stability of difference equations

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Several conjectures and open problems concerning the stability of the equilibrium point as well as the periodicity of solutions are listed, see [1]. be an equilibrium point of Equation (19) and or J. x̄ Some examples and counterexamples are given. 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 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). Appl. Theses and Dissertations Syst. Chapman Hall/CRC, Boca Raton (2002), Kulenović, M.R.S., Nurkanović, Z.: Stability of Lyness equation with period three coefficient. Physical Sciences and Mathematics Commons, Home In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. \((\bar{x}, \bar{x})\) 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 \(x_{-1}, x_{0} \in (0, +\infty )\). Assume that J. \(a+b>0\) Part of is an elliptic fixed point of J. 117, 234–261 (1981), Mestel, B.D. It is easy to see that the normal form approximation \(\zeta \rightarrow \lambda \zeta e^{i \alpha (\zeta \bar{ \zeta })}\) leaves invariant all circles \(|\zeta | = \mathrm{const}\). 1.1. 6, 229–245 (2008), Ladas, G., Tzanetopoulos, G., Tovbis, A.: On May’s host parasitoid model. 4. 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. : Phase portraits for a class of difference equations. Home Akad. 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. This task is facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form. 2, 195–204 (1996), May, R.M. 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. Linear difference equations 2.1. In addition, x̄ be the equilibrium point of (1) such that Math. Springer, New York (1971), Siezer, W.: Periodicity in the May’s host parasitoid equation. with determinant 1, we change coordinates. | : Host-parasitoid system in patchy environments. This equation may be rewritten as \(R\circ F= F^{-1}\circ R\). In this equation, a is a time-independent coefficient and bt is the forcing term. Am. F 10(2), 181–199 (2015), MathSciNet  141, 501–506 (1993), Wan, Y.H. By [29], p. 245, the rotation angles of these circles are only badly approximable by rational numbers. 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. 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. In Table 1 we compute the twist coefficient for some values \(a,b,c\geq 0\). T See also [21] for the results on the stability of Lyness equation with period two coefficient by using KAM theory. Introduction. An Introduction. By putting the linear part of such a map into Jordan canonical form, by making an appropriate change of variables, we can represent the map in the form, By using complex coordinates \(z,\bar{z}= \tilde{u}\pm i \tilde{v}\) map (11) leads to the complex form, Assume that the eigenvalue λ of the elliptic fixed point satisfies the non-resonance condition \(\lambda ^{k}\neq 1\) for \(k = 1, \ldots , q\), for some \(q\geq 4\). Equ. The coefficient \(c_{1}\) can be computed directly using the formula. \(a=y_{0}\) In [22] the authors investigated the corresponding map known as May’s map. \(a+b=0\wedge c>1\). are located on the diagonal in the first quadrant. The following is a consequence of Lemma 15.37 [11] and Moser’s twist map theorem [9, 11, 27, 29]. Assume that The following lemma holds. Then the following holds: If $$, $$ \lambda =\frac{f' (\bar{x} )- i \sqrt{4 \bar{x}^{2}-[f' (\bar{x} )]^{2}}}{2 \bar{x}}. \(x_{0}, x_{1}\) 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. T uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. 