A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

This is the author accepted manuscript. The final version is available from IOP Publishing via the DOI in this recordThis paper studies a novel class of two-dimensional maps with infinitely many coexisting attractors. Firstly, the mathematical model of these maps is formulated by introducing a sinusoidal function. The existence and the stabilities of the fixed points in the model are studied indicating that they are infinitely many and all unstable. In particular, a computer searching program is employed to explore the chaotic attractors in these maps, and a simple map is exemplified to show their complex dynamics. Interestingly, this map contains infinitely many coexisting attractors which has been rarely reported in the past literature. Further studies of these coexisting attractors are carried out by investigating their time histories, phase trajectories, basins of attraction, Lyapunov exponents spectrum and Lyapunov (Kaplan-Yorke) dimension. Bifurcation analysis reveals that the map has periodic and chaotic solutions, and more importantly, exhibits extreme multi-stability.National Natural Science Foundation of ChinaNatural Science Foundation of Jiangsu Province of China5th 333 High-level Personnel Training Project of Jiangsu Province of ChinaExcellent Scientific and Technological Innovation Team of Jiangsu Universit

Modeling phase synchronization of interacting neuronal populations:from phase reductions to collective behavior of oscillatory neural networks

Synchronous, coherent interaction is key for the functioning of our brain. The coordinated interplay between neurons and neural circuits allows to perceive, process and transmit information in the brain. As such, synchronization phenomena occur across all scales. The coordination of oscillatory activity between cortical regions is hypothesized to underlie the concept of phase synchronization. Accordingly, phase models have found their way into neuroscience. The concepts of neural synchrony and oscillations are introduced in Chapter 1 and linked to phase synchronization phenomena in oscillatory neural networks. Chapter 2 provides the necessary mathematical theory upon which a sound phase description builds. I outline phase reduction techniques to distill the phase dynamics from complex oscillatory networks. In Chapter 3 I apply them to networks of weakly coupled Brusselators and of Wilson-Cowan neural masses. Numerical and analytical approaches are compared against each other and their sensitivity to parameter regions and nonlinear coupling schemes is analysed. In Chapters 4 and 5 I investigate synchronization phenomena of complex phase oscillator networks. First, I study the effects of network-network interactions on the macroscopic dynamics when coupling two symmetric populations of phase oscillators. This setup is compared against a single network of oscillators whose frequencies are distributed according to a symmetric bimodal Lorentzian. Subsequently, I extend the applicability of the Ott-Antonsen ansatz to parameterdependent oscillatory systems. This allows for capturing the collective dynamics of coupled oscillators when additional parameters influence the individual dynamics. Chapter 6 draws the line to experimental data. The phase time series of resting state MEG data display large-scale brain activity at the edge of criticality. After reducing neurophysiological phase models from the underlying dynamics of Wilson-Cowan and Freeman neural masses, they are analyzed with respect to two complementary notions of critical dynamics. A general discussion and an outlook of future work are provided in the final Chapter 7

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory