Chaos & Oscilations I

Delayed differential equations (DDEs) are class of dynamical systems in which the current states strongly depend on their previous states. They received considerable attention due to their rich dynamical properties such as hyperchaotic attractors. One of the most useful models is the Lang-Kobayashi (LK) equations for a semiconductor laser with optical feedback. The chaotic dynamics of semiconductor laser have been exploited for applications such as random bit generation [1].

The dynamics of the solar magnetic field can be studied using the mean-field dynamo theory, which is based on the approximation of two scales, which suggests that the magnetic field consists of a large-scale field with small-scale fluctuations. The large-scale field can be generated by the α2 effect, related to the fluid's kinetic helicity. In the dynamo model, the α2 effect is responsible for the regeneration of both the poloidal and toroidal components of the field.

In nature, there is a vast variety of systems that cannot be modeled by the same set of equations and parameters as time passes. This may be caused either by the contact with its environment or due to internal factors. In this talk, we present the study of dynamical systems subjected to parameter drift and its implications in their evolution. For that purpose, we show the analysis of two multi-stable systems with chaotic attractors: the Lorenz system and the time-delayed Duffing oscillator. In the first case, the drift is contained in the Rayleigh number.

Identifying key players in coupled individual systems is a fundamental problem in network theory. We investigate synchronizable network-coupled dynamical systems such as high-voltage electric power grids and coupled oscillators on complex networks. We define key players as nodes that, once perturbed, generate the largest excursion away from synchrony. A spectral decomposition of the coupling matrix gives an elegant solution to this identification problem.

Labyrinth chaos was discovered by Otto Rossler and Rene Thomas in their endeavour to identify the necessary mathematical conditions for the appearance of chaotic and hyperchaotic motion in continuous flows.
This is a generic and minimal model of a dynamical system which turned out that, even though it is simple, it is full of surprising properties. Simple and elegant as it is, it still holds great promise for elucidating aspects of chaotic dynamics that are not evident in other systems.