Adam Krawiec

Top ten accelerating cosmological models

Recent astronomical observations indicate that the Universe is presently almost flat and undergoing a period of accelerated expansion. Basing on Einsteins general relativity all these observations can be explained by the hypothesis of a dark energy component in addition to cold dark matter (CDM). Because the nature of this dark energy is unknown, it was proposed some alternative scenario to explain the current accelerating Universe. The key point of this scenario is to modify the standard FRW equation instead of mysterious dark energy component. The standard approach to constrain model parameters, based on the likelihood method, gives a best-fit model and confidence ranges for those parameters. We always arbitrary choose the set of parameters which define a model which we compare with observational data. Because in the generic case, the introducing of new parameters improves a fit to the data set, there appears the problem of elimination of model parameters which can play an insufficient role. The Bayesian information criteria of model selection (BIC) is dedicated to promotion a set of parameters which should be incorporated to the model. We divide class of all accelerating cosmological models into two groups according to the two types of explanation acceleration of the Universe. Then the Bayesian framework of model selection is used to determine the set of parameters which gives preferred fit to the SNIa data. We find a few of flat cosmological models which can be recommend by the Bayes factor. We show that models with dark energy as a new fluid are favoured over models featuring a modified FRW equation.

The Kaldor‐Kalecki business cycle model

The question of the determination of investment decisions and their links with economicactivity leads us to formulate a new business cycle model. It is based on the dynamic multiplierapproach and the distinction between investment and implementation. The study of thenonlinear behaviour of the Kaldor‐Kalecki model represented by the second‐order delaydifferential equations is presented. It is shown that the dynamics depends crucially on thetime‐delay parameter T ‐ the gestation time period of investment. We apply the Poincaré‐Andronov‐Hopf bifurcation theorem generalized for functional differential equations. Itallows us to predict the occurrence of a limit cycle bifurcation for the time‐delay parameterT = Tbif. The dependence of T = Tbif on the parameters of our model is discussed. As T is increased, the system bifurcates to limit cycle behaviour, then to multiply periodic andaperiodic cycles, and eventually tends towards chaotic behaviour. Our analysis of the dynamicsof the Kaldor‐Kalecki model gives us that the limit cycle behaviour is independent of theassumption of nonlinearity of the investment function. The limit cycle is created only due tothe time‐delay parameter via the Hopf bifurcation mechanism. We also show that for a smalltime‐delay parameter, the Kaldor‐Kalecki model assumes the form of the Liénard equation.

Nonlinear oscillations in business cycle model with time lags

Abstract In this paper we analyse the dynamics of the Kaldor–Kalecki business cycle model. This model is based on the classical Kaldor model in which capital stock changes are caused by past investment decisions. This lag is connected with time delay needed for new capital to be installed. The dynamics of the model is reduced to the form of damped oscillator with negative feedback connected with lag parameter and next it is analysed in terms of bifurcation theory. We find conditions for existence and persistence of oscillatory behaviour which is represented by limit cycle on some central manifold in phase space, i.e., single Hopf bifurcation. We demonstrate that the Hopf cycles may be exhibited for nonzero measure set of the parameter space. The conditions for bifurcation of co-dimension two connected with interaction of bifurcations as well as bifurcation diagrams are also given. Finally, we obtain numerical values describing an amplitude and a period of oscillation for different parameter of the system. It is also proved that while the investment function is not nonlinear a quasi-periodic solution (a 1:2 resonant double Hopf point) can appear. The source of such a behaviour is rather a consequence of time lag than nonlinearity of the investment function. Our results confirm the existence of asymmetric (two periodic) cycles in the Kaldor–Kalecki model with time-to-build.

The Kaldor­Kalecki Model of Business Cycle as a Two-Dimensional Dynamical System

Abstract In the paper we analyze the Kaldor–Kalecki model of business cycle. The time delay is introduced to the capital accumulation equation according to Kalecki’s idea of delay in investment processes. The dynamics of this model is represented in terms of time delay differential equation system. In the special case of small time-to-build parameter the general dynamics is reduced to two-dimensional autonomous dynamical system. This system is examined in details by methods of qualitative analysis of differential equations. It is shown that there is a Hopf bifurcation leading to a limit cycle. Additionally stability of this solution is discussed.

Constraints on a Cardassian Model from Type Ia Supernova Data, Revisited

We discuss some observational constraints, resulting from Type Ia supernova (SN Ia) observations, imposed on the behavior of the original flat Cardassian model and on its extension with the curvature term included. We test the models using the Perlmutter SN Ia data, as well as the new Knop and Tonry samples. We estimate the Cardassian model parameters using the best-fit procedure and the likelihood method. In the fitting procedure we use density variables for matter, Cardassian fluid, and curvature and include the errors in redshift measurement. For the Perlmutter sample in the nonflat Cardassian model we obtain a high-density or normal (Ωm,0 ≈ 0.3) universe, while for the flat Cardassian model we have a high-density universe. For sample A in the high-density universe, we find negative values of estimates of n, which can be interpreted as a phantom fluid effect. For the likelihood method we find that a nearly flat universe is preferred. We show that if we assume that matter density is 0.3, then n ≈ 0 in the flat Cardassian model, which corresponds to the Perlmutter model with a cosmological constant. Tests with the Knop and Tonry SN Ia samples show no significant differences in results.

Phantom cosmology as a simple model with dynamical complexity.

We study the Friedman-Robertson-Walker model with phantom fields modeled in terms of scalar fields. We apply the Ziglin theory of integrability and find that the flat model is nonintegrable. Then we cannot expect to determine simple analytical solutions of the Einstein equations. We demonstrate that there is only a discrete set of parameters where this model is integrable. For comparison we describe the phantoms fields in terms of the barotropic equation of state. It is shown that in contrast to the phantoms modeled as scalar fields, the dynamics is always integrable and phase portraits are contracted. In this case we find the duality relation.

The Hopf Bifurcation in the Kaldor-Kalecki Model

Abstract By using the theory of dynamical systems and especially the theory of Hopf bifurcations the existence of limit cycles in the Kaldor-Kalecki model is discussed. This model is a modified version of the 1940 Kaldor model of business cycle with introduced the time-to-build parameter in investment function in the Kalecki way. In this paper the model is investigated in an approximation of a smalllag. It is shown that such a model can be reduced to the Lienard equation which has a bifurcation value of lag parameter. From the Hopf theorem the sufficient conditions for generation of the limit cycle on the phase space are obtained. It is demonstrated how the amplitude of oscillation depends on the value of a lag parameter.

AIC, BIC, Bayesian evidence against the interacting dark energy model

Recent astronomical observations have indicated that the Universe is in a phase of accelerated expansion. While there are many cosmological models which try to explain this phenomenon, we focus on the interacting

Simple dynamics on the brane

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