Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control

Quasiperiodic graphs: structural design, scaling and entropic properties

A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy

Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators

In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in Physics Please Lakshmanan for figures (e-mail: [email protected]

Quasiperiodic graphs at the onset of chaos

We examine the connectivity fluctuations across networks obtained when the horizontal visibility (HV) algorithm is used on trajectories generated by nonlinear circle maps at the quasiperiodic transition to chaos. The resultant HV graph is highly anomalous as the degrees fluctuate at all scales with amplitude that increases with the size of the network. We determine families of Pesin-like identities between entropy growth rates and generalized graph-theoretical Lyapunov exponents. An irrational winding number with pure periodic continued fraction characterizes each family. We illustrate our results for the so-called golden, silver and bronze numbers.Comment: arXiv admin note: text overlap with arXiv:1205.190

Hyper-chaotic magnetisation dynamics of two interacting dipoles

The present work is a numerical study of the deterministic spin dynamics of two interacting anisotropic magnetic particles in the presence of a time-dependent external magnetic field using the Landau–Lifshitz equation. Particles are coupled through the dipole–dipole interaction. The applied magnetic field is made of a constant longitudinal amplitude component and a time-dependent transversal amplitude component. Dynamical states obtained are represented by their Lyapunov exponents and bifurcation diagrams. The dependence on the largest and the second largest Lyapunov exponents, as a function of the magnitude and frequency of the applied magnetic field, and the relative distance between particles, is studied. The system presents multiple transitions between regular and chaotic behaviour depending on the control parameters. In particular, the system presents consistent hyper-chaotic states

Empirical exploration of air traffic and human dynamics in terminal airspaces

Air traffic is widely known as a complex, task-critical techno-social system, with numerous interactions between airspace, procedures, aircraft and air traffic controllers. In order to develop and deploy high-level operational concepts and automation systems scientifically and effectively, it is essential to conduct an in-depth investigation on the intrinsic traffic-human dynamics and characteristics, which is not widely seen in the literature. To fill this gap, we propose a multi-layer network to model and analyze air traffic systems. A Route-based Airspace Network (RAN) and Flight Trajectory Network (FTN) encapsulate critical physical and operational characteristics; an Integrated Flow-Driven Network (IFDN) and Interrelated Conflict-Communication Network (ICCN) are formulated to represent air traffic flow transmissions and intervention from air traffic controllers, respectively. Furthermore, a set of analytical metrics including network variables, complex network attributes, controllers' cognitive complexity, and chaotic metrics are introduced and applied in a case study of Guangzhou terminal airspace. Empirical results show the existence of fundamental diagram and macroscopic fundamental diagram at the route, sector and terminal levels. Moreover, the dynamics and underlying mechanisms of "ATCOs-flow" interactions are revealed and interpreted by adaptive meta-cognition strategies based on network analysis of the ICCN. Finally, at the system level, chaos is identified in conflict system and human behavioral system when traffic switch to the semi-stable or congested phase. This study offers analytical tools for understanding the complex human-flow interactions at potentially a broad range of air traffic systems, and underpins future developments and automation of intelligent air traffic management systems.Comment: 30 pages, 28 figures, currently under revie

Entropies for severely contracted configuration space

We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical-mechanical systems that experience an exceptional contraction of their configuration space. The entropic index $\alpha>1$ describes the contraction process, while the dual index $\alpha ^{\prime }=2-\alpha<1$ defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place