Invariants and related Liapunov functions for difference equations

Consider the difference equation x n+1 = f(x n , where x n is in R k and f : D → D is continuous where D ⊂ R k . Suppose that I : R k → R is a continuous invariant, that is, I( f(x) ) = I( x ) for every x ∈ D . We will show that if I attains an isolated minimum or maximum value at the equilibrium (f...

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Bibliographic Details
Published inApplied mathematics letters Vol. 13; no. 7; pp. 1 - 8
Main Author Kulenović, M.R.S
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.10.2000
Subjects
Invariants
Lyness' equation
Stability
Lyness' equation
Invariants
Stability
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Summary:Consider the difference equation x n+1 = f(x n , where x n is in R k and f : D → D is continuous where D ⊂ R k . Suppose that I : R k → R is a continuous invariant, that is, I( f(x) ) = I( x ) for every x ∈ D . We will show that if I attains an isolated minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function, namely ±(I( x ) − I( p)) and so the equilibrium p is stable. This result is then applied to some difference equations appearing in different fields of applications.
ISSN:0893-9659
1873-5452
DOI:10.1016/S0893-9659(00)00068-9