3 Problem 3: The Verhulst model and the logistic map A discrete-time Verhulst model X t = X t ( X t ) t X t =X t ( X t )t (4) can be reformulated as the so-called logistic map - one of the classical examples of systems that admit chaos. This is obtained by re-grouping terms and expressing (4) as an update equation for a dimensionless variable u t u t obtained from X t X t by a linear transformation: u t + 1 = a u t ( 1 u t ) u t+1 =au t (1u t ) (5) This equation is called the logistic map. For any value of 0 a 4, the transform (8) maps a unit interval [0, 1] onto itself. The logistic map is a particular case of a more general class of map functions u t + 1 = f ( u t ) u t+1 =f(u t ) with f ( u ) = a u ( 1 u ) f(u)=au(1u). Find the expressions for the dimensionless variable u t u t and parameter a in terms of the original parameters of the Verhulst model. Select all correct answers: 10 points u t = t 1 + t X t u t = 1+t t X t and a = t a=t u t = 1 + t t X t u t = t 1+t X t and a = t a=t u t = t 1 + t X t u t = 1+t t X t and a = 1 + t a=1+t