Topology of multiplex heterogeneous networks of Hodgkin-Huxley-type of models with bistability leading to stabilization stable equilibrium

The dynamics of a multiplex heterogeneous networks of oscillators is studied. Two types of very similar models based on the Hodgkin-Huxley formalism are used as the basic elements of the network: the first one demonstrates bursting oscillations; the second one manifests bistability between bursting oscillations and stable equilibrium. Multiplex networks were developed and investigated, assuming more active communication between models with bistability. Different topologies of the networks are studied. It is shown that in this case it is enough to have one element with bistability in the subnetworks in order to stabilize the equilibrium state in the entire network.

Hidden and self-excited attractors in radiophysical and biophysical models

One of the central tasks of investigation of dynamical systems is the problem of analysis of the steady (limiting) behavior of the system after the completion of transient processes, i.e., the problem of localization and analysis of attractors (bounded sets of states of the system to which the system tends after transient processes from close initial states). Transition of the system with initial conditions from the vicinity of stationary state to an attractor corresponds to the case of a self-excited attractor. However, there exist attractors of another type: hidden attractors are attractors with the basin of attraction which does not have intersection with a small neighborhoods of any equ…

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

AbstractAfter the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real exi…

Synchronization of hidden chaotic attractors on the example of radiophysical oscillators

In the present paper we consider the problem of synchronization of hidden and self-excited attractors in the context of application to a system of secure communication. The system of two coupled Chua models was studied. Complete synchronization was observed as for self-excited, as hidden attractors. Beside it for hidden attractors some special type of dynamic was revealed.

Scenario of the Birth of Hidden Attractors in the Chua Circuit

Recently it was shown that in the dynamical model of Chua circuit both the classical selfexcited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of selfexcited and hidden attractors is studied. It is shown a pitchfork bifurcation in which a pair of symmetric attractors coexists and merges into one symmetric attractor through an attractormerging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

Mixed-mode oscillation-incrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer-van der Pol oscillator