Nataliya Stankevich

This study discusses an approach for estimation of the largest Lyapunov exponent for the mathematical model of the cardiovascular system. The accuracy was verified using the confidence intervals approach. The algorithm was used to investigate the effects of noises with different amplitudes and spectral compositions on the dynamics of the model. Three sets of parameters are considered, corresponding to different states of the human cardiovascular system model. It is shown that, in each case, the model exhibited chaotic dynamics. The model gave different responses to the changes in the characteristics of the noise, when using different sets of parameters. The noise had both constructive and destructive effects, depending on the parameters of the model and the noise, by, respectively, amplifying or inhibiting the chaotic dynamics of the model.

The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Their interaction is simulated via a chemical coupling that is implemented with a sigmoid function. It is shown that the model may exhibit complex behavior: quasiperiodic, chaotic and hyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaos associated with the appearance of a discrete Shilnikov attractor is described. It is shown that the formation of these attractors leads to the appearance of in-phase bursting oscillations.

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.

A generalized model with bifurcations associated with blue sky catastrophes is introduced. Depending on an integer index $m$, different kinds of attractors arise, including those associated with quasi-periodic oscillations and with hyperbolic chaos. Verification of the hyperbolicity is provided based on statistical analysis of intersection angles of stable and unstable manifolds.

We study bifurcation scenarios leading to the so-called Shilnikov singular attractors as a result of breakdown of closed invariant curves in the Chialvo map, which is a twodimensional endomorphism demonstrating neuron-like dynamics. We show that two different routes, soft and hard, of the emergence of such closed invariant curves can be traced here: the soft one corresponds to the birth of an invariant curve as a result of a supercritical Neimark – Sacker bifurcation, and the hard one relates to the immediately occurring big invariant curve after disappearance of a stable fixed point at the saddle-node bifurcation. We study both these mechanisms and trace subsequent scenarios of breakdown of the invariant curve and chaos development leading to the emergence of Shilnikov singular attractors. Additionally, we study geometrical peculiarities of these attractors, such as structures of rotating patterns of orbits inside the Shilnikov singular funnel and present a two-parametric analysis with Lyapunov exponents and minimal a distance between the chaotic attractor and the unstable focus.