A generalized Landau scenario for the emergence of complex oscillations in N-dimensional dynamical systems
- Gaspar Orriols
- Depto. Física, Universitat Autònoma de Barcelona
- Jan. 31, 2003, 3 p.m.
- Sala de seminarios IMEDEA, Esporles
We will present experimental and numerical results showing how certain classes of dynamical systems are able to describe complex time evolutions based on the nonlinear superposition of oscillations generated through Hopf bifurcations. The mechanism represents a way for the emergence of irregularity and complexity independent of chaos, and it evokes the scenario proposed by Landau for tentatively explaining the onset of turbulence.
The experiments have been done with a family of opto-thermal devices of effective dimension varying from 1 to 6. The mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that, in its turn, is a linear combination of all the dynamic variables.
A generic mechanism of nonlinear mode mixing occurs through which the attractor incorporates localized helical motions related to the influence of neighboring saddles and, in this way, the observed time dynamics describes an irregular succession of oscillatory trains based on the N-1 characteristic frequencies. The nonlinear mechanism produces robust complex waveforms exhibiting similarity with respect to both time scale and system dimension.
We suspect that this kind of behavior may be relevant for certain real-world systems where a number of oscillatory processes with different time scales and strong interrelations coexist together. For instance, we think on the onset of turbulence in fluids and the background activity of a brain.