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Alexey Kazakov

Publications and source records attributed to Alexey Kazakov.

15 recordsLinked to original sources
arXiv2025

Cascades of Lorenz attractors in the Shimizu-Morioka model

The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. As in the classical Lorenz system, a part of the boundary is associated with the curve $l_{A=0}$, where the first tangency between some Lyapunov subspaces occurs along orbits of the attractor. However, in the Lorenz system, the curve $l_{A=0}$ forms the exact boundary of the Lorenz attractor existence region. Beyond this curve, the attractor is not robustly chaotic, although it may be indistinguishable from the Lorenz attractor in simple numerical experiments. In the Shimizu-Morioka model, the curve $l_{A=0}$ is divided into two parts. The Lorenz attractor existence region adjoins $l_{A=0}$ along the first part of this curve, as in the Lorenz system. Near the second part, as we show, the region of the existence of the Lorenz attractor is fractal. We describe two infinite cascades of disjoint subregions with the Lorenz attractor. One cascade occurs along the curve $l_{A=0}$, another -- in the transversal direction. We show that along the cascades, the Lorenz attractor undergoes ``doubling bifurcations'', leading to a complication of its topological structure.

math.DS↗
arXiv2024

Multi-winged Lorenz attractors due to bifurcations of a periodic orbit with multipliers $(-1,i,-i)$

We show that bifurcations of periodic orbits with multipliers $(-1,i,-i)$ can lead to the birth of pseudohyperbolic (i.e., robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ``Sim\'o angels''. These three attractors exist in an orientation-reversing, three-dimensional, quadratic H\'enon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simo angels. Both for the normal form and the 3D H\'enon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent.

nlin.CD↗
arXiv2024

Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry

We study bifurcations of a symmetric equilibrium state in systems of differential equations invariant with respect to a $\mathbb{Z}_4$-symmetry. We prove that if the equilibrium state has a triple zero eigenvalue, then pseudohyperbolic attractors of different types can arise as a result of the bifurcation. The first type is the classical Lorenz attractor and the second type is the so-called Sim\'o angel. The normal form of the considered bifurcation also serves as the normal form of the bifurcation of a periodic orbit with multipliers $(-1, i, -i)$. Therefore, the results of this paper can also be used to prove the emergence of discrete pseudohyperbolic attractors as a result of this codimension-3 bifurcation, providing a theoretical confirmation for the numerically observed Lorenz-like attractors and Sim\'o angels in the three-dimensional H\'enon map.

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arXiv2024

Robust chaos in a totally symmetric network of four phase oscillators

We provide conditions on the coupling function such that a system of 4 globally coupled identical oscillators has chaotic attractors, a pair of Lorenz attractors or a 4-winged analogue of the Lorenz attractor. The attractors emerge near the triple instability threshold of the splay-phase synchronization state of the oscillators. We provide theoretical arguments and verify numerically, based on the pseudohyperbolicity test, that the chaotic dynamics are robust with respect to small, e.g. time-dependent, perturbations of the system. The robust chaoticity should also be inherited by any network of weakly interacting systems with such attractors.

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arXiv2024

Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agents

We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyaponov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.

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arXiv2023

Machine learning-based detection of cardiovascular disease using ECG signals: performance vs. complexity

Cardiovascular disease remains a significant problem in modern society. Among non-invasive techniques, the electrocardiogram (ECG) is one of the most reliable methods for detecting abnormalities in cardiac activities. However, ECG interpretation requires expert knowledge and it is time-consuming. Developing a novel method to detect the disease early could prevent death and complication. The paper presents novel various approaches for classifying cardiac diseases from ECG recordings. The first approach suggests the Poincare representation of ECG signal and deep-learning-based image classifiers (ResNet50 and DenseNet121 were learned over Poincare diagrams), which showed decent performance in predicting AF (atrial fibrillation) but not other types of arrhythmia. XGBoost, a gradient-boosting model, showed an acceptable performance in long-term data but had a long inference time due to highly-consuming calculation within the pre-processing phase. Finally, the 1D convolutional model, specifically the 1D ResNet, showed the best results in both studied CinC 2017 and CinC 2020 datasets, reaching the F1 score of 85% and 71%, respectively, and that was superior to the first-ranking solution of each challenge. The paper also investigated efficiency metrics such as power consumption and equivalent CO2 emissions, with one-dimensional models like 1D CNN and 1D ResNet being the most energy efficient. Model interpretation analysis showed that the DenseNet detected AF using heart rate variability while the 1DResNet assessed AF pattern in raw ECG signals.

