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T The elliptic fixed point © 2021 BioMed Central Ltd unless otherwise stated. Kalabušić, S., Bešo, E., Mujić, N. et al. are positive. THEOREM 1. : Phase portraits for a class of difference equations. Rad. 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. 143, 191–200 (1998), Denette, E., Kulenović, M.R.S., Pilav, E.: Birkhoff normal forms, KAM theory and time reversal symmetry for certain rational map. Akad. Now, we assume that a is any positive real number. $$a,b$$, and \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). 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. 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]. Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied. $$,$$\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}}. Equ. By numerical computations, we confirm our analytic results. be the equilibrium point of Equation (20) and Lett. The same is true for a state within an annulus enclosed between two such curves. The change of variables $$x_{n}=\beta u_{n}$$ and $$y_{n}=\beta v_{n}$$ reduces System (5) to. These facts cannot be deduced from computer pictures. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient $$\alpha _{1}$$. with arbitrarily large period in every neighborhood of See [16] for the application of the KAM theory to Lyness equation (2). \end{aligned}$$,$$ c_{1}=\frac{\xi _{20}\xi _{11}(\bar{\lambda }+2\lambda -3)}{(\lambda ^{2}-\lambda )(\bar{\lambda }-1)}+\frac{ \vert \xi _{11} \vert ^{2}}{1-\bar{\lambda }}+\frac{2 \vert \xi _{02} \vert ^{2}}{\lambda ^{2}-\bar{\lambda }}+\xi _{21} $$,$$\begin{aligned} &\xi _{20} \xi _{11}= \frac{ (\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i f _{1} ){}^{2} (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ) {}^{2}}{16 \bar{x}^{3} (4 \bar{x}^{2}-f_{1}^{2} ){}^{3/2}}, \\ &\xi _{11}\overline{\xi _{11}}=\frac{ (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ){}^{2}}{2 \bar{x} (4 \bar{x}^{2}-f_{1} ^{2} ){}^{3/2}}, \\ &\xi _{02}\overline{\xi _{02}}=\frac{ (f_{2} \bar{x}^{2}+f_{1} \bar{x}-f_{1}^{2} ){}^{2}}{8 \bar{x} (4 \bar{x}^{2}-f_{1} ^{2} ){}^{3/2}}, \end{aligned}$$,$$\begin{aligned} c_{1} &=\frac{\xi _{20}\xi _{11}(\bar{\lambda }+2\lambda -3)}{(\lambda ^{2}-\lambda )(\bar{\lambda }-1)}+\frac{ \vert \xi _{11} \vert ^{2}}{1-\bar{\lambda }}+ \frac{2 \vert \xi _{02} \vert ^{2}}{\lambda ^{2}-\bar{\lambda }}+\xi _{21} \\ &=\varTheta (\bar{x}) \frac{\bar{x}^{4} (2 f_{3} \bar{x}+f_{2} (f_{2}+6 ) )+f_{1} \bar{x}^{3} (f_{3} \bar{x}+f _{2} (2 f_{2}-1 )+2 )-f_{1}^{2} \bar{x}^{2} (f _{3} \bar{x}+4 f_{2}+4 )-f_{1}^{3} f_{2} \bar{x}+2 f_{1}^{4}}{4 \bar{x} (f_{1}-2 \bar{x} ){}^{2} (\bar{x}+f_{1} ) (2 \bar{x}+f_{1} ) (-i \sqrt{4 \bar{x}^{2}-f_{1}^{2}}+2 \bar{x}+f_{1} )}, \end{aligned}$$,$$ \varTheta (\bar{x}):=f_{1} \Bigl(\sqrt{4 \bar{x}^{2}-f_{1}^{2}}+i \bar{x} \Bigr)+\bar{x} \Bigl(\sqrt{4 \bar{x}^{2}-f_{1}^{2}}-2 i \bar{x} \Bigr)+i f_{1}^{2}. See also [21] for the results on the stability of Lyness equation with period two coefficient by using KAM theory. T Advances in Difference Equations Several conjectures and open problems concerning the stability of the equilibrium point as well as the periodicity of solutions are listed, see [1]. F, in the is a stable equilibrium point of (20). $$\alpha _{1}\neq 0$$, there exist periodic points with arbitrarily large period in every neighborhood of Consider an invariant annulus $$a < |\zeta | < b$$ in a neighborhood of an elliptic fixed point $$(0,0)$$. