The Limit Cycle Existence Theorem

Finally, the solutions of the equations of motion (5.53, 54) have to be investigated. A complete survey of all the possible types of solution to the equations in dependence on the parameters k, (5q, di, and y will not, however, be made. The immediate aim is to derive sufficient conditions for those parameters which lead to the existence of an asymptotically periodic type of solution this is the type of solution which has been anticipated qualitatively in terms of the investment cycle and in which the main interest lies. In other words It is necessary for the substantial aspects of the model being considered to derive an existence theorem for solutions approaching a limit cycle under certain conditions of the parameters ir, 6q, di, p and y. 

A limit cycle C( ) is defined as a closed trajectory, i.e. a periodic solution to the equations of motion, with the property that there exists a domain 9)c around C( ) so that all trajectories starting within % approach C(t) as oo.% can be denoted as the domain of attraction and C(t) as an attractor . A special case of an attractor is an infinitesimally small limit cycle , i.e. a stable focus. 

Assumptions Consider two autonomous first order differential equations for the variables d (r) and x (r) and suppose that a finite domain Si) exists in the d-x plane such that  

Theorem In this case there must exist a (at least one) limit cycle within Si) and all trajectories in Si) either are, or approach as oo, a limit cycle. 

Assumptions Suppose that the parameters k, (5q, yS, y, di = 0 satisfy the conditions 

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Experiment 3. Sustained cycles occur for a parameter combination that satisfies the assumptions of the limit cycle existence theorem. Figure 5.6a exhibits the limit cycle (see also Fig. 5.6 b). Again, the origin is the only singular point which, however, has now become an unstable focus of the motion of the economy. 

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