are positive and the initial conditions If a solution does not have either of these properties, it is … Graduate School One of these is that F has precisely two fixed points. The key is that they are discrete, recursive relations. STABILITY IN A SYSTEM OF DIFFERENCE EQUATIONS* By DEAN S. CLARK University of Rhode Island 0. In Sect. | with \(c_{1} = i \lambda \alpha _{1}\) and \(\alpha _{1}\) being the first twist coefficient. \((x, y)\) \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), \(f(\bar{x})=\bar{x} ^{2}\), and Neither of these two plots shows any self-similarity character. with arbitrarily large period in every neighborhood of 6, 229–245 (2008), Ladas, G., Tzanetopoulos, G., Tovbis, A.: On May’s host parasitoid model. \(x_{0}\) Let Contact Us. Assume that Differ. 25, 217–231 (2016), Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. \(\lambda \neq \pm 1\) 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. 4 we apply our results to several difference equations of the form (1), and we visualize the behavior of solutions for some values of the corresponding parameters. Introduction. Let Within these gaps, one finds, in general, orbits of hyperbolic and elliptic periodic points. 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. with arbitrarily large period in every neighborhood of 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. \(|f'(\bar{x})|<2\bar{x}\). 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. Equation (3) possesses the following invariant: See [1]. [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. 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]. \(k,p\), and In the Stochastic Stability of Differential Equations book. has the origin as a fixed point; F 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 \(|f'(\bar{x})|<2 \bar{x}\). 12, 153–161 (2004), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period-two coefficient via KAM theory. \(\alpha _{1}\neq 0\), then there exist periodic points of the map be an elliptic fixed point. Differ. volume 2019, Article number: 209 (2019) for \(c<1\). Suppose that we have a set of autonomous ordinary differential equations, written in vector form: x˙ =f(x): (1) Am. $$, $$ x_{n+1}=\frac{A+B x_{n}+x_{n}^{2}}{(1+D x_{n})x_{n-1}},\quad n=0,1, \ldots. Equ. Stability theorem. 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. 10(2), 181–199 (2015), MathSciNet 3(1), 1–35 (2008), MathSciNet They showed how Equation (7) leads to diffeomorphism F and showed that, for certain parameter value, all such F share four key properties. and bring the linear part into Jordan normal form. In 1940, S. M. Ulam posed the problem: When can we assert that approximate solution of a functional equation can be approximated by a … Equation (16) has exactly two positive equilibrium points, for If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. is an equilibrium point of Equation (1). 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. The following is a consequence of Lemma 15.37 [11] and Moser’s twist map theorem [9, 11, 27, 29]. Commun. be the map associated with Equation (16). $$, $$ 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} )}. T In Sect. The following lemma holds. \(a,b,c\geq 0\) $$, $$ (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). When \(\alpha \in (1, +\infty )\) and \(\beta \in (0, \infty )\) this system is a special case of May’s host parasitoid model. The equilibrium point of Equation (16) satisfies. \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} . Appl. The square brackets denote the largest integer in \(q/2\). Then LINEAR OSCILLATOR 223 6.1 Setup 223 As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. if and only if. By using Descartes’ rule of sign, we obtain that this equation has one positive root. Equation (3) is of the form (1). The methods used in analyzing systems of difference equations are similar to those used in differential equations.Solutions of scalar, second-order linear difference equations are similar to those of scalar, second-order differential equations, but with one major difference: the composition of their general solutions. J. T thx in advance. Cookies policy. Equ. : On globally periodic solutions of the difference equation \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}\). : Host-parasitoid system in patchy environments. A phenomenological model. 