RESUMEN
Hidden attractors are present in many nonlinear dynamical systems and are not associated with equilibria, making them difficult to locate. Recent studies have demonstrated methods of locating hidden attractors, but the route to these attractors is still not fully understood. In this Research Letter, we present the route to hidden attractors in systems with stable equilibrium points and in systems without any equilibrium points. We show that hidden attractors emerge as a result of the saddle-node bifurcation of stable and unstable periodic orbits. Real-time hardware experiments were performed to demonstrate the existence of hidden attractors in these systems. Despite the difficulties in identifying suitable initial conditions from the appropriate basin of attraction, we performed experiments to detect hidden attractors in nonlinear electronic circuits. Our results provide insights into the generation of hidden attractors in nonlinear dynamical systems.
RESUMEN
The influence of higher nervous activity on the processes of autonomic control of the cardiovascular system and baroreflex regulation is of considerable interest, both for understanding the fundamental laws of the functioning of the human body and for developing methods for diagnostics and treatment of pathologies. The complexity of the analyzed systems limits the possibilities of research in this area and requires the development of new tools. Earlier we propose a method for studying the collective dynamics of the processes of autonomic control of blood circulation in the awake state and in different stages of sleep. The method is based on estimating a quantitative measure representing the total percentage of phase synchronization between the low-frequency oscillations in heart rate and blood pressure. Analysis of electrocardiogram and invasive blood pressure signals in apnea patients in the awake state and in different sleep stages showed a high sensitivity of the proposed measure. It is shown that in slow-wave sleep the degree of synchronization of the studied rhythms is higher than in the awake state and lower than in sleep with rapid eye movement. The results reflect the modulation of the processes of autonomic control of blood circulation by higher nervous activity and can be used for the quantitative assessment of this modulation.
RESUMEN
In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as examples and the results are consistent with the predictions of theorems.
RESUMEN
In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration.
