Topological and Measurable Dynamics of Lorenz Maps

Keller, Gerhard; Pierre, Matthias St.

doi:10.1007/978-3-642-56589-2_15

Cited by 15 publications

(7 citation statements)

References 26 publications

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“…In order to find phaselocked orbits on the attractor, we consider a one-dimensional map that describes how points on K 0 map back to K 0 ; see also [39]. This map will be an induced map constructed using the Hofbauer towers [20,22]. We subdivide the line K 0 into intervals from which different numbers of iterations with M are necessary to come back to K 0 .…”

Section: 2mentioning

confidence: 99%

“…We find that this can also be the case when the dynamics is phase locked, so that the situation cannot be reduced to the dynamics on a circle map. Fortunately, we are able to reduce the system to a one-dimensional induced map based on the construction of Hofbauer towers [20,22]. The phase-locked solutions can then be found in a way that is similar to the approach in [36], but the use of the induced map avoids the need for checking admissibility of these solutions.…”

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confidence: 99%

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Arnol$'$d Tongues Arising from a Grazing-Sliding Bifurcation

Szalai

Osinga²

2009

SIAM J. Appl. Dyn. Syst.

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The Neȋmark-Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At a Neȋmark-Sacker bifurcation a periodic orbit loses stability and, except for certain so-called strong resonances, an invariant torus is born; the dynamics on the torus can be either quasi-periodic or phase locked, which is organized by Arnol d tongues in parameter space. Inside the Arnol d tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries. In this paper we investigate whether a piecewisesmooth system with sliding regions may exhibit an equivalent of the Neȋmark-Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next and the dividing so-called switching surface contains a sliding region if the vector fields on both sides point towards the switching surface. The existence of a sliding region has a superstabilizing effect on periodic orbits interacting with it. In particular, the associated Poincaré map is non-invertible. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region. We provide conditions under which the grazing-sliding bifurcation can be thought of as a Neȋmark-Sacker bifurcation. We give a normal form of the Poincaré map derived at the grazing-sliding bifurcation and show that the resonances are again organized in Arnol d tongues. The associated periodic orbits typically bifurcate in border-collision bifurcations that can lead to dynamics that is more complicated than simple quasi-periodic motion. Interestingly, the Arnol d tongues of piecewise-smooth systems look like strings of connected sausages and the tongues close at double border-collision points. Since in most models of physical systems non-smoothness is a simplifying approximation, we relate our results to regularized systems. As one expects, the phase-locked solutions deform into smooth orbits that, in a neighborhood of the Neȋmark-Sacker bifurcation, lie on a smooth torus. The deformation of the Arnol d tongues is more complicated; in contrast to the standard scenario, we find several coexisting pairs of periodic orbits near the points where the Arnol d tongues close in the piecewisesmooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems.

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Section: 2mentioning

confidence: 99%

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confidence: 99%

Arnol$'$d Tongues Arising from a Grazing-Sliding Bifurcation

Szalai

Osinga²

2009

SIAM J. Appl. Dyn. Syst.

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“…Williams (see [1,5,7,6,21,19]). Later on, main contributions include M. Martens and W. de Melo [14], G. Keller and M. St. Pierre [10,20], D. Berry and B. Mestel [3], and R. Labarca and C. G. Moreira [12,11].…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…So, by Lemma 4.7 we get O + f (x) ∩ (0, c) c ∈ (c, 1) ∩ O + f (x), ∀ x ∈ (0, 1). (10) As a consequence, ω f (x) ⊃ ω f (c − ) ∪ ω f (c + ), ∀ x ∈ (0, 1). (11) In particular, c − ∈ ω f (c − ) and c + ∈ ω f (c + ).…”

Section: Appendixmentioning

confidence: 98%

Topological attractors of contracting Lorenz maps

Brandão

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We study the non-wandering set of contracting Lorenz maps. We show that if such a map f doesn't have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set Λ such that ω f (x) = Λ for a residual set of points x ∈ [0, 1]. Furthermore, we classify the possible kinds of attractors that may occur.Date: August 28, 2018. Partially supported by FAPERJ, CNPq and CAPES.

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“…can be found in the PhD thesis of M. St. Pierre, see [ 4 ] (cf. [ 5 ]) or the PhD thesis of B. Winckler, see [ 6 ] (cf. [ 7 , 8 ]).…”

Section: Introductionmentioning

confidence: 99%

On α-Limit Sets in Lorenz Maps

Cholewa

Oprocha

2021

Entropy

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The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

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Topological and Measurable Dynamics of Lorenz Maps

Cited by 15 publications

References 26 publications

Arnol$'$d Tongues Arising from a Grazing-Sliding Bifurcation

Arnol$'$d Tongues Arising from a Grazing-Sliding Bifurcation

Topological attractors of contracting Lorenz maps

On α-Limit Sets in Lorenz Maps

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