ECONOMIC INVESTMENT

A simplified economic model of a national economy with investment is represented by the dynamical system

where y is the national product and is the investment function.

Maple Solution

Stability Analysis

It is clear that the system has a single equilibrium point at the origin for all r. The Jacobian matrix is given by

and , and hence the equilibrium point is an attractor if and a repeller if where .

The eigenvalues of the Jacobian matrix are given by

which are given by at . Also

Thus all the conditions of the Hopf Bifurcation theorem are satisfied except for the asymptotic stability of the origin at .

Asymptotic Stability of Origin

Introduce the Lyapunov function then on the trajectories

Now choose to eliminate the cross term and

Thus at

But

hence we have the plot shown below.

Thus, if , and if , and thus with inequalityonly on the line which is not a trajectory of the system. Hence the origin is asymptotically stable and there must be a supercritical Hopf bifurcation at .

Limit Cycle

The limit cycle is shown below for parameter value .

Unstable Focus + Stable Limit Cycle

Limit Cycle