last modified
February 12, 2014  15:25 CET
Discrete time dynamical systems
The analysis of discretetime dynamical systems has been carried out in three different cases of study.
 A chaotic circuit based on hysteresis
Under a specific hypothesis on the circuit parameters characterizing the oscillator, the threedimensional flow corresponding to the system of normalized circuit equations turns out to be piecewise linear (PWL). However, in contrast with common invertible threedimensional flows (giving rise to invertible twodimensional returnmaps), it can be studied through a suitable discretetime dynamical system (onedimensional map). As the flow is noninvertible and noncontinuous with respect to the initial conditions, the map is, in turn, noninvertible and can exhibit both discontinuities and lack of smoothness. Unlike what occurs in several continuous models, many of the properties of the map can be analytically proved, among which the ranges of continuity and the appearance/disappearance of discontinuities. Through the map, many dynamic features of the flow can be ultimately described, such as local and global bifurcations, regular or chaotic asymptotic behavior.
The following figure displays the second iterate of a discontinuous map (left panel) with two invariant intervals (I and J), corresponding to the two chaotic attractors shown in the right panel. Also the presence of two (unstable) fixed points is evidenced, corresponding to the limit cycles C1 and C2 in the right panel.
 A onedimensional map for the generation of pseudorandom sequences
The bifurcation analysis of a piecewiseaffine discretetime dynamical system has been carried out. Such a system derives from a wellknown map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudorandom sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. Several different dynamic situations may arise, which are completely characterized as a function of three parameters. In particular, it has been shown that chaotic intervals may coexist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces have been obtained either analytically or numerically by studying the critical points of the map (i.e. extremum points and discontinuity points) and their iterates. This bifurcation analysis, in view of a robust design of the map, should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties.
This onedimensional bifurcation diagram evidences the presence of chaotic intervals.
 A onedimensional map to study a second order impact model for forest fire prediction
An exhaustive bifurcation analysis of a onedimensional piecewisesmooth map naturally induced by the twodimensional flow has been carried out with respect to four pairs of system parameters. The results of the analysis confirm those reported in the literature and extend them, pointing out the different roles played by several system parameters in influencing the qualitative behavior of fire regimes. The main advantage of the analysis of the second order impact model through this onedimensional map is that important dynamic characteristics can be retrieved by analyzing the (critical) points in the map domain corresponding to local maxima/minima and discontinuities.
Below a twodimensional bifurcation diagram is shown.
Involved People:
Major Publications:

Bizzarri F, Storace M and Colombo A (2008), "Bifurcation analysis of an impact model for forest fire prediction", International Journal of Bifurcation and Chaos., Aug., 2008. Vol. 18(8), pp. 22752288.
[BibTeX]

Bizzarri F, Storace M and Gardini L (2006), "Bifurcation analysis of a circuitrelated generalization of the shipmap", International Journal of Bifurcation and Chaos. Vol. 16, pp. 24352452.
[BibTeX]

Bizzarri F and Storace M (2002), "Twodimensional bifurcation diagrams of a chaotic circuit based on hysteresis", International Journal of Bifurcation and Chaos. Vol. 12, pp. 4369.
[BibTeX]

Bizzarri F, Storace M, Gardini L and Lupini R (2001), "Bifurcation analysis of a PWL chaotic circuit based on hysteresis through a onedimensional map", International Journal of Bifurcation and Chaos. Vol. 11, pp. 19111927.
[BibTeX]
Funding:
 Italian Ministry of University and Research Grant  20052007 (PRIN2005). Research network with the Universities of Milan (Polytechnic), Genoa, Naples (Second University), and Rome "La Sapienza". Title: "Design and implementation methodologies of free running and synthesised oscillators characterised by low phase noise"
 Italian Ministry of University and Research Grant  20032006 (FIRB2001). Research network with the Universities of Ferrara, Genoa, Turin, and Milan. Title: "Innovative methods for analysis and design of chaotic circuits".
 Italian National Research Council (CNR)  20012002, in the framework of the "Young researchers projects" initiative. Title: "Analysis, implementation and possible applications of an electronic oscillator based on hysteresis able to produce chaotic dynamics".