arXiv subjects

Ivan Ovsyannikov

Publications and source records attributed to Ivan Ovsyannikov.

7 recordsLinked to original sources

arXiv2023

Appearance of discrete Lorenz attractors in the transitions from saddle to saddle-focus

Triply degenerate fixed points appear in global bifurcations -- homoclinic and heteroclinic tangencies. In order to get Lorenz-like attractors, the dynamics of the first return map along the homoclinic or heteroclinic cycle should be effectively at least three-dimensional, i.e. there should not exist lower-dimensional invariant manifolds. This can be achieved by adding some special conditions, global or local. Global degeneracies are related to the existence of non-simple homoclinic tangencies or non-simple heteroclinic orbit in a cycle, these cases were studied before. Local conditions either require the cycle to contain at least one saddle-focus, or add certain relations on the multipliers of the fixed point such that the leading stable direction of the saddle either disappears or alternates. All these cases were already studied before except one, related to the transition from saddle to saddle-focus. In the present paper this case is investigated, and the existence of a cascade of parameter domains containing systems with discrete Lorenz attractors is proved. In particular, it includes the Belyakov bifurcation, when the saddle becomes a saddle-focus through a collision of the eigenvalues, and the 3DL bifurcation, when the dimension of the leading stable subset alternates between 1 and 2. This paper completes the list of the simplest bifurcations of homoclinic and heteroclinic tangencies by studying the last possible case.

math.DS↗

arXiv2023

Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

Revisiting the Lorenz '63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite $ρ$, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average `transport', which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the non-zero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the non-zero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

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arXiv2022

Classification of Codimension-1 Singular Bifurcations in Low-dimensional DAEs

The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before.

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arXiv2021

On birth of discrete Lorenz attractors under bifurcations of 3D maps with nontransversal heteroclinic cycles

Lorenz attractors are important objects in the modern theory of chaos. The reason from one side is that they are met in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). At the same time, Lorenz attractors are robust, in the sense that they are generally not destroyed by small perturbations (autonomous, non-autonomous, stochastic). This allows us to be sure that the observed in the experiment object is exactly the chaotic attractor, rather than a long-time periodic orbit. Discrete-time analogs of the Lorenz attractor possess even more complicated structure -- they allow homoclinic tangencies of invariant manifolds within the attractor. Thus, discrete Lorenz attractors belong to the class of wild chaotic attractors. These attractors can be born in codimension-three local and certain global (homoclinic and heteroclinic) bifurcations. While various homoclinic bifurcations leading to such attractors were studied, for heteroclinic cycles only cases when at least one of the fixed points is saddle-focus were considered to date. In the present paper the case of a heteroclinic cycle consisting of saddle fixed points with a quadratic tangency of invariant manifolds, is considered. It is shown that in order to have a three-dimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lower-dimensional global invariant manifolds. Thus, it is assumed that either the quadratic tangency or the transversal heteroclinic orbit is non-simple. The main result of the paper is the proof that the original system is the limiting point in the space of dynamical systems of a sequence of domains in which the diffeomorphism possesses discrete Lorenz attractors.

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arXiv2021

Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors

Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. However, to date, only partial cases were investigated. In this paper bifurcations of homoclinic and heteroclinic cycles with quadratic tangencies of invariant manifolds are studied. A full list of such bifurcations, leading to the appearance of discrete Lorenz attractors is provided. In addition, with help of numerical techniques, it was proved that if one reverses time in the diffeomorphisms described above, the resulting systems also have such attractors. This result is an important step in the systematic studies of chaos and hyperchaos.

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arXiv2019

Inhomogeneous domain walls in spintronic nanowires

In case of a spin-polarized current, the magnetization dynamics in nanowires are governed by the classical Landau-Lifschitz equation with Gilbert damping term, augmented by a typically non-variational Slonczewski term. Taking axial symmetry into account, we study the existence of domain wall type coherent structure solutions, with focus on one space dimension and spin-polarization, but our results also apply to vanishing spin-torque term. Using methods from bifurcation theory for arbitrary constant applied fields, we prove the existence of domain walls with non-trivial azimuthal profile, referred to as inhomogeneous. We present an apparently new type of domain wall, referred to as non-flat, whose approach of the axial magnetization has a certain oscillatory character. Additionally, we present the leading order mechanism for the parameter selection of flat and non-flat inhomogeneous domain walls for an applied field below a threshold, which depends on anisotropy, damping, and spin-transfer. Moreover, numerical continuation results of all these domain wall solutions are presented.

math.AP↗

arXiv2017

Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.

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