More Looney Math

First, homogenize the dynamics by translating the state-variables appropriately, i.e. find real $\alpha, \beta$ such that with the substitution $r(t)=r'(t)+\alpha$ and $c(t)=c'(t)+\beta$ , the constant part on the right hand side vanishes. Thus, we require $\alpha=0.8 \alpha - 0.7 \beta + 200, \hspace{10pt} \beta=0.7 \alpha + 0.8 \beta -170$ Solving this, we get $\alpha=300, \beta=200$ . With the new state-variables $(r'(t),c'(t))$ , by direct substitution, we have $r'(t+1)^2 + c'(t+1)^2= 1.13 (r'(t)^2 + c'(t)^2)$ with $r'(0)=10, c'(0)= 0$ . Thus it is clear that the limit diverges.