Marek Szydlowski

This paper covers an application of nonlinear mechanics in macrodynamic model of the business cycle. The time-to-build is introduced into the capital accumulation equation according to Kalecki's idea of delay in investment processes. The dynamics of this model is represented in terms of a time delay differential equation system. It is found that there are two causes which generate cyclic behaviour in the model. Apart from the standard Kaldor proposition of special nonlinearity in the investment function, the cycle behaviour is due to the time delay parameter. In both scenarios, cyclic behaviour emerges from the Hopf bifurcation to the periodic orbit.
In the special case of a small time-to-build parameter the general dynamics is reduced to a two-dimensional autonomous dynamical system. This system is examined in detail by methods of qualitative analysis of differential equations. Then cyclic behaviour in the system is represented by a limit cycle on the plane phase. It is shown that there is a certain bifurcation value of the time delay parameter which leads to a periodic orbit. We discuss the problem of the existence of a global attractor in $2$-dimensional phase space whose counterpart for the Kaldor model was considered by Chang and Smyth. It is shown that the presence of time-to-build excludes the asymptotically stable global critical point. Additionally, we analyse the question of uniqueness of the limit cycles of the model.

It is proved that trajectories of any mechanical system $(M,g,V)$, where $M$ is the configuration space, $g$ the metric defined by the kinetic energy form, and $V$ a potential function, with the natural Lagrangian, are pregeodesics with respect to the Jacobi metric $g_E=2|E-V|g$, $E$ being the total energy of the system.

The property of sensitive dependence on initial conditions is formulated as the local instability of nearby geodesics. In studying sectional curvature, the bivector formalism is applied. We also show that the trajectories of simple mechanical systems can be put into one-to-one correspondence with geodesics of suitable $N+1$ dimensional space with the Lorentzian signature ($N$ is a dimension of the configuration space). This illustrates the fact that the simple relativistic mechanical systems can be used not only in applications to general relativity and cosmology where the kinetic energy form with the Loretzian signature is indefinite at the very beginning.