Christos Douskos

Numerical Exploration of Kaldorian Macrodynamics: Hopf-Neimark Bifurcations and Business Cycles with Fixed Exchange Rates

We explore numerically a three-dimensional discrete-time Kaldorian macrodynamic model in an open economy with fixed exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods market α, and the degree of capital mobility β on the stability of equilibrium and on the existence of business cycles. We determine the stability region in the parameter space and find that increase of α destabilizes the equilibrium more quickly than increase of β. We determine the Hopf-Neimark bifurcation curve along which business cycles are generated, and discuss briefly the occurrence of Arnold tongues. Bifurcation and Lyapunov exponent diagrams are computed providing information on the emergence, persistence, and amplitude of the cycles and illustrating the complex dynamics involved. Examples of cycles and other attractors are presented. Finally, we discuss a two-dimensional variation of the model related to a “wealth effect,” called model 2, and show that in this case, α does not destabilize the equilibrium more quickly than β, and that a Hopf-Neimark bifurcation curve does not exist in the parameter space, therefore model 2 does not produce cycles.

The linguistic argument for intellectualism

A central argument against Ryle’s (The concept of mind, University of Chicago Press, Chicago, 1949) distinction between propositional and non propositional knowledge has relied on linguistic evidence. Stanley and Williamson (J Philos 98:411–444, 2001) have claimed that knowing-how ascriptions do not differ in any relevant syntactic or semantic respect from ascriptions of propositional knowledge, concluding thereby that knowing-how ascriptions attribute propositional knowledge, or a kind thereof. In this paper I examine the cross-linguistic basis of this argument. I focus on the linguistic analysis of practical knowledge ascriptions in Modern Greek, although the issues raised are not restricted to one language. It is relatively straightforward to show that none of the three types of practical knowledge ascriptions in Modern Greek is an embedded question configuration, and hence Stanley and Williamson original claim is confined to certain languages only. This is not the end of the matter, however, since Stanley (Nous 45:207–238, 2011) argues that the equivalents of ‘knowing-how’ ascriptions in certain languages should be semantically analyzed as embedded questions despite their syntactic form. I argue that this fallback position faces a host of empirical and theoretical problems, in view of which it cannot bear the weight Stanley puts on it, supporting a conclusion about the kind of knowledge thereby attributed.

Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

We explore numerically a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods market and the degree of capital mobility , on the stability of equilibrium, and on the possible existence of business cycles. We determine by a numerical grid-search method the stability region in the parameter space and find that, by comparison to fixed exchange rates, flexible exchange rates cause increased stability of equilibrium with respect to variations of the model parameters. We identify analytically the Hopf-Neimark bifurcation curve along which business cycles may be generated, as well as the flip bifurcation curve along which period-doubling cascades leading to chaotic behavior are generated. We find that period-doubling cascades leading to chaos are the dominant behavior of the system of flexible exchange rates outside the stability region, persisting up to large values of the degree of capital mobility . Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate on the stability of equilibrium is also briefly discussed. Finally, we explore the special case (Model 2) incorporating the so-called wealth effect, which in the present case of flexible exchange rates is found to behave similarly to the basic model, contrary to the case of fixed exchange rates in which incorporation of the wealth effect causes an entirely different behavior of the system.

An accurate numerical solver on second order PDEs with variable coefficients in three dimensions

In the present paper a numerical technique is developed for the approximate solution of second-order partial differential equations (PDEs) with variable coefficients in three dimensions. With the temporary introduction of two unknown auxiliary functions of the coordinate system the initial equation is separated into three parts that are reduced to ordinary differential equations, one for each dimension, that are discretized with a finite difference scheme. The use of suitable manipulations and the elimination of the unknown auxiliary functions, gives finally a linear system of algebraic equations where the matrix of the coefficients of the unknowns is diagonally dominant, a prerequisite for the rapid convergence of the iterative procedure. The efficiency and accuracy of the proposed numerical scheme is validated by its application to two test problems of fluid mechanics which have exact solutions. The numerical results based on the present technique are more accurate than those obtained by either the standard relaxation treatment with central differences or the ADI method when the contribution of the first-derivative terms in the initial equation is dominant. In all cases the comparison of the numerical results with those of the analytical solution, demonstrates the reliability of the presented numerical code.

Complete Coefficient Criteria for Five-Dimensional Hopf Bifurcations, with an Application to Economic Dynamics

Paper presents a complete mathematical characterization of coefficient criteria for five-dimensional Hopf bifurcations and an example of the application of these criteria to a model of economic dynamics. The application illustrates that the proposed criteria are practical and useful in determining the existence or nonexistence of Hopf bifurcations of five-dimensional dynamical systems in entire ranges of the system’s parameters.

Numerical exploration of Kaldorian interregional macrodynamics: enhanced stability and predominance of period doubling under flexible exchange rates.

We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movement . By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is required together with high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.

Asymptotic Orbits in Hill’s Problem When the Larger Primary is a Source of Radiation

A modification of the Hill problem when the larger primary is a source of radiation is considered and asymptotic motions around the collinear equilibrium points are studied. Our work focuses on the computation of homoclinic orbits to the collinear equilibrium points themselves or to the Lyapunov orbits emanating from each equilibrium point. These orbits depart asymptotically from an equilibrium point (or a Lyapunov orbit) and return to the same point (or orbit) asymptotically. In both cases, semi-analytical solutions have been obtained in order to determine appropriate initial conditions which have been used as suitable seed for the numerical computation of the asymptotic orbits with a predetermined accuracy. In addition, for homoclinic orbits to the Lyapunov periodic orbits, transversality is achieved by the construction of appropriate surface of section portraits of the unstable manifolds.

An Accurate Finite‐Difference Scheme on Second Order Partial Differential Equations in 3D

A numerical technique is presented for the approximate solution of second‐order partial differential equations in three dimensions. With the temporary introduction of two unknown functions of the coordinates the initial equation is separated into three parts that are reduced to ordinary differential equations, one for each dimension, associated with a finite‐difference scheme. The use of suitable manipulations and the elimination of the unknown functions, gives finally a linear system of algebraic equations that is solved using an iterative method. The numerical technique is tested on an example of fluid mechanics: the equation of steady‐state molecular diffusion of a substance in a continuous medium moving in a straight tube of square cross‐section. The numerical results based on the present technique are more accurate than those of the standard relaxation treatment and the ADI method, when the contribution of the first‐derivative terms in the initial equation is dominant. In all cases the comparison of ...