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arXiv2022

Scenarios for the Creation of Hyperchaotic Attractors of 3D Maps

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e., such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In order to possess this property periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. For realization of this possibility, we propose several bifurcation scenarios that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of the three-dimensional Mirá map.

math.DS↗
arXiv2022

New drugs and stock market: how to predict pharma market reaction to clinical trial announcements

Pharmaceutical companies operate in a strictly regulated and highly risky environment in which a single slip can lead to serious financial implications. Accordingly, the announcements of clinical trial results tend to determine the future course of events, hence being closely monitored by the public. In this work, we provide statistical evidence for the result promulgation influence on the public pharma market value. Whereas most works focus on retrospective impact analysis, the present research aims to predict the numerical values of announcement-induced changes in stock prices. For this purpose, we develop a pipeline that includes a BERT-based model for extracting sentiment polarity of announcements, a Temporal Fusion Transformer for forecasting the expected return, a graph convolution network for capturing event relationships, and gradient boosting for predicting the price change. The challenge of the problem lies in inherently different patterns of responses to positive and negative announcements, reflected in a stronger and more pronounced reaction to the negative news. Moreover, such phenomenon as the drop in stocks after the positive announcements affirms the counterintuitiveness of the price behavior. Importantly, we discover two crucial factors that should be considered while working within a predictive framework. The first factor is the drug portfolio size of the company, indicating the greater susceptibility to an announcement in the case of small drug diversification. The second one is the network effect of the events related to the same company or nosology. All findings and insights are gained on the basis of one of the biggest FDA (the Food and Drug Administration) announcement datasets, consisting of 5436 clinical trial announcements from 681 companies over the last five years.

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arXiv2022

On Shilnikov attractors of three-dimensional flows and maps

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results are illustrated with various specific examples.

math.DS↗
arXiv2020

Homoclinic chaos in the Rössler model

We study the origin of homoclinic chaos in the classical 3D model proposed by O. Rössler in 1976. Of our particular interest are the convoluted bifurcations of the Shilnikov saddle-foci and how their synergy determines the global unfolding of the model, along with transformations of its chaotic attractors. We apply two computational methods proposed, 1D return maps and a symbolic approach specifically tailored to this model, to scrutinize homoclinic bifurcations, as well as to detect the regions of structurally stable and chaotic dynamics in the parameter space of the Rössler model.

nlin.CD↗
arXiv2020

Wild pseudohyperbolic attractor in a four-dimensional Lorenz system

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.

math.DS↗
arXiv2020

On discrete pseudohyperbolic attractors of Lorenz type

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.

math.DS↗
arXiv2018

On a scenario of onset of strongly dissipative mixed dynamics

In this paper we present the scenario of the occurrence of strongly dissipative mixed dynamics in two-dimensional reversible diffeomorphisms, using as an example the system describing a motion of two point vortices under the influence of wave perturbation and shear flow. For mixed dynamics of this type the chaotic attractor intersects with the chaotic repeller, but their intersection forms a "thin" set. The main stage of this scenario is the appearance of homoclinic structures for a symmetric saddle orbit which arise after crisis of a homoclinic attractor and repeller.

math.DS↗
arXiv2014

Simple scenarios of onset of chaos in three-dimensional maps

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincare maps in models of nonholonomic mechanics.

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