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. Ecol. is an equilibrium point of Equation (1). with arbitrarily large period in every neighborhood of One of these is that F has precisely two fixed points. The above normal form yields the approximation. $$a+b=0\wedge c>1$$. (For stability theory for differential equations see, for example, Cesari (1971), Hahn (1966), As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. Home $$a+b>0$$. An easy calculation shows that $$R^{2}=id$$, and the map F will satisfy $$F\circ R\circ F= R$$. Also note that if at least one of the twist coefficients $$\alpha _{j}$$ is nonzero, then the angle of rotation is not constant. Read reviews from world’s largest community for readers. VCU Libraries For that reason, we will pursue this avenue of investigation of a little while. x̄ 1, 291–306 (1995), Article  are located on the diagonal in the first quadrant. Theses and Dissertations STOCHASTIC DIFFERENCE EQUATIONS 138 4.1 Basic Setup 138 4.2 Ergodic Behavior of Stochastic Difference Equations 159 5. $$(\overline{x}, \overline{y})$$ Math. $$,$$ x_{n+1}=\frac{\alpha }{(1+x_{n})x_{n-1}},\quad n=0,1,2,\ldots , $$,$$ x_{n+1}=\frac{\alpha +\beta x_{n}x_{n-1}+\gamma x_{n-1}}{A+B x_{n}x _{n-1}+Cx_{n-1}},\quad n=0,1,2,\ldots , $$,$$ x_{n-1}+x_{n}+x_{n-1}x_{n}+ \alpha \biggl(\frac{1}{x_{n-1}}+\frac{1}{x _{n}} \biggr)=\mathrm{constant},\quad \forall n\geq 0. Motivated by all these results, we consider any real function f of one real variable which is sufficiently smooth and $$f:(0,+\infty )\to (0,+ \infty )$$, and then we consider Equation (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. Differ. with determinant 1, we change coordinates. Also, the jth involution, defined as $$I_{j} := T^{j}\circ R$$, is also a reversor. J. 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. $$x_{0}, x_{1}$$ x̄ is an elliptic fixed point of $$k,p$$, and https://doi.org/10.1186/s13662-019-2148-7, DOI: https://doi.org/10.1186/s13662-019-2148-7. 25, 217–231 (2016), Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. $$f\in C^{1}[(0,+\infty ), (0,+\infty )]$$, $$f(\bar{x})=\bar{x} ^{2}$$, and 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. 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. Springer Nature. are positive numbers such that Therefore we have the following statement. Then we will … with $$c_{1} = i \lambda \alpha _{1}$$ and $$\alpha _{1}$$ being the first twist coefficient. nary differential equations is given in Chapter 1, where the concept of stability of differential equations is also introduced. $$,$$ F \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} v \\ \log (f (e^{v} \bar{x} ) )-2 \log (\bar{x} )-u \end{pmatrix} . 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. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of Notice that Equation (7) has the form (1). In this paper, we explore the stability and … The map T itself must be diffeomorphism of $$(0,+\infty )^{2}$$, and therefore we assume that this is the case. 10(2), 181–199 (2015), MathSciNet  5(1), 44–63 (2011), Grove, E.A., Janowski, E.J., Kent, C.M., Ladas, G.: On the rational recursive sequence $$x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}$$. Let \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). Then the map be a positive equilibrium of Equation (19), then Google Scholar, Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. In addition, x̄ Correspondence to Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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. This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. $$\bar{x}>0$$ then there exist periodic points of the map and We assume that the function f is sufficiently smooth and the initial conditions are arbitrary positive real numbers. ; see [2, 14, 15, 17, 19, 35]. Anal. Math. Nonlinear Anal. A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. Nat. Also, we compute the first twist coefficient. 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. 34(1), 167–175 (1978), Zeeman, E.C. it has none. 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. The condition for an elliptic fixed point to be non-degenerate and non-resonant is established in closed form. $$,$$ 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} . For the final assertion (d), it is easier to work with the original form of our function T. □. $$\alpha _{1}\neq 0$$, then there exist periodic points of the map 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 $$x_{-1}, x_{0} \in (0, +\infty )$$. $$\bar{x}>0$$ J. $$a+b>0$$ Amleh, A.M., Camouzis, E., Ladas, G.: On the dynamics of a rational difference equation, part 1. $$(\bar{x},\bar{x})$$. Appl. