On attractors of piecewise linear 2-endomorphisms

Dobrynskiy, V. A.

doi:10.1016/s0362-546x(97)00629-9

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…The aim of this paper is to extend the range of examples for which two-dimensional attractors can be proved to exist using two-dimensional Markov partitions. Unlike the example of Dobryiskiy [12] the attractor contains a repeller.…”

Section: Introductionmentioning

confidence: 85%

“…Note that such fully two-dimensional attractors have been observed numerically [11], and the existence of two-dimensional trapping regions has been established in many cases [21]. However, in the two situations we are aware of for which the existence of a two-dimensional strange attractor has been proved the attractor is either the closure of a one-dimensional unstable manifold [10] (which is not the case in the examples described here) or it can be treated by analyzing an appropriate one-dimensional map ( [7,11,21], the 'Cournot' case, see below). We aim to develop techniques which do not rely on one-dimensional techniques, and hence have a broader application.…”

Section: Introductionmentioning

confidence: 93%

“…two-dimensional regions with infinitely many unstable periodic orbits and dense orbits. We know of two exceptions to this statement, the Cournot map cases (T L = T R = 0) covered in [7] and the special case considered by Dobryiskiy [12]. The former case reduces to analysis based on the standard theory of one-dimensional maps, and the latter is based on a proof that the unstable manifold of a saddle is dense.…”

Section: Introductionmentioning

confidence: 99%

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Two-dimensional attractors in the border-collision normal form

Glendinning

Wong

2011

Nonlinearity

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Abstract. New techniques are developed to show that the two-dimensional normal form for codimension one border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems has attractors which are themselves two dimensional. This makes it possible to prove the existence of these attractors for a countable set of parameter values which cannot be treated using the essentially onedimensional methods in the literature.PACS numbers: 05.45.-a

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Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Two-dimensional attractors in the border-collision normal form

Glendinning

Wong

2011

Nonlinearity

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“…two-dimensional regions on which the map is transitive and periodic orbits are dense. However, proofs of the existence of these regions are either restricted to cases where the dynamics can be analyzed by related one-dimensional maps [Bischi et al, 2000], or to a specially chosen countable set of parameters at which there is a finite Markov partition with sufficient expansion in the map [Glendinning & Wong, 2011], see also [Dobrynskii, 1998;Dobrynskiy, 1999]. Glendinning and Jeffrey [2012] show that analogous attractors exist in the n-dimensional version of the border collision normal form due to di Bernardo [di Bernardo, 2003].…”

Section: Introductionmentioning

confidence: 99%

Invariant Measures for the n-Dimensional Border Collision Normal Form

Glendinning

2014

Int. J. Bifurcation Chaos

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The border collision normal form is a continuous piecewise affine map of R n with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence that the attractors have Hausdorff (fractal) dimension n. If n = 2 the attractors have topological dimension two, i.e. they contain open sets, and if n > 2 then they have topological dimension n generically.

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On the Structure of Generalized Hyperbolic Attractors of Mappings That Are Not One-to-One

Dobrynskii

2005

Diff Equat

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Pesin [1] introduced a class of diffeomorphisms with singularities which have generalized hyperbolic attractors and established a number of properties of such diffeomorphisms. Later, Sataev [2] developed the theory of diffeomorphisms with singularities (under somewhat more restrictive conditions) up to so-called "natural" limits. However, the following problem remains open: how completely do the above-mentioned results describe the properties of nontrivial generalized hyperbolic attractors of piecewise smooth mappings that are not one-to-one? This problem is nontrivial and has been little studied yet. To the best of the author's knowledge, the paper [3] is the only one in which, among other things, the existence of a nontrivial generalized hyperbolic attractor was proved for a certain piecewise linear endomorphism of a plane. The structure of this attractor differs dramatically from that of generalized hyperbolic attractors of diffeomorphisms of the plane with singularities. The latter are one-dimensional sets with the local structure of the direct product of the Cantor set by a straight-line segment; on the opposite, the former attractor is a two-dimensional set, namely, a polygon bounded by segments of the images of the critical set of the mapping and the unstable manifold at a fixed point lying on its boundary. Therefore, by merely extending the results obtained for diffeomorphisms with singularities to the case of similar piecewise smooth endomorphisms, one cannot solve the problem posed above on the whole. The family of piecewise smooth endomorphisms of the unit square to be studied below is an example of non-oneto-one mappings whose generalized attractors have properties mainly similar to those of attractors of diffeomorphisms with singularities.Thus we consider the following family of endomorphisms of the unit square I 2 = [0, 1] ⊗ [0, 1]:Here f (t) = 2 min{t, 1 − t}, t ∈ [0, 1], T stands for transposition, and λ, 0 < λ < 1, and s > 0 are parameters. Although, in this case, the function f is only piecewise smooth, it can obviously be included in the class of so-called "unimodular mappings of the unit square" [4][5][6][7][8][9]. It is also obvious that F is the simplest representative of such mappings. 1 Before continuing the exposition of the results, we note that "unimodular mappings of the unit square" include sets of isomorphisms whose properties (for λ ∼ = 1 and for sufficiently small s > 0) are similar to those of the Henon diffeomorphism (x, y) → (1 − ax 2 + y, bx) [10] and the Lozi homeomorphism (x, y) → (1 − a|x| + y, bx) [1] (with a ∼ = 2 and sufficiently small |b| > 0). This permits one to consider the above-mentioned mappings as a possible counterpart of the Lozi and Henon mappings in the class of non-one-to-one mappings. Our main result concerning the properties of F is the existence of parameter values for which this mapping has a nontrivial one-dimensional strange attractor. From the viewpoint of dynamics, it is a generalized hyperbolic attractor, and from the viewpoint of topology, it is a...

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On attractors of piecewise linear 2-endomorphisms

Cited by 6 publications

References 7 publications

Two-dimensional attractors in the border-collision normal form

Two-dimensional attractors in the border-collision normal form

Invariant Measures for the n-Dimensional Border Collision Normal Form

On the Structure of Generalized Hyperbolic Attractors of Mappings That Are Not One-to-One

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