Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

Abstract

In this paper we present an example of a new strange attractor. We show that it 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.

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

“…All these stability windows indicate that chaotic dynamics in system (1) are not hyperbolic and even not pseudohyperbolic [31][32][33]. In other words, in the accordance with PQ-hypothesis [33] strange attractors in the system under investigation belong to a class of quasiattractors introduced by Afraimovich and Shilnikov in [34]. Stable periodic orbits of high periods and with narrow absorbing domains exist inside such attractors or appear with arbitrarily small perturbations.…”

“…Such stability windows indicate the existence of the specific homoclinic bifurcations (cubic homoclinic tangencies or symmetrical pairs of homoclinic tangencies) in the system [30]. All these stability windows indicate that chaotic dynamics in system (1) are not hyperbolic and even not pseudohyperbolic [31][32][33]. In other words, in the accordance with PQ-hypothesis [33] strange attractors in the system under investigation belong to a class of quasiattractors introduced by Afraimovich and Shilnikov in [34].…”

On discrete pseudohyperbolic attractors of Lorenz type

“…Speaking shortly, an attractor is pseudohyperbolic if, in some its neighborhood D (one can think that D is some absorbing region), a "weak" version of hyperbolicity is fulfilled, i.e. there is an exponential contraction along some invariant directions and exponential expansion of volumes on the subspaces transversal to them, for more details see [55,56].…”

“…This field can be calculated in various ways. In particular, one of such methods, the so-called LMP-method (abbreviation for "Light Method of Pseudohyperbolicity" checking), was proposed in [55], see also [56]. This method allows to construct a field of vectors corresponding to the strongest contractions and to display, in form of LMP-graph, the dependence of the angles dϕ between vectors N 1 (x) and N 1 (x 2 ) on the distance dx between the corresponding points x 1 and x 2 on the attractor.…”

“…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].…”