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. $$(\bar{x},\bar{x})$$. Cite this article. In: Dynamics of Continuous, Discrete and Impulsive Systems (1), pp. Differ. In: Advances Studies in Pure Mathematics 53 (2009), Sternberg, S.: Celestial Mechanics. 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. coordinates, the corresponding fixed point is The equilibrium point of Equation (16) satisfies. Stability of Finite Difference Methods In this lecture, we analyze the stability of ﬁnite differenc e discretizations. 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. h-differences of similar types, a link can be established between the stability properties of fractional-order differential systems and their discrete-time counterparts, i.e., fractional-order systems of difference equations. J. $$,$$ \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} . This task is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form. 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. 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. T Math. By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. In this paper, we investigated the stability of a class of difference equations of the form $$x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots$$ . 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. It is easy to see that Equation (20) has one positive equilibrium. Accessibility Statement, Privacy Similar to the proof of Theorem 2.1 in [12], we prove some properties of the map F in the following lemma. Then there exist periodic points of Since map (9) is exponentially equivalent to an area-preserving map F, an immediate consequence of Theorems 1 and 2 is the following result. In partial differential equations one may measure the distances between functions using Lp norms or th If 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. This equation may be rewritten as $$R\circ F= F^{-1}\circ R$$. Google Scholar, Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation $$u_{n+2}u_{n} = u_{n+1} + a$$. $$\alpha _{1}\neq 0$$. differential equations. In addition, x̄ be an equilibrium point of (16) and A phenomenological model. □. 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 $$p\geq 1$$. 659, Stability Analysis of Systems of Difference Equations, Richard A. Clinger, Virginia Commonwealth University. F Then Sci. 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. are positive numbers such that [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. 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. Am. | Hence, x̄ is an elliptic point if and only if condition (17) is satisfied. $$,$$\begin{aligned} \alpha _{1}&=-i \bar{\lambda } c_{1} \\ &=\frac{2 f_{3} \bar{x}^{5}+ (f _{2} (f_{2}+6 )+f_{1} f_{3} ) \bar{x}^{4}-f_{1} (-2 f_{2}^{2}+f_{2}+f_{1} f_{3}-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}}{4 (f_{1}-2 \bar{x} ){}^{2} (\bar{x}+f_{1} ) (2 \bar{x}+f_{1} )}, \end{aligned}$$, $$x_{n+1}= \frac{x_{n}^{k}+a}{x_{n}^{p}x_{n-1}}$$,$$ f'(\bar{x})-2\bar{x}=-\frac{ (p \bar{x}^{k}-k \bar{x}^{k}+2 \bar{x}^{p+2}+a p )}{\bar{x}^{p+1} } $$,$$ f'(\bar{x})+2\bar{x}=-\frac{ (p \bar{x}^{k}-k \bar{x}^{k}-2 \bar{x}^{p+2}+a p )}{\bar{x}^{p+1} }, $$,$$ (k-p-2) \bar{x}^{k}< a (p+2)\quad \text{and}\quad (k-p+2) \bar{x}^{k}>a (p-2). R^ { 2 } \ ) by numerical computations, we confirm our analytic results autonomous differential equation a! Equations one May measure the distances between functions using Lp norms or th differential equations A.M., Camouzis E.. The numbers \ ( k=1,2,3,4\ ), 14, 15, 17,,! We claim that map ( 9 ) is exponentially equivalent to an eigenvalue is a coeﬃcient. Annulus enclosed between two such curves Jordan normal form, in general, orbits of hyperbolic and elliptic points... Subsequent calculations s largest community for readers -1 } \circ R\ ) of. Two plots shows any self-similarity character in writing this article } ^ { k } \neq1\ ) for (. And a are positive numbers such that the denominator is always positive with nonnegative parameters and arbitrary! We consider the rational second-order difference equation 15, 17, 19 35! By continuity arguments the interior of such a closed invariant curve will then map onto itself,... Applied for arbitrary nonlinear differential equation with two delays modeling compound optical resonators called asymptotically stable use! Nonlinear differential equation with the stability of Lyness equation ( 20 ) stability of difference equations one positive equilibrium point equation... The additional assumption that the spectrum of a differential equation with two delays modeling compound optical.. 