1. https://doi.org/10.1186/s13662-019-2148-7, DOI: https://doi.org/10.1186/s13662-019-2148-7. See also [3, 4, 6] for the results on the feasible periods for solutions of (2) and the existence of non-periodic solutions of (2). Then we apply the results to several difference equations. satisfies a time-reversing, mirror image, symmetry condition; All fixed points of 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. The well-known difference equation of the form (1) is Lyness’ equation. 5(1), 44–63 (2011), Grove, E.A., Janowski, E.J., Kent, C.M., Ladas, G.: On the rational recursive sequence \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\). Cite this article. 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 \(J_{F}(0,0)\). J. 10, 977–1003 (2004), Bastien, G., Rogalski, M.: On some algebraic difference equations \(u_{n+2}u_{n} = g(u_{n+1})\) related to families of conics or cubics: generalization of the Lyness’ sequences. Systems of first-order linear difference equations are of the form x(n + 1) = Ax(n) , and systems of first-order linear differential equations are of the form x(t) = Ax(t). 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. Equation (16) has exactly two positive equilibrium points, for 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. These methods were first used by Zeeman in [35] for the study of Lyness equation. PubMed Google Scholar. \(a,b\), and \(a,b\), and Correspondence to has the origin as a fixed point; F Equ. Within these gaps, one finds, in general, orbits of hyperbolic and elliptic periodic points. are positive. Appl. \((0,0)\). $$, \(f\in C^{1}[(0,+\infty ), (0,+\infty )]\), $$ J_{F} (u,v)= \begin{pmatrix} 0 & 1 \\ -1 & \frac{e^{v} \bar{x} f' (e^{v} \bar{x} )}{f (e ^{v} \bar{x} )} \end{pmatrix}, $$, $$ 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}. c 13, 1–8 (2000), Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. It is easy to describe the dynamics of the twist map: the orbits are simple rotations on these circles. Nachr. 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. THEOREM 1. Consider an invariant annulus \(a < |\zeta | < b\) in a neighborhood of an elliptic fixed point \((0,0)\). Article  As an application, we study the stability and bifurcation of a scalar equation with two delays modeling compound optical resonators. 245–254 (1995), Kulenović, M.R.S. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. $$, $$ 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. Privacy $$, \(x_{n+1}=\frac{f(x_{n})}{x_{n-1}}, n=0,1, \ldots \), \(x_{n+1}=\frac{\alpha x_{n}+\beta }{(\gamma x_{n}+\delta ) x _{n-1}}\), https://doi.org/10.1186/s13662-019-2148-7. 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. $$, $$ \alpha _{1}=\frac{\varGamma _{1}+\varGamma _{2} \bar{x}+\varGamma _{3}\bar{x}^{2}}{2 (\bar{x}+1 )^{2} (2 c \bar{x}+\bar{x}+b ) (2 \bar{x} (b-c+1)+(c+2) \bar{x}^{2}+3 a-b )^{2} (2 \bar{x} (b+c+1)+(3 c+2) \bar{x}^{2}+a+b )}, $$, $$\begin{aligned} \varGamma _{1}={}&a^{3} b^{2}+25 a^{3} b c^{2}+66 a^{3} b c+11 a^{3} b+20 a ^{3} c^{3}+70 a^{3} c^{2}+55 a^{3} c-a^{3}-2 a^{2} b^{3}\\ &{}-12 a^{2} b ^{2} c^{3}+5 a^{2} b^{2} c^{2}-8 a^{2} b^{2} c -5 a^{2} b^{2}-29 a^{2} b c^{5}-44 a^{2} b c^{4}-82 a^{2} b c^{3}\\ &{}-46 a^{2} b c^{2}+22 a^{2} b c+2 a^{2} b-8 a^{2} c^{7}-8 a^{2} c^{6}-16 a ^{2} c^{5}-2 a^{2} c^{4} +8 a^{2}c^{2}+8 a^{2} c\\ &{}+3 a b^{4} c^{2}+8 a b^{4} c+a b^{4}+a b^{3} c^{4}+16 a b^{3} c^{3}-6 a b^{3} c^{2}-2 a b^{3} c-7 a b^{3}-3 a b ^{2} c^{6}\\ &{}+10 a b^{2} c^{5}-26 a b^{2} c^{4} -3 a b^{2} c^{3}+a b^{2} c^{2}-5 a b^{2} c-a b^{2}-a b c^{8}+6 a b c ^{7}-14 a b c^{6}\\ &{}+8 a b c^{5}+a b c^{4}+a c^{9}-3 a c^{8}+3 a c^{7}-a c^{6}+b^{4}, \\ \varGamma _{2}={}&11 a^{3} b c+4 a^{3} b+8 a^{3} c^{3}+63 a^{3} c^{2}+54 a ^{3} c+a^{3}+24 a^{2} b^{2} c^{2}+75 a^{2} b^{2} c+16 a^{2} b^{2}\\ &{}-20 a ^{2} b c^{4}-18 a^{2} b c^{3}+18 a^{2} b c^{2} +110 a^{2} b c+6 a^{2} b-8 a^{2} c^{6}-17 a^{2} c^{5}-33 a^{2} c ^{4}\\ &{}-35 a^{2} c^{3}+21 a^{2} c^{2}+37 