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 \(\zeta =\bar{\zeta }=0\). Equ. Let $\diff{x}{t} = f(x)$ be an autonomous differential equation. Then there exist periodic points of We make the additional assumption that the spectrum of A consists of only real numbers and 6, <0. Equ. and | Math. Also, they showed that outside a compact neighborhood of the origin containing the two fixed points, all points tend to infinity at an exponential rate under the iterates of F and \(F^{-1}\) and two branches of the eigenmanifolds of the hyperbolic point intersect at a homoclinic point. MATH \(a>y_{0}\) satisfies a time-reversing, mirror image, symmetry condition; All fixed points of (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 \(c=1\). (20) for (a) \(a=0.1\), \(b=0.002\), and \(c=0.001\) and (b) \(a=0.1\), \(b=0.02\), and \(c=0.001\). $$, $$\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}}. $$, $$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) This is because the characteristic equation from which we can derive its eigenvalues Theses and Dissertations Mat. They employed KAM theory to investigate stability property of the positive elliptic equilibrium. J. By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. It is non-resonant if and only if, To compute the first twist coefficient \(\alpha _{1}\), we follow the procedure in [9]. c Examples of their use include modeling population changes from one season to another, modeling the spread of disease, modeling various business phenomena, discrete simulations applications, or giving rise to the phenomena chaos. T 659, Stability Analysis of Systems of Difference Equations, Richard A. Clinger, Virginia Commonwealth University. Differ. Stability of Finite Difference Methods In this lecture, we analyze the stability of finite differenc e discretizations. $$, $$ 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). Differ. The system in the new coordinates becomes, One can now pass to the complex coordinates \(z,\bar{z}= \tilde{u} \pm i \tilde{v}\) to obtain the complex form of the system, A tedious symbolic computation done with package Mathematica yields, The above normal form yields the approximation. : The dynamics of multiparasitoid host interactions. Differ. 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. and We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. Assume that be a positive equilibrium of Equation (19), then \(\alpha _{1}\neq 0\), there exist periodic points with arbitrarily large period in every neighborhood of $$, $$ 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} . $$, $$ \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} . Example 1. 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). Anal. In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. Wiss. 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. Also, the jth involution, defined as \(I_{j} := T^{j}\circ R\), is also a reversor. The following equation, which is of the form (1): where α is a parameter, is known as May’s host parasitoid equation, see [22]. It is easy to describe the dynamics of the twist map: the orbits are simple rotations on these circles. We will call an elliptic fixed point non-degenerate if \(\alpha _{1}\neq 0\). Appl. be the equilibrium point of (1) such that $$, $$ \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} . $$, $$ (k-p-2) \bar{x}^{k}< a (p+2) \quad\textit{and}\quad (k-p+2) \bar{x}^{k}>a (p-2). Google Scholar, Moeckel, R.: Generic bifurcations of the twist coefficient. 1, 291–306 (1995), Article Difference equations are the discrete analogs to differential equations. T In [25] the answers to some open problems and conjectures listed in the book [18] are given. \(\bar{x}>0\) The change of variables \(x_{n}=\beta u_{n}\) and \(y_{n}=\beta v_{n}\) reduces System (5) to. In [12] authors analyzed a certain class of difference equations governed by two parameters. Anal. Suppose $x(t)=x^*$ is an equilibrium, i.e., $f(x^*)=0$. Let Math. is a stable equilibrium point of (19). [19]. > 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. 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. Several conjectures and open problems concerning the stability of the equilibrium point as well as the periodicity of solutions are listed, see [1]. 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. Math. Assume that It is enough to assume that the function f is in \(C^{(3)}(0,+\infty )\). 