Periodic cycles in the Solow model with a delay effect. (English) Zbl 1478.34090
Summary: The three natural modifications of the known mathematical macroeconomics model of macroeconomics are studied in which a delay factor is presumed. This led to the replacement of the ordinary differential equation, which cannot exhibit periodic cycles on the equations with a deviating argument (functional-differential equations). It was possible to show the existence of periodic solutions that can and are intended to describe the periodic cycles in the market economy in two of the three variants of such changes in the classical form of the model.
The mathematical portion is based on the application of the modern theory of dynamical systems with an infinite-dimensional space of initial conditions. This will allow us to apply the Andronov-Hopf Theorem for equations with a deviating argument in such a form that the parameters of the cycles are located. We will also apply the well-known Krylov-Bogolyubov algorithm that is extended to infinite-dimensional dynamical systems that is used and reduces the problem to the analysis of the finite-dimensional system of ordinary differential equations-the normal Poincare-Dulac form.
The mathematical portion is based on the application of the modern theory of dynamical systems with an infinite-dimensional space of initial conditions. This will allow us to apply the Andronov-Hopf Theorem for equations with a deviating argument in such a form that the parameters of the cycles are located. We will also apply the well-known Krylov-Bogolyubov algorithm that is extended to infinite-dimensional dynamical systems that is used and reduces the problem to the analysis of the finite-dimensional system of ordinary differential equations-the normal Poincare-Dulac form.
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 91B64 | Macroeconomic theory (monetary models, models of taxation) |
| 34K18 | Bifurcation theory of functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
| 34K13 | Periodic solutions to functional-differential equations |
| 34K17 | Transformation and reduction of functional-differential equations and systems, normal forms |
| 34K19 | Invariant manifolds of functional-differential equations |
Keywords:
Solow model; functional-differential equations; stability; bifurcations; normal form; asymptotic formulasReferences:
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