Discrete systems of equations229showed that the canonical form ofut=Aut−1iszt=Jtz0In the case of a repeated root this iszt=±λttλt−10λt²z0Hencez1t=λtz10+tλt−1z20z2t=λtz20Therefore if|λ|<1, then³³λt³³→0ast→∞, consequentlyz1t→0 andz2t→0ast. The system is asymptotically stable. If, on the other hand,|λ|>1, then³³λt³³ast, andz1t→±∞andz2tast. The system isasymptotically unstable.We can conclude for repeated roots, therefore, that(a)if|λ|<1 the system is asymptotically stable(b)if|λ|>1 the system is asymptotically unstable.Example 5.11Consider the following systemxt+1=4+xt−ytyt+1=−20+xt+3ytThenx∗=12 andy∗=4. Representing the system as deviations from equilibrium,we havext+1−x∗=(xt−x∗)−(yt−
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.