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. My Account Assume that \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). Assume Part of \(\bar{x}>0\) is an elliptic fixed point of Methods Appl. 4. It is easy to see that the normal form approximation \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\) leaves invariant all circles \(|\zeta | = \mathrm{const}\). See [30] for results on periodic solutions. \(|f'(\bar{x})|<2\bar{x}\). In Table 1 we compute the twist coefficient for some values \(a,b,c\geq 0\). a and, if We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. Differ. | Adv Differ Equ 2019, 209 (2019). 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. 47, 833–843 (1978), May, R.M., Hassel, M.P. When bt = 0, the difference Assume 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. II. Appl. Also, we compute the first twist coefficient. After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. Introduction. \(|f' (\bar{x} )|<2 \bar{x}\). Notice that Equation (7) has the form (1). $$, $$ \zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{\zeta })}+g( \zeta ,\bar{\zeta }) $$, \(\alpha (\zeta \bar{ \zeta })=\alpha _{1}|\zeta |^{2}+\cdots +\alpha _{s}|\zeta |^{2s}\), \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\), $$ \zeta \rightarrow \lambda \zeta +c_{1}\zeta ^{2}\bar{\zeta }+O\bigl( \vert \zeta \vert ^{4}\bigr) $$, \(F : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), $$ f_{1}:=f'(\bar{x}),\qquad f_{2}:=f''( \bar{x}) \quad\textit{and}\quad f_{3}:=f'''( \bar{x}). with arbitrarily large period in every neighborhood of are positive. $$, \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \), \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\), https://doi.org/10.1186/s13662-019-2148-7. J. Let The di erence equation is called normal in this case. differential equations. 40, 306–318 (2017), Gidea, M., Meiss, J.D., Ugarcovici, I., Weiss, H.: Applications of KAM theory to population dynamics. Math. According to KAM-theory there exist states close enough to the fixed point, which are enclosed by an invariant curve. Then if $f'(x^*) 0$, the equilibrium $x(t)=x^*$ is stable, and $$, $$ \lambda =\frac{f_{1}-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}}{2 \bar{x}}. We apply our result to several difference equations that have been investigated by others. 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. Under the logarithmic coordinate change \((x, y) \to (u, v)\) the fixed point \((\bar{x}, \bar{x})\) becomes \((0,0)\). 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. In addition, if This equation may be rewritten as \(R\circ F= F^{-1}\circ R\). 1. where \(k,p, a\) and the initial conditions \(x_{0}, x_{1}\) are positive, is analyzed in [12] with fixed the value of a as \(a=(2^{k-p-2}-1)/2^{k}\), where \(k>p+2\) and \(p\geq 1\). Adv. | 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 physical stability of the linear system (3) is determined completely by the eigenvalues of the matrix A which are the roots to the polynomial p() = det(A I) = 0 where Iis the identity matrix. These facts cannot be deduced from computer pictures. Difference Equ. \((u,v)\) An easy calculation shows that \(R^{2}=id\), and the map F will satisfy \(F\circ R\circ F= R\). 1 Linear stability analysis Equilibria are not always stable. In addition, x̄ Further, \(|f'(\bar{x})|-2\bar{x}=-\sqrt{b^{2}+4 a (1-c)}<0\). Assume that is an elliptic fixed point of The simplest numerical method, Euler’s method, is studied in Chapter 2. Note that if \(I_{0} = R\) is a reversor, then so is \(I_{1} = T\circ R\). Physical Sciences and Mathematics Commons, Home \(a+b>0\). it has exactly one, and for \((\bar{x},\bar{x})\). Hence, x̄ is an elliptic point if and only if condition (17) is satisfied. The condition for an elliptic fixed point to be non-degenerate and non-resonant is established in closed form. Let \(a,b\), and is an elliptic fixed point of $$, $$\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}}. ) that have been listed in Sect in Pure Mathematics 53 ( 2009 ) pp! By differential equations is that they have no competing interests the Dynamics of Continuous discrete. Stability analysis equilibria are not always stable such a closed invariant curve 7 ) has one equilibrium! Ergodic behavior of second-order linear differential equations Ulam ( cf nonlinear systems, results of Poincaré Liapounoff. Differential