Dynamics of Second Order Rational Difference Equations

1. Auflage 2001
ISBN: 978-1-58488-275-6
Verlag: Chapman and Hall/CRC

Buch, Englisch, 232 Seiten, Format (B × H): 163 mm x 243 mm, Gewicht: 503 g

This self-contained monograph provides systematic, instructive analysis of second-order rational difference equations. After classifying the various types of these equations and introducing some preliminary results, the authors systematically investigate each equation for semicycles, invariant intervals, boundedness, periodicity, and global stability. Of paramount importance in their own right, the results presented also offer prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. The techniques and results in this monograph are also extremely useful in analyzing the equations in the mathematical models of various biological systems and other applications. Each chapter contains a section of open problems and conjectures that will stimulate further research interest in working towards a complete understanding of the dynamics of the equation and its functional generalizations-many of them ideal for research projects or Ph.D. theses. Clear, simple, and direct exposition combined with thoughtful uniformity in the presentation make Dynamics of Second Order Rational Difference Equations valuable as an advanced undergraduate or a graduate-level text, a reference for researchers, and as a supplement to every textbook on difference equations at all levels of instruction.

Zielgruppe

Advanced undergraduate and graduate students as well as researchers in pure and applied mathematics, mathematical biology, economics, engineering, and physics.

Autoren/Hrsg.

Kulenovic, Mustafa R. S.

Ladas, G.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

INTRODUCTION AND CLASSIFICATION OF EQUATION TYPESPRELIMINARY RESULTSDefinitions of Stability and Linearized Stability AnalysisThe Stable Manifold Theorem in the PlaneGlobal Asymptotic Stability of the Zero EquilibriumGlobal Attractivity of the Positive EquilibriumLimiting SolutionsThe Riccati EquationSemicycle AnalysisLOCAL STABILITY, SEMICYCLES, PERIODICITY, AND INVARIANT INTERVALSEquilibrium PointsStability of the Zero EquilibriumLocal Stability of the Positive EquilibriumWhen is Every Solution Periodic with the same Period?Existence of Prime Period Two SolutionsLocal Asymptotic Stability of a Two CycleConvergence to Period Two Solutions when C=0Invariant IntervalsOpen Problems and Conjectures(1,1)-TYPE EQUATIONSIntroductionThe Case a=g=A=B=0: xn+1= b xn/C xn-1The Case a=b=A=C=0: xn+1=g xn-1/B xnOpen Problems and Conjectures(1,2)-TYPE EQUATIONSIntroductionThe Case b=g=C=0: xn+1= a /(A+ B xn)The Case b=g=A=0: xn+1= a /(B xn+ C xn-1)The Case a=g=B=0: xn+1= b xn/(A + C xn-1)The Case a=g=A=0: xn+1= b xn/(B xn+ C xn-1)The Case a=b=C=0: xn+1= g xn-1/(A+ B xn)The Case a=b=A=0: xn+1= g xn-1/(B xn+ C xn-1)Open Problems and Conjectures(2,1)-TYPE EQUATIONSIntroductionThe Case g=A=B=0: xn+1=(a + b xn)/(C xn-1)The Case g=A=C=0: xn+1=(a + b xn)/B xnOpen Problems and Conjectures(2,2)-TYPE EQUATIONS(2,2)- Type EquationsIntroductionThe Case g=C=0: xn+1=(a + b xn)/(A+ B xn)The Case g=B=0: xn+1=(a + b xn)/(A + C xn-1)The Case g=A=0: xn+1=(a + b xn)/(B xn+ C xn-1)The Case b=C=0: xn+1=(a + g xn-1)/(A+ B xn)The Case b=A=0: xn+1=(a + g xn-1)/(B xn+ C xn-1)The Case a=C=0: xn+1=(b xn+ g xn-1)/(A+ B xn)The Case a=B=0: xn+1=(b xn+ g xn-1)/(A + C xn-1)The Case a=A=0: xn+1=(b xn+ g xn-1)/(B xn+ C xn-1)Open Problems and Conjectures(2,3)-TYPE EQUATIONSIntroductionThe Case g=0: xn+1=(a + b xn)/(A+ B xn+ C xn-1)The Case b=0: xn+1=(a + g xn-1)/(A+ B xn+ C xn-1)The Case a=0: xn+1=(b xn+ g xn-1)/(A+ B xn+ C xn-1)Open Problems and Conjectures(3,2)-TYPE EQUATIONSIntroductionThe Case C=0: xn+1=(a + b xn+ g xn-1)/(A+ B xn )The Case B=0: xn+1=(a + b xn+ g xn-1)/(A+ C xn-1)The Case A=0: xn+1=(a + b xn+ g xn-1)/(B xn+ C xn-1)Open Problems and ConjecturesTHE (3,3)-TYPE EQUATION The (3,3)- Type Equation: xn+1=(a + b xn+ g xn-1 )/(A+ B xn+ C xn-1)Linearized Stability AnalysisInvariant IntervalsConvergence ResultsOpen Problems and ConjecturesAPPENDIX: Global Attractivity for Higher Order EquationsBIBLIOGRAPHY

Über Autor(innen)

Kulenovic\, Mustafa R.S.; Ladas\, G.