Convergence in homogeneous difference equations of degree 1

Mazrooei‐Sebdani, Reza

doi:10.1080/10236198.2011.611508

Cited by 5 publications

(7 citation statements)

References 10 publications

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“…(b) If ( − ) 2 > 3 ( − ), > , > , and condition (14) is satisfied then the equilibrium point 1 is locally asymptotically stable and the equilibrium point 2 = + is nonhyperbolic.…”

Section: Local Stability Analysismentioning

confidence: 99%

“…The first systematic study of global dynamics of a special quadratic fractional case of (3) where = = = = = = 0 was performed in [5,6]. Dynamics of some related quadratic fractional difference equations was considered in the papers [7][8][9][10][11][12][13][14][15][16][17]. In this paper we will perform the local stability analysis of all three equilibrium points of (1) and we will give the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a saddle point, a repeller, or a nonhyperbolic equilibrium.…”

Section: Introductionmentioning

confidence: 99%

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Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

Hrustić¹,

Kulenović

Nurkanović³

2016

Discrete Dynamics in Nature and Society

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“…(b) If ( − ) 2 > 3 ( − ), > , > , and condition (14) is satisfied then the equilibrium point 1 is locally asymptotically stable and the equilibrium point 2 = + is nonhyperbolic.…”

Section: Local Stability Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

Hrustić¹,

Kulenović

Nurkanović³

2016

Discrete Dynamics in Nature and Society

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“…Homogeneous difference equations are considered in [1][2][3][4][5][6][7][8], where these equations are equivalent to lower order equations using semiconjugate factors. In [2], we considered second and third order homogeneous rational difference equations, and then, we obtained monotonic, non-monotonic and oscillated solutions.…”

Section: Introductionmentioning

confidence: 99%

Planar homogeneous systems of difference equations

Mazrooei‐Sebdani

2014

Math. Meth. Appl. Sci.

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This paper is concerned to additive and multiplicative systems of homogeneous difference equations of non‐negative degree. We apply a reduction in order for both additive and multiplicative systems. Then, we consider convergence and monotony of positive solutions. In fact, using convergence results on factor maps, we obtain convergence results on homogeneous systems. We will conclude that monotonic behaviour on the invariant ray (i.e. xMathClass-rel=falsemml-overliner¯y for multiplicative systems and xMathClass-rel=falsemml-overlines¯MathClass-bin+y for additive systems) may or may not be the representative of other solutions. To illustrate our results, some examples are presented by multiplicative and additive homogeneous systems of rational equations. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Homogeneous difference equations considered in . Now, consider the difference equation of order k + 1

x_{n MathClass-bin+ 1} MathClass-rel= f_{n} (x_{n} MathClass-punc, x_{n MathClass-bin- 1} MathClass-punc, MathClass-op\dots MathClass-punc, x_{n MathClass-bin- k}) MathClass-punc, n MathClass-rel= 0 MathClass-punc, 1 MathClass-punc, 2 MathClass-punc, MathClass-op\dots MathClass-punc,

where

f_{n} MathClass-punc: D MathClass-rel\to double-struckR

and

D MathClass-rel\subseteq {double-struckR}^{k MathClass-bin+ 1}

.…”

Section: Introductionmentioning

confidence: 99%

“…Homogeneous difference equations considered in [1][2][3][4][5][6][7]. Now, consider the difference equation of order k C 1…”

Section: Introductionmentioning

confidence: 99%

Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Mazrooei‐Sebdani

2013

Math. Meth. Appl. Sci.

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Consider the non‐autonomous equations: xnMathClass-bin+1MathClass-rel=xn()AnxnMathClass-bin+BnxnMathClass-bin−1αnxnMathClass-bin+βnxnMathClass-bin−1MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspαnxnMathClass-bin+βnxnMathClass-bin−1MathClass-rel>0MathClass-punc, xnMathClass-bin+1MathClass-rel=xnMathClass-bin−1()BnxnMathClass-bin+DnxnMathClass-bin−2αnxnMathClass-bin+γnxnMathClass-bin−2MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspαnxnMathClass-bin+γnxnMathClass-bin−2MathClass-rel>0MathClass-punc, xnMathClass-bin+1MathClass-rel=xn()BnxnMathClass-bin−1MathClass-bin+CnxnMathClass-bin−2βnxnMathClass-bin−1MathClass-bin+γnxnMathClass-bin−2MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspβnxnMathClass-bin−1MathClass-bin+γnxnMathClass-bin−2MathClass-rel>0MathClass-punc, where AnMathClass-rel≥01emnbsp1emnbspMathClass-punc,BnMathClass-rel≥01emnbsp1emnbspMathClass-punc,CnMathClass-rel≥01emnbsp1emnbspMathClass-punc,αnMathClass-rel≥01emnbsp1emnbspMathClass-punc,βnMathClass-rel≥01emnbsp1emnbspMathClass-punc,γnMathClass-rel≥0MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbspnMathClass-rel=0MathClass-punc,1MathClass-punc,2MathClass-punc,MathClass-op…MathClass-punc, and also msubnormallimnMathClass-rel→MathClass-rel∞AnMathClass-rel=AMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞BnMathClass-rel=BMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞CnMathClass-rel=CMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞αnMathClass-rel=αMathClass-rel>01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞βnMathClass-rel=βMathClass-rel>01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞γnMathClass-rel=γMathClass-rel>0MathClass-punc.These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.

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Convergence in homogeneous difference equations of degree 1

Cited by 5 publications

References 10 publications

Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

Local Dynamics and Global Stability of Certain Second-Order Rational Difference Equation with Quadratic Terms

Planar homogeneous systems of difference equations

Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

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