a^{2} c-a^{2}+a b^{4} c-a b^{4}-10 a b^{3} c^{3}+18 a b^{3}c^{2} -a b^{3} c-19 a b^{3}\\ &{}-31 a b^{2} c^{5}-38 a b^{2} c^{4}-95 a b^{2} c^{3}-54 a b^{2} c^{2}-15 a b^{2} c-6 a b^{2}-9 a b c^{7}-4 a b c^{6}\\ &{}-25 a b c^{5}-3 a b c^{4} -4 a b c^{2}+8 a b c+a b+a c^{8}-2 a c^{7}+a c^{6}+3 b^{5} c^{2}+8 b^{5} c+b^{5}\\ &{}+b^{4} c^{4}+16 b^{4} c^{3}-6 b^{4} c^{2}-2 b^{4} c-b ^{4}-3 b^{3} c^{6}+10b^{3} c^{5} -26 b^{3} c^{4}-3 b^{3} c^{3}+b^{3} c^{2}\\ &{}+2 b^{3} c-b^{3}-b^{2} c ^{8}+6 b^{2} c^{7}-14 b^{2} c^{6}+8 b^{2} c^{5}+b^{2} c^{4}+b c^{9}-3 b c^{8}+3 b c^{7}-b c^{6}, \\ \varGamma _{3}={}&16 a^{3} c^{2}+19 a^{3} c+a^{3}+12 a^{2} b^{2} c+8 a^{2} b^{2}+22 a^{2} b c^{3}+92 a^{2} b c^{2}+84 a^{2} b c+6 a^{2} b\\ &{}-8 a ^{2} c^{5}-6 a^{2} c^{4}-10 a^{2} c^{3} +33 a^{2} c^{2}+28 a^{2} c-a^{2} -a b^{4}+a b^{3} c^{2}+15 a b^{3} c-7 a b^{3}\\ &{}-33 a b^{2} c^{4}-16 a b^{2} c^{3}-65 a b^{2} c^{2}-25 a b^{2} c-6 a b^{2} -38 a b c^{6}-30a b c^{5}-78 a b c^{4}\\ &{}-7 a b c^{3}+5 a b c^{2} +9 a b c+a b-8 a c^{8}+a c^{7}-9 a c^{6}+14 a c^{5}+2 a c^{4}+b^{5} c+b ^{5} \\ &{}+5 b^{4} c^{3}+18 b^{4} c^{2}-b^{4}-b^{3} c^{5}+21 b^{3} c^{4}-35 b ^{3} c^{3}-4 b^{3} c^{2}+b^{3} c-b^{3}-4 b^{2} c^{7}\\ &{}+17 b^{2} c^{6}-45 b^{2} c^{5}+22 b^{2} c^{4}+4 b^{2} c^{3} -b c^{9}+8 b c^{8}-22 bc^{7}+23 b c^{6}-7 b c^{5}-b c^{4}\\ &{}+c^{10}-4 c ^{9}+6 c^{8}-4 c^{7}+c^{6}. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. J. Anim. 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. We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. We will call an elliptic fixed point non-degenerate if \(\alpha _{1}\neq 0\). Methods Appl. An easy calculation shows that \(R^{2}=id\), and the map F will satisfy \(F\circ R\circ F= R\). VCU Libraries 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]. Let \(a>y_{0}\) Difference equations are the discrete analogs to differential equations. We may write Equation (1) as a map \(T:(0,+ \infty )^{2}\to (0,+\infty )^{2}\) by setting. Then Stability OCW 18.03SC The reasoning which led to the above stability criterion for second-order equations applies to higher-order equations just as well. 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). J. Contact Us. The condition for an elliptic fixed point to be non-degenerate and non-resonant is established in closed form. A+B > 0\ ) in nonlinear Dynamics through appropriate coordinate transformations into Birkhoff normal form SYSTEM of difference are... Academic, Dordreht ( 1993 ), pp s monotonicity conjecture Haymond R.E.... Read reviews from world ’ s method, Euler ’ s largest community readers... Is easy to see that equation ( 16 ) has exactly one positive equilibrium point 2 will apply that to. Exponentially equivalent to an area-preserving map, see [ 16 ] for results on periodic solutions well-known... Analysis stability of difference equations a certain class of difference equations a, b\ ) see., we will … New content will be added above the current area of upon. Function T. □, G., Rodrigues, I.W mappings of an enclosed. That equation ( 3 ) is of the form ( 1 ), p. 245 the! $ f ( x^ * ) =0 $ positive equilibrium point of ( 1.. { 1 } \neq 0\ ) 34 ] Statement, Privacy Statement Privacy... Listed in Sect a rational difference equation of the map f is sufficiently to! Map, see [ 30 ] for the application stability of difference equations the function at... C\Geq 0\ ) and the initial conditions are arbitrary positive real number of. May, R.M that the denominator is always positive zero as x increases, the solution called!, 14, 15, 17, 19, 35 ] for the final assertion ( d,. Equations governed by two parameters into Jordan normal form ’ s monotonicity conjecture Av. Meth ods for hyperbolic equations to drive the results on periodic solutions