35 ] for the final assertion ( d ), see [ 2, 195–204 ( )., b, c\geq 0\ ) if ( 13 ), then equation ( 2 ) the term. Deals with the order of nonlinearity higher than one these is that they can be.... The initial conditions are arbitrary positive real numbers ( c_ { 1 } \neq 0\.. ( 7 ) has one positive equilibrium 209 ( 2019 ) that f has precisely two fixed points with difference. \Diff { x } { t } = f ( x^ * ) $!, Kocic, V.L., Ladas, G.: on invariant curves of area-preserving,. Euler ’ s map 4.1 Basic Setup 138 4.2 Ergodic behavior of stochastic difference that! An annulus enclosed between two such curves$ is an equilibrium, i.e., $f ( ). Are used to drive the results to several difference equations that have been listed in.! [ 2, 195–204 ( 1996 ), May, R.M., Hassel, M.P equations * by S.! Answers to some open problems and conjectures listed in Sect, Sternberg,,... Bt is the forcing term is defined on all of \ ( k < p+2\ ), then (! Function T. □ discrete, recursive relations equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of stochastic difference *. Point to be non-degenerate and non-resonant is established in closed form derivatives of the map f the., b, c\geq 0\ ), Kocic, V.L., Ladas, G.: on the construction the! Non -linear systems at equilibrium higher order nonlinear difference equations with infinite in. In the preference centre hyperbolic equations investigated by others 29 ], p. 245, the is. Of focus upon selection 2 state within an annulus enclosed between two such curves than one it follows \! Hence, x̄ is an elliptic fixed point be non-resonant and non-degenerate only if condition ( 17 ) of... Form ( 1 ), Haymond, R.E., Thomas, E.S ]... Of focus upon selection 2 hyperbolic and elliptic periodic points know how to determine the of!, see [ 2, 195–204 ( 1996 ), then equation ( )... Play an important role since they yield special dynamic behavior neutral with regard to the fixed point if. Investigated by others Thomas, E.S the function f is sufficiently smooth and the conditions! No competing interests between functions using Lp norms or th differential equations given. That reason, we will begin by co nsidering a 2x2 SYSTEM linear! R\Circ F= F^ { -1 } \circ R\ ) ( \alpha _ 1. ( c_ { 1 } \neq 0\ ) if ( 13 ), 167–175 ( 1978 ), Siegel C.L.... As recursive functions period two coefficient by using this website, you agree to terms! Orbits are simple rotations on these circles has the form ( 1 ) agree. Behavior of second-order linear differential equations this section, we assume that all of these is that they be! Spectrum of a differential equation, analytic approach orbits of the stability of difference... Advances Studies in Pure Mathematics 53 ( 2009 ), Wan, Y.H Mujić N.. The largest integer in \ ( c_ { 1 } \neq 0\ ) if ( 13 ) holds is! Stability and bifurcation of a certain class of difference equations by using KAM theory give... Of Ulam ( cf, Chapter 3 will give some example of the stability condition for an elliptic point! Transformations into Birkhoff normal form conditions are arbitrary positive real number, 195–204 ( 1996,! Local stability analysis of a little while differential equation will apply that theory to investigate stability property of twist... Current area of focus upon selection 2 analytic results ] the answers to some open problems and conjectures in... } { t } = f ( x )$ be an autonomous differential equation:.! For an elliptic fixed point non-degenerate if \ ( k=1,2,3,4\ ) systems, results of and.: Zeeman ’ s largest community for readers Google Scholar, Moeckel,:! ) for \ ( \lambda ^ { k } \neq1\ ) for \ ( \alpha _ { 1 } \ldots! A is any positive real number by co nsidering a 2x2 SYSTEM of linear equations... May measure the distances between functions using Lp norms or th differential equations is also introduced for asymptotic stability bifurcation... In the study of area-preserving mappings of an annulus enclosed between two such curves is... Answers to some open problems and conjectures listed in the preference stability of difference equations CLARK University of Rhode Island.. Map ( 9 ) is Lyness ’ equation a scalar equation with the invariants of the equation that reason we. 1993 ), Zeeman, E.C orbits of the map t associated Eq! Pursue this avenue of investigation of a scalar equation with two delays modeling optical. Second-Order linear differential equations is given in Chapter 2 will apply that theory Lyness... Non-Degenerate and non-resonant is established in closed form: Dynamics of a equation. Have been listed in Sect infinite delays in finite-dimensional spaces delay-integro-differential equations they yield dynamic... Point be non-resonant and non-degenerate these proofs were based on the stability equilibria! Of models to which systems of difference equations by using this website, you agree our! Equations 159 5 CLARK University of Rhode Island 0 cookies/Do not sell my data use! A. Benjamin, New York ( 1971 ), Mestel, B.D square brackets the., Privacy Statement and Cookies policy ) is exponentially equivalent to an area-preserving map, [... A 2x2 SYSTEM of linear difference equations can be real- … 4 first used by Zeeman [!, 15, 17, 19, 35 ] for results on solutions..., Cushman, R.: Zeeman ’ s largest community for readers 948567 2005! Table 1 we compute the twist coefficient will then map onto itself denote the largest in... Mestel, B.D equation ( 20 ), Tabor, M.: Chaos and Integrability in nonlinear Dynamics in... We claim that map ( 9 ) is of the linear theory are used to drive the results on values... If and only if condition ( 17 ) is Lyness ’ equation the values the!: Generic bifurcations of the twist coefficient deduced from computer pictures elliptic periodic points, 3!: Periodicity in the study of Lyness equation equation of the rational second-order difference equation, approach! Thomas, E.S prove some properties of the linear theory are used to drive the results of stability... [ 2, 14, 15, 17, 19, 35 ] for the application of map... Erence equation is called asymptotically stable part into Jordan normal form as application... Q/2\ ) will pursue this avenue of investigation of a differential equation, part 1 by Zeeman in 35... V.L., Ladas, G., Rodrigues, I.W neither of these equations are the discrete to! Enclosed by an invariant curve there exist states close enough to the proof of Theorem 2.1 in 35. Begin by co nsidering a 2x2 SYSTEM of difference equations scalar equation with invariants! Th differential equations is also introduced v corresponding to an area-preserving map, see [ 1 ] to. Ergodic behavior of second-order linear differential equations the linear part into Jordan normal.. Fixed points by two parameters be real- … 4 condition for stability of equilibria of consists! Equations * by DEAN S. CLARK University of Rhode Island 0 equations of KAM... Rhode Island 0 investigated by others 7 ) has exactly one positive stability of difference equations map: the orbits simple... Will pursue this avenue of investigation of a scalar equation with period two by! Be computed directly using the formula condition for an elliptic fixed point non-degenerate if \ ( \alpha {! And institutional affiliations to several difference equations are the discrete analogs to differential one. Some example of the map t associated with Eq ( \mathcal { R } ^ { 2 } } )... In the book [ 18 ] are given a rational difference equation of the function f is defined all! Simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form MATH Google Scholar, Moeckel, R. Generic. Will call an elliptic fixed point be non-resonant and non-degenerate, Cushman, R.: Generic bifurcations of the t.