Global behavior in rational difference equations

Yevgeniy Kostrov, University of Rhode Island

Abstract

We investigate the global behavior of the solutions of several rational difference equations. In particular, we study the global stability, the periodic nature, and the boundedness character of their solutions. We investigate four equations in this dissertation: ^ In the first manuscript we study the equation xn+1=a+bxn+g xn-1+dxn-2A+Bxn +Cxn-1+Dxn-2, n=0,1,... and we determine all special cases of this equation which possess an "essentially unique" period-two solution, and we pose several open problems and conjectures about their behavior. ^ In the second manuscript, we exhibit a range of parameters and a set of initial conditions where the rational difference equation xn+1=a+i=0

2k

bixn-i A+j=0

k

B2j xn-2j, n=0,1,... has unbounded solutions. ^ In the third manuscript, we give a detailed analysis of the "forbidden set" of the Riccati difference equation xn+1=an+b nxnAn+Bnx n,n=0,1,... where the coefficient sequences an ^∞n=0, bn^∞ n=0,A n^∞n=0 ,Bn ^∞n=0 are periodic sequences of real numbers with (not necessarily prime) period-2, and where the initial condition x₀ ∈ R. The character of the solutions through the "good points" of the equation is also presented. ^ In the fourth manuscript, we give a detailed analysis of the character of solutions of the Riccati difference equation zn+1=a+bzn A+Bzn, n=0,1,... where the coefficients α, β, A, B are complex numbers, and where the initial condition z₀ is a complex number. ^

Subject Area

Mathematics

Recommended Citation

Yevgeniy Kostrov, "Global behavior in rational difference equations" (2009). Dissertations and Master's Theses (Campus Access). Paper AAI3367996.
http://digitalcommons.uri.edu/dissertations/AAI3367996

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