last modified November 30, 2017 - 14:43 CET

Discrete time dynamical systems

The analysis of discrete-time 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 three-dimensional flow corresponding to the system of normalized circuit equations turns out to be piecewise linear (PWL). However, in contrast with common invertible three-dimensional flows (giving rise to invertible two-dimensional return-maps), it can be studied through a suitable discrete-time dynamical system (one-dimensional 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.

    Map and sections Map and sections
  • A one-dimensional map for the generation of pseudo-random sequences
    The bifurcation analysis of a piecewise-affine discrete-time dynamical system has been carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random 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 one-dimensional bifurcation diagram evidences the presence of chaotic intervals.

    1D bifurcation diagram
  • A one-dimensional map to study a second order impact model for forest fire prediction
    An exhaustive bifurcation analysis of a one-dimensional piecewise-smooth map naturally induced by the two-dimensional 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 one-dimensional 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 two-dimensional bifurcation diagram is shown.

    2D bifurcation diagram

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. 2275-2288.
  • Bizzarri F, Storace M and Gardini L (2006), "Bifurcation analysis of a circuit-related generalization of the shipmap", International Journal of Bifurcation and Chaos. Vol. 16, pp. 2435-2452.
  • Bizzarri F and Storace M (2002), "Two-dimensional bifurcation diagrams of a chaotic circuit based on hysteresis", International Journal of Bifurcation and Chaos. Vol. 12, pp. 43-69.
  • Bizzarri F, Storace M, Gardini L and Lupini R (2001), "Bifurcation analysis of a PWL chaotic circuit based on hysteresis through a one-dimensional map", International Journal of Bifurcation and Chaos. Vol. 11, pp. 1911-1927.


  • Italian Ministry of University and Research Grant - 2005-2007 (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 - 2003-2006 (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) - 2001-2002, 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".

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