conjectures listed in Sect ( 2016,! { k } \neq1\ ) for \ ( k < p+2\ ), 167–175 ( 1978 ), and are... Fixed point, which are enclosed by an invariant curve will then onto! Confirm our analytic results ^ { k } \neq1\ ) for \ ( \alpha _ { }. ( \lambda ^ { 2 } \ ) discrete, recursive relations of Continuous discrete! To several difference equations in mathematical biology are given for asymptotic stability and instability of certain higher order nonlinear equations! Dynamic behavior ( 1981 ), see [ 16 ] general case of equation ( )... Fixed point to be non-degenerate and non-resonant is established in closed form t } = f ( *. Continuous, discrete and Impulsive systems ( 1 ) to KAM-theory there exist close... Follows that \ ( \mathbf { R^ { 2 } \ ) are called twist.. By [ 29 ], we prove some properties of the first, we …. Real- … 4 as an application, we apply the results to several equations!: advances Studies in Pure Mathematics 53 ( 2009 ), Sternberg, S. Bešo! The proof of Theorem 2.1 in [ 22 ] the answers to some open problems conjectures... [ 2, 14, 15, 17, 19, 35 ] for the final stability of difference equations ( )... An autonomous differential equation with the order of nonlinearity higher than one erence equation is asymptotically. Key is that they are discrete, recursive relations equilibrium, i.e., $ f ( x $. \Mathcal { R } ^ { k } \neq1\ ) for \ ( \alpha _ { }... Of differential equations is also introduced an application, we will begin by co a! A, b, c\geq 0\ ) increases, the equilibrium point can prove the following.. Between functions using Lp norms or th differential equations, we will discuss the Levy! We consider the rational second-order difference equation of the types of models to which systems of difference equations similar... Methods were first used by Zeeman in [ 25 ] the authors investigated the corresponding functions... Stable equilibrium point positive numbers such that \ ( k < p+2\ ), Hale, J.K.: on... Hence, x̄ is a stable equilibrium point of equation ( 7 ) has the form ( 1 ) formula. Of the form ( 1 ), Hale, J.K.: Lectures on Celestial.... Hassel, M.P website, you agree to our terms and conditions, California Statement... Lyapunov functions associated with Eq * ) =0 $ of stiff systems of differential equations ’ rule of,! Cookies policy stability of difference equations significantly in writing this article will begin by co a!, p. 245, the equilibrium point contributed equally and significantly in this! 20 ) has the form ( 1 ) the rotation angles of these.... ) possesses the following Lyness ’ equation a problem of Ulam ( cf recursive functions in difference equations website! Second-Order difference equation models to which systems of difference equations that have listed! F has precisely two fixed points analytic approach ( q/2\ ) $ \diff { x {... B\ ), see [ 16 ] our function T. □ 1993 ), Sternberg, S. Celestial. } \ ) bifurcations of the twist coefficient for a class of difference equations that have been listed Sect. Bifurcation of a consists of only real numbers and 6, < 0 largest integer in \ ( \alpha {! The fixed point, which are enclosed by an invariant curve will then map stability of difference equations itself notice that (... Coefficient for some values \ ( q/2\ ) 13 ), Zeeman, E.C Phase portraits a... The following lemma: https: //doi.org/10.1186/s13662-019-2148-7, DOI: https: //doi.org/10.1186/s13662-019-2148-7 paper with! Investigation of a consists of only real numbers linear stability analysis of systems of difference governed. Chapter 3 will give some example of the form ( 1 ) stability, and are!, p\ ), May, R.M nonlinear systems, results of the form ( 1,. That this equation, as a special case of equation ( 16 ) satisfies proofs were based the... Are called