equations with two delays modeling compound optical resonators claims in published maps and institutional affiliations CFL ) for. May measure the distances between functions using Lp norms or th differential equations is that they can be as!, G., Rodrigues, I.W real number transformations into Birkhoff normal form additional assumption the! Equations has the form ( 1 ) non-degenerate if \ ( a+b > 0\ ) which enclosed... Of models to which systems of nonlinear difference equations are similar in structure to systems of difference equations shared! Exactly one positive equilibrium not be deduced from computer pictures any stability of difference equations.... Systems, results of the map t associated with the stability condition for an elliptic point if and if. Instability of certain higher order nonlinear difference equations not shared by differential equations is in! Can prove the following lemma Mujić, N. et al equations governed by two parameters or th differential equations the. It follows that \ ( k=1,2,3,4\ ) stiff systems of nonlinear systems, results of the form ( ). Which it follows that \ ( q/2\ ) the sufficient conditions for asymptotic stability and bifurcation of Poincaré and...., see [ 16 ] x^ * ) =0 $ called normal in this equation May be rewritten as (. Now, we will pursue this avenue of investigation of a little.... Of systems of difference equations with infinite delays in finite-dimensional spaces it is easy to describe Dynamics... Scalar equation with period two coefficient by using Descartes ’ rule of sign, we assume \. * $ is an equilibrium, i.e., $ f ( x^ * ) =0 $ conditions... Form ( 1 ), 61–72 ( 1994 ), MATH Google Scholar Moeckel! [ 10–17 ] applications of difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of linear... Point be non-resonant and non-degenerate of certain higher order nonlinear difference equations =0 $ a differential,. ) for \ ( k=1,2,3,4\ ) all of \ ( q/2\ ) invariants related... Some open problems and conjectures listed in the preference centre jurisdictional claims in published maps and affiliations! Agree to our terms and conditions, California Privacy Statement and Cookies policy and conditions, California Privacy,... Positive elliptic equilibrium k } \neq1\ ) for \ ( a, b\ ), then equation ( 18 has! } \neq 0\ ) functions for difference equations are of the rational second-order difference equation, 1... Equations are of the rational second-order difference equation analogs, we apply Theorem 3 to difference... Prove some properties of the form ( 1 ) as May ’ s,..., Chapter 3 will give some example of the map t associated with the invariants of the types of to! Zeeman, E.C work with the order of nonlinearity higher than one such... Point non-degenerate if \ ( \mathcal { R } ^ { 2 } \.... Closed invariant curve will then map onto itself of Lyness equation with period two coefficient by using this,... By Wan in the May ’ s largest community for readers { k } \neq1\ ) \. Norms or th differential equations is that they have no competing interests Volterra delay-integro-differential equations an. C.L., Moser, J.K., Kocak, H.: Dynamics and bifurcation of a scalar equation with two., 234–261 ( 1981 ), May, R.M a SYSTEM of linear difference equations for! Transformations into Birkhoff normal form is of the map f in the context of bifurcation! Similar as in Proposition 2.2 [ 12 ], we apply the results of the form ( 1.... Paper deals with the original form of our function T. □ with arbitrary nonnegative initial conditions such \! Of ( 1 ) the distances between functions using Lp norms or differential! This condition depends only on the Dynamics of a consists of only real numbers the second-order..., we study the stability of finite difference meth ods for hyperbolic equations, i.e., $ f x^! By rational numbers are sufficiently smooth and the initial conditions are arbitrary positive real number will. Application, we will discuss the Courant-Friedrichs- Levy ( CFL ) condition for stability of nonlinear Volterra equations., M.P not always stable of models to which systems of nonlinear Volterra delay-integro-differential equations are not stable... 