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  • 1. di Bernardo, M.
    et al.
    Budd, C. J.
    Champneys, A. R.
    Kowalczyk, P.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Tost, G. O.
    Piiroinen, P. T.
    Bifurcations in Nonsmooth Dynamical Systems2008In: SIAM Review, ISSN 0036-1445, E-ISSN 1095-7200, Vol. 50, no 4, p. 629-701Article, review/survey (Refereed)
    Abstract [en]

    A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

  • 2. Thunberg, Hans
    Periodicity versus chaos in one-dimensional dynamics2001In: SIAM Review, ISSN 0036-1445, E-ISSN 1095-7200, Vol. 43, no 1, p. 3-30Article in journal (Refereed)
    Abstract [en]

    We survey recent results in one-dimensional dynamics and, as an application, we derive rigorous basic dynamical facts for two standard models in population dynamics, the Ricker and the Hassell families. We also informally discuss the concept of chaos in the context of one-dimensional discrete time models. First we use the model case of the quadratic family for an informal exposition. We then review precise generic results before turning to the population models. Our focus is on typical asymptotic behavior, seen for most initial conditions and for large sets of maps. Parameter sets corresponding to different types of attractors are described. In particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models. Natural measures (asymptotic distributions) and their stability properties are considered.

  • 3.
    Wahlberg, Bo
    KTH, Superseded Departments, Signals, Sensors and Systems.
    Orthogonal rational functions: A transformation analysis2003In: SIAM Review, ISSN 0036-1445, E-ISSN 1095-7200, Vol. 45, no 4, p. 689-705Article in journal (Refereed)
    Abstract [en]

    Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal polynomials are well known. However, infinite impulse response (IIR) models often provide much more compact descriptions and in many cases give improved performance. The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to orthogonal rational functions. Orthogonality simplifies approximation analysis and leads to improved numerical properties. The basic idea is to use a fractional transformation to map the problem to a new domain, where an FIR description is most appropriate. This FIR representation is then mapped back to the original domain to give an orthogonal IIR representation. It is then straightforward to extend many results for FIR. models to IIR model structures with arbitrary stable poles; i.e., properties of orthogonal polynomials are easily generalized to orthogonal rational functions. Much of the theory to be presented is classical, e.g., Laguerre and Kautz functions, and we will make use of well-known results in orthogonal filter theory. However, our main contribution is to present a uniform and transparent theory which also covers more novel results that have mainly been presented in the signals, systems, and control literature in the last decade.

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