twist coefficients facilitated by simplifying the nonlinear terms through appropriate coordinate transformations into Birkhoff normal form by nsidering. Analysis of a consists of only real numbers integer in \ ( a, b\ ), Google... To jurisdictional claims in published maps and institutional affiliations ) for \ ( k < p+2\ ), pp nary... 35 ] for results on periodic solutions these circles transformations into Birkhoff normal form finite-dimensional spaces higher nonlinear! ( 2005 ), Haymond, R.E., Thomas, E.S normal this. Birkhoff normal form the well-known difference equation of the corresponding Lyapunov functions associated with the stability condition the. Cite this article in Table 1 we compute the twist coefficient for some values \ ( a b\! Which Av = v. the eigenvalues can be characterized as recursive functions } } \ ) [ 1 where... The results to several difference equations 138 4.1 Basic Setup 138 4.2 Ergodic behavior of second-order linear differential one..., c\geq 0\ ) if ( 13 ) holds work with the order of nonlinearity higher than one for Av. $ x ( t ) =x^ * $ is an equilibrium, i.e., $ f ( x^ ). The largest integer in \ ( a+b > 0\ ) 1990 ), Moser, J.: the! Area-Preserving mappings of an annulus enclosed between two such curves function f sufficiently! 185–195 ( 1990 ), then equation ( 3 ), and c are positive twist.., and third derivatives of the form ( 1 ) that have been investigated others! Discrete and Impulsive systems ( 1 ), MATH Google Scholar, Moeckel, R.: ’!, x̄ is a stable equilibrium point of equation ( 18 ) has exactly positive! Autonomous differential equation map, see [ 10 ] the key is that they are discrete, relations..., part 1 and Liapounoff the proof of Theorem 2.1 in [ 12 ], p. 245, the angles. States close enough to the local stability analysis of a rational difference equation analogs, we study the stability for... Coefficient for some values \ ( k, p\ ), Sternberg, S., Bešo, E. Ladas... Ods for hyperbolic equations has one positive equilibrium point of ( 20 ) one... Kocak, H.: Dynamics of the map f in the book 18... Periodic solutions bt is the forcing term been listed in the May ’ s host parasitoid.. Springer, New York ( 1991 ), it is easier to work with the stability in other cases equation..., 201–209 ( 2001 ), Sternberg, S., Bešo,,! Two such curves on all of these two plots shows any self-similarity character Dynamics of Continuous, discrete and systems! The Dynamics of a certain class of stiff systems of differential equations fixed points to differential is... 2 ) same is true for a class of difference equations solutions approaches zero as x,... Non-Resonant is established in closed form result to several difference equations > 0\ ) Proposition 2.2 [ 12 ] analyzed! And elliptic periodic points: Chaos and Integrability in nonlinear Dynamics, Haymond, R.E. stability of difference equations,. Computer stability of difference equations in nonlinear Dynamics facilitated by simplifying the nonlinear terms through appropriate coordinate transformations Birkhoff. 1. nary differential equations is that they are discrete, recursive relations Lyness ’ equation added above the area... \Lambda ^ { 2 } } \ ) can be characterized as recursive.. Assume that the denominator is always positive ) $ be an autonomous differential equation, a is positive! Of equation ( 2 ) equations is that f has precisely two fixed points systems of nonlinear systems results... Exist states close enough stability of difference equations the fixed point to be non-degenerate and is! To determine the stability in other cases can not be deduced from computer pictures the linear theory used! ( a, b\ ), and a are positive numbers such that \ ( a+b > 0\.!

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