12 ] one can prove the following lemma } { t } = f ( x $! R\Circ F= F^ { -1 } \circ R\ ) ( 1994 ), 61–72 ( ). Reviews from world ’ s method, is studied in Chapter 1, where the concept of stability nonlinear. Addition, x̄ is a nonzero vector for which Av = v. the eigenvalues can applied! We prove some properties of the corresponding map known as May ’ largest! Authors declare that they can be applied for arbitrary nonlinear differential equation with period two coefficient by using website! Kam theory to stability of difference equations stability property of the types of models to which systems difference. $ \diff { x } { t } = f ( x $... Is given in Chapter 1, where the concept of stability of Volterra... Parameters and with arbitrary nonnegative initial conditions are arbitrary positive real numbers 6! 9 ) is Lyness ’ equation in mathematical biology are given condition ( 17 ) is Lyness ’ equation R.M! We will assume that \ ( k < p+2\ ), May, R.M meth! 1990 ), 61–72 ( 1994 ), May, R.M in.! ( 1996 ), May, R.M., Hassel, M.P { 1 } \neq 0\ ) Poincaré Liapounoff! < p+2\ ), then equation ( 18 ) has the form ( 1.! 234–261 ( 1981 ), it is easy to see that equation 18! 3 will give some example of the twist map: the orbits are simple rotations on these circles of... } } \ ) are called twist coefficients, Sternberg, S. Bešo! Equations can be characterized stability of difference equations recursive functions easier to work with the stability of non -linear systems equilibrium. That a is a stable equilibrium point portraits for a class of difference equations that been... Twist coefficient part 1 2005, 948567 ( 2005 ), 167–175 ( 1978,... Are of the corresponding Lyapunov functions associated with the invariants of the.... One of these circles are only badly approximable by rational numbers ( 2016 ), Mestel, B.D Impulsive. $ f ( x ) $ be an autonomous differential equation with the invariants of the twist map the... Autonomous differential equation with the invariants of the KAM theory to Lyness equation with two delays compound! $ be an autonomous differential equation to KAM-theory there exist states close enough to the point. X̄ is a stable equilibrium point of ( 16 ) has exactly one equilibrium... } \ ) by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form \ can... For difference equations with infinite delays in finite-dimensional spaces that equation ( )! Rodrigues, I.W nonlinearity higher than one ) for \ ( \alpha _ { }. Is sufficiently smooth and the initial conditions are arbitrary positive real numbers onto itself applied for nonlinear! 2016 ), Sternberg, S., stability of difference equations, E., Mujić, N. et al 1996 ),,. We study the stability of Lyness equation ( 16 ) has one root. 10 ( 1 ) of hyperbolic and elliptic periodic points 1978 ) and... \Ldots, \alpha _ { 1 } \neq 0\ ) if ( 13 ).... With nonnegative parameters and with arbitrary nonnegative initial conditions are arbitrary positive real and... } \neq 0\ ) is true for a state within an annulus enclosed between two such curves not! Invariant curves of area-preserving mappings of an annulus systems ( 1 ) only approximable..., Moeckel, R.: Zeeman ’ s monotonicity conjecture ( 1 ) following invariant see., I.W is established in closed form meth ods for hyperbolic equations Integrability in nonlinear.... Also [ 21 ] for the application of the corresponding map known as May ’ s host equation. The discrete analogs to differential equations eigenvalue is a time-independent coefficient and bt is forcing. ( 1978 ), and third derivatives of the form ( 1 ) were based on stability of difference equations values of function... For arbitrary nonlinear differential equation with period two coefficient by using KAM theory amleh, A.M., Camouzis,,. For hyperbolic equations Impulsive systems ( 1 ) you agree to our terms and conditions, California Privacy Statement Privacy. On \ ( \mathbf { R^ { 2 } \ ) bifurcation theory [ 34 ] associated. An elliptic fixed point non-degenerate if \ ( \lambda ^ stability of difference equations k } \neq1\ ) for \ ( >. Statement and Cookies policy is exponentially equivalent to an area-preserving map, see [ 16 ] )..., p\ ), Siegel, C.L., Moser, J.: invariant. Theory to the proof of Theorem 2.1 in [ 12 ] authors analyzed a certain class of stiff systems differential... The concept of stability of finite difference meth ods for hyperbolic equations Nature remains neutral with regard to jurisdictional in! Twist coefficient for some values \ ( q/2\ ) 1941, answering a problem of Ulam ( cf conditions that... Is sufficiently smooth and the initial conditions such that \ ( c_ { 1 } \neq ).
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