Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

Гонченко, С. В.; Kazakov, Alexey O.; Turaev, Dmitry

doi:10.1088/1361-6544/abc794

Cited by 29 publications

(13 citation statements)

References 96 publications

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“…In a series of papers [25][26][27] we have shown that heterodimensional cycles can be a part of a pseudohyperbolic chain-transitive attractor which appears in systems with Shilnikov loops [18,32,38] or after a periodic perturbation of the Lorenz attractor [39]. It follows from our results in the current paper that the attractor in such systems remains heterodimensional for an open set of parameter values.…”

Section: Introductionsupporting

confidence: 60%

“…where the coefficients v+ , ŷ− , μ, a, and a ij change uniformly continuously when the system f is perturbed, and for the original system f , we have v+ = v + , ŷ− = y − , and μ = 0. Since it does not cause ambiguity, in further references to (18) we use v + and y − instead of v+ and ŷ− . Note that the coefficient a = 0 is exactly the derivative defined in (6).…”

Section: First-return Mapsmentioning

confidence: 99%

“…After that, we substitute these formulas into the rest of equations ( 13), (18) and find that there exist…”

Section: First-return Mapsmentioning

confidence: 99%

“…We first prove the second part of the statement describing the situation at µ ≤ 0. It suffices to show that, at µ ≥ 0, the image F m • F 12 (W u loc (O 1 )) does not enter the domain of F 21 , i.e., it is outside a small neighborhood of M − 2 = (u − , 0, w − ) for any sufficiently large m. Since W u loc (O 1 ) is given by the equation (x = 0, z = 0), this is equivalent (see (18) and ( 15)) to showing that the system of equations…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Persistence of Heterodimensional Cycles

Li¹,

Turaev²

2021

Preprint

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A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C 2 , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family always create robust heterodimensional dynamics, i.e., chain-transitive sets which contain coexisting orbits with different numbers of positive Lyapunov exponents and persist for an open set of parameter values. In particular, we solve the so-called C r -stabilization problem for the coindex-1 heterodimensional cycles in any regularity class r = 2, . . . , ∞, ω. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

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Section: Introductionsupporting

confidence: 60%

Section: First-return Mapsmentioning

confidence: 99%

“…After that, we substitute these formulas into the rest of equations ( 13), (18) and find that there exist…”

Section: First-return Mapsmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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Persistence of Heterodimensional Cycles

Li¹,

Turaev²

2021

Preprint

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“…This attractor was called in [34] wild spiral attractor, since it contains a saddle-focus equilibrium together with wild hyperbolic subsets and, hence, it allows homoclinic tangencies. 2 Despite the fact that such attractors were predicted (as well as geometrically constructed) in the late 90s, the first example of such an attractor in a system of four differential equations was found much recently in [35], see also [36].…”

Section: Introductionmentioning

confidence: 99%

On discrete pseudohyperbolic attractors of Lorenz type

Gonchenko,

Kazakov

2020

Preprint

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We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors, that can be observed in three-dimensional diffeomorphisms. We propose new phenomenological scenarios of their appearance in one parameter families of such maps. We pay especially our attention to such a scenario that can lead to period-2 Lorenz-like attractors. These attractors have very interesting dynamical properties and we show that their crises can lead, in turn, to the emergence of pseudohyperbolic discrete Lorenz shape attractors of new types. We also show examples of all these attractors in three-dimensional generalized Hénon maps.

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Persistence of heterodimensional cycles

Li,

Turaev

2024

Invent. math.

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A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least $C^{2}$ C 2 , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class $r=2,\ldots ,\infty ,\omega $ r = 2 , … , ∞ , ω . The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

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Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

Abstract: We present an example of a new strange attractor which, as we show, belongs to a class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential equations which can be represented as an extension of the Lorenz system.

Cited by 29 publications

References 96 publications

Persistence of Heterodimensional Cycles

Persistence of Heterodimensional Cycles

On discrete pseudohyperbolic attractors of Lorenz type

Persistence of heterodimensional cycles

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