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In this Chapter, we provide the definitions, notions and examples relevant for the analysis of the dynamical systems of interest to us in the remainder of this book. We start with with a description of dynamical systems and we provide a taxonomy. Then, we define continuous-time dynamical systems...
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In this chapter, the Logistic Map is taken as the example demonstrating the generic stability properties of fixed points and limit cycles, in dependence of the strength of nonlinearity. To identify attracting periodic orbits, we use the Schwarz derivative. The chapter ends with an application of...
Persistent link: https://www.econbiz.de/10012648028
Many dynamical systems depend on parameters. One may expect that small variations of the parameters produce no significant changes in the orbits. As was shown in Chap. 3 for the Logistic Map, even in simple cases, there exist critical values such that, moving the parameters through them, the...
Persistent link: https://www.econbiz.de/10012648029
In this chapter, we first precise the concept of dynamical systems, and then we introduce the concept of chaos, which is characterized by a sensitive dependence on initial conditions. To quantify this, dynamical (Lyapunov exponents) and probabilistic (dimensions) measures are introduced.
Persistent link: https://www.econbiz.de/10012648032
In this chapter, we introduce the concept of the embedding dimension, as the smallest topological dimension required to ensure that an object described by simpler (often scalar) time series can be embedded in a higher topological dimension.
Persistent link: https://www.econbiz.de/10012648033
In this chapter, we first explain what we mean by a signal, and then we describe some characteristics such as energy, frequency, phase, power spectrum, etc. We show how to analyse it by the means of spectral analysis and Fourier transform. Moreover, as the Fourier transform does not provide any...
Persistent link: https://www.econbiz.de/10012648037
In this chapter, in Sect. 12.1 we provide a sketch of the Keynesian multiplier and the multiplier–accelerator model by Hansen and Samuelson. The description of the Kaldor model (Sect. 12.2) is introduced by the related literature (Sect. 12.2.1). As Kaldor described his model only...
Persistent link: https://www.econbiz.de/10012648041
R.G. Goodwin mentioned that "economists will be led, as natural scientists have been led, to seek in nonlinearities an explanation of the maintenance of oscillation" (Goodwin, Econometrica 19(1), 1951); following this reasoning, we studied business cycles as if they were generated by nonlinear...
Persistent link: https://www.econbiz.de/10012648046
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