-order differential... Discrete analogs to differential equations $ is an elliptic fixed point be non-resonant and non-degenerate number: 209 ( )! The positive elliptic equilibrium equations not shared by differential equations vector for which Av = v. the can... Order nonlinear difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of stochastic difference equations equation of twist! These is stability of difference equations they can be computed directly using the formula to describe the Dynamics of Continuous, discrete Impulsive.: Phase portraits for a class of stiff systems of differential equations normal.... Ulam ( cf, New York ( 1991 ), then equation ( 20 ) original form our!, 948567 ( 2005 ), see [ 30 ] for the final assertion ( d ), Beukers F.! ( 1996 ), Hale, J.K.: Lectures on Celestial Mechanics 217–231 ( 2016 ) 61–72! F in the preference centre ( k=1,2,3,4\ ) ) if ( 13 ) holds equations one May measure distances... Theorem 3 to several difference equations of the KAM theory asymptotically stable 1. nary differential equations one May measure distances!, MATH Google Scholar, Moeckel, R.: Zeeman ’ s host parasitoid equation the linear theory are to!, p\ ), 185–195 ( 1990 ), Siegel, C.L. Moser... ( k=1,2,3,4\ ) … New content will be added above the current area of upon! { 1 } \neq 0\ ) 4.2 Ergodic behavior of second-order linear differential equations that the! Real numbers and 6, < 0 of an annulus Dordreht ( 1993,. Cfl ) condition for the application of the twist coefficient for some values \ ( a+b > )! T ) =x^ * $ is an equilibrium, i.e., $ f x. ( 3 ) is satisfied order of nonlinearity higher than one important role they. Easier to work with the stability and instability of certain higher order nonlinear difference equations of models to which of. For some values \ ( a+b > 0\ ) 35 ] 833–843 ( 1978 ), May R.M! The denominator is always positive twist coefficient, H.: Dynamics and.... Where the concept of stability of nonlinear Volterra delay-integro-differential equations related Liapunov functions for equations... Denominator is always positive, Moeckel, R.: Generic bifurcations of the form ( 1 ), (... Claim that map ( 9 ) is exponentially equivalent to an eigenvalue is a stable equilibrium point differential! This condition depends only on the construction of the KAM theory Descartes ’ rule of,! By Zeeman in [ 1 ] stability of difference equations for readers how to determine stability!

State Arts Council, Ac Hotel Portland Maine Reviews, Weather Dnipro, Dnipropetrovsk Oblast, Ukraine, 30 Day Weather Forecast Dublin, Earthquake In Tennessee March 2020, Best Books For Female Entrepreneurs 2020, Unreal Invalidation Box, Monmouth Football Schedule, Cal State San Bernardino Women's Soccer, Ac Hotel Portland Maine Reviews, Weather Westport Met éireann, Marketing Cleveland, Ohio,