| 1. |
- di Bernardo, M., et al.
(author)
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Bifurcations in Nonsmooth Dynamical Systems
- 2008
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In: SIAM Review. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1445 .- 1095-7200. ; 50:4, s. 629-701
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Research review (peer-reviewed)abstract
- 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.
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| 2. |
- Kowalczyk, P., et al.
(author)
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Two-parameter discontinuity-induced bifurcations of limit cycles : Classification and open problems
- 2006
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In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 16:3, s. 601-629
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Journal article (peer-reviewed)abstract
- This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincare map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.
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| 3. |
- Nordmark, Arne B., et al.
(author)
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Simulation and stability analysis of impacting systems with complete chattering
- 2009
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In: Nonlinear dynamics. - : Springer Science and Business Media LLC. - 0924-090X .- 1573-269X. ; 58:1-2, s. 85-106
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Journal article (peer-reviewed)abstract
- This paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chattering-accumulation of impacts-for which local discontinuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations.
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| 4. |
- Piiroinen, P. T., et al.
(author)
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Breaking symmetries and constraints : Transitions from 2D to 3D in passive walkers
- 2003
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In: Multibody system dynamics. - 1384-5640 .- 1573-272X. ; 10:2, s. 147-176
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Journal article (peer-reviewed)abstract
- The inherent dynamics of bipedal, passive mechanisms are studied to investigate the relation between motions constrained to two-dimensional (2D) planes and those free to move in a three-dimensional (3D) environment. In particular, we develop numerical and analytical techniques using dynamical-systems methodology to address the persistence and stability changes of periodic, gait-like motions due to the relaxation of configuration constraints and the breaking of problem symmetries. The results indicate the limitations of a 2D analysis to predict the dynamics in the 3D environment. For example, it is shown how the loss of constraints may introduce characteristically non-2D instability mechanisms, and how small symmetry-breaking terms may result in the termination of solution branches.
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| 5. |
- Piiroinen, P. T., et al.
(author)
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On a normal-form analysis for a class of passive bipedal walkers
- 2001
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In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - 0218-1274. ; 11:9, s. 2411-2425
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Journal article (peer-reviewed)abstract
- This paper implements a center-manifold technique to arrive at a normal-form for the natural dynamics of a passive, bipedal rigid-body mechanism in the vicinity of infinite foot width and near-symmetric body geometry. In particular, numerical schemes are developed for finding approximate forms of the relevant invariant manifolds and the near-singular dynamics on these manifolds. The normal-form approximations are found to be highly accurate for relatively large foot widths with a range of validity extending to widths on the order of the mechanisms' height.
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