Occurrence of Big Bang Bifurcations in Discretized Sliding-mode Control Systems

Fossas, Enric; Granados, Albert

doi:10.1007/s12591-012-0121-y

Cited by 5 publications

(10 citation statements)

References 11 publications

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“…In this example we recover the results presented in [51], which extend the example shown in Section 7.1.2 to higher order linear systems.…”

Section: Higher Order Sliding-mode Controlled System With Relayssupporting

confidence: 81%

“…In our next example we consider a first order one-dimensional system 2 , y = f (y), (7.4) such that f is monotonically decreasing and has a simple zero at the origin, f (0) = 0, which is a stable equilibrium point of system (7.4). We generalize the results presented in [50] for a linear system and in [51] for second order linear system. We wish to stabilize system (7.4) at new equilibrium point y * by performing a control action u(t, y):ẏ = f (y) + u(t, y).…”

Section: First Order Sliding-mode Controlled System With Relaysmentioning

confidence: 64%

“…In the recent years, piecewise-smooth maps, both in R and in R n , have been widely investigated and very interesting bifurcation phenomena (border collisions, big bangs, period adding, period incrementing,...) have been reported by many authors. This included both works by a more applied community, like in power electronics ( [7,8,9,10,19,30,43,76,87,115]), control theory ( [46,50,51,102,135,133,134,136]), biology ( [57,106]), neuroscience ( [73,74,84,88,89,103,122,123,124,125]) or economy ( [126]) but also researchers from a more theoretical and/or computational perspective in non-smooth systems ( [12,13,14,15,17,18,20,21,22,23,24,25,26,27,28,…”

Section: Discussionmentioning

confidence: 99%

“…Beyond its relation with homoclinic bifurcations for flows, such type of bifurcation scenarios have been observed in more applied contexts modeled by both one and n-dimensional piecewise-smooth maps. Examples of such applications where these bifurcation scenarios appear are power electronics ( [45,29,113,67,136,7,85,134]), control theory ( [46,58,51,133,135,50,51]), economics ( [126]) or neuroscience ( [57,74,84,89,103,122,116,124,125,123]). Typically, these bifurcation scenarios are numerically observed in two-dimensional parameter spaces near codimension-two bifurcation points.…”

Section: Introductionmentioning

confidence: 99%

“…Often, due to the complexity of real applications, one-dimensional maps are not enough and one needs to consider maps in higher dimensions. Examples of two (or higher)-dimensional piecewise-smooth maps are found as Poincaré (or stroboscopic) maps in non-autonomous mechanical systems ( [37,109,44]), but their are also found in power electronics ( [45,29,113,67,136,7]), control theory ( [46,133,135,51]) or mathematical-neuroscience or biology ( [124,125,57,103,123,116]). Piecewise-smooth maps in higher dimensions are also obtained when studying Filippov flows in R n (n ≥ 3) ( [49]); they appear as half-return Poincaré maps, and become discontinuous close to grazing and Hopf bifurcations ( [56,108,94,111,39,38]).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

Granados¹,

Alsedà²,

Krupa³

2017

SIAM Rev.

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This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasicontractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and "rotation" numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.

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“…In this example we recover the results presented in [51], which extend the example shown in Section 7.1.2 to higher order linear systems.…”

Section: Higher Order Sliding-mode Controlled System With Relayssupporting

confidence: 81%

Section: First Order Sliding-mode Controlled System With Relaysmentioning

confidence: 64%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

Granados¹,

Alsedà²,

Krupa³

2017

SIAM Rev.

Self Cite

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Lyapunov exponent and topological entropy plateaus in piecewise linear maps

Botella-Soler

Oteo

Ros

et al. 2013

J. Phys. A: Math. Theor.

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We consider a two-parameter family of piecewise linear maps in which the moduli of the two slopes take different values. We provide numerical evidence of the existence of some parameter regions in which the Lyapunov exponent and the topological entropy remain constant. Analytical proof of this phenomenon is also given for certain cases. Surprisingly however, the systems with that property are not conjugate as we prove by using kneading theory.

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Canards in piecewise-linear systems: explosions and super-explosions

et al. 2013

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We show that a planar slow-fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion.

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Occurrence of Big Bang Bifurcations in Discretized Sliding-mode Control Systems

Cited by 5 publications

References 11 publications

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

Lyapunov exponent and topological entropy plateaus in piecewise linear maps

Canards in piecewise-linear systems: explosions and super-explosions

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