Rate-induced critical transitions

This thesis focuses on rate-induced critical transitions or tipping points (R-tipping points), where the system undergoes a critical transition if the time-varying external conditions vary faster than some critical rate. Such a critical transition is usually a sudden and unexpected change of the system state. The change can be either irreversible: a permanent tipping point with no return to the original state, or reversible: a temporary tipping point with self-recovery back to the original state, both of which may cause significant consequences in applications. Indeed, R-tipping is an ubiquitous nonlinear phenomenon in nature that remains largely unexplored by the scientists. From a mathematical viewpoint, it is a genuine nonautonomous instability that cannot be explained by the classical (autonomous) bifurcation theory and requires an alternative approach. The first part of the thesis focuses on a mathematical framework for R-tipping in systems of nonautonomous differential equations, where the nonautonomous terms representing time-varying external conditions decay asymptotically. In particular, special compactification techniques for asymptotically autonomous systems are developed to simplify analysis of R-tipping. In the second part of the thesis, the main concepts of edge states and thresholds are introduced to define the R-tipping phenomenon. Then, simple testable criteria for the occurrence of reversible and irreversible R-tipping in arbitrary dimension are given. This part extends the previous results on irreversible R-tipping in one dimension. The third part of the thesis identifies canonical examples of R-tipping based on the system dimension, timescales and the threshold type. These examples are relatively simple low-dimensional nonlinear systems that capture different R-tipping mechanisms. R-tipping analysis of canonical examples, which is underpinned by the compactification framework developed in the second part, reveals intricate R-tipping diagrams with multiple critical rates and transitions between different types of R-tipping

In-plane noncollinear exchange coupling mediated by helical edge states in Quantum Spin Hall system

We study the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction mediated by helical edge states in quantum spin hall system. The helical edge states induce an in-plane noncollinear exchange coupling between two local spins, in contrast to the isotropic coupling induced in normal metal. The angle between the two local spins in the ground state depends on the Fermi level. This property may be used to control the angle of spins by tuning the electric gate.Comment: 4 pages, 1 figur

Fractal model and Lattice Boltzmann Method for Characterization of Non-Darcy Flow in Rough Fractures.

The irregular morphology of single rock fracture significantly influences subsurface fluid flow and gives rise to a complex and unsteady flow state that typically cannot be appropriately described using simple laws. Yet the fluid flow in rough fractures of underground rock is poorly understood. Here we present a numerical method and experimental measurements to probe the effect of fracture roughness on the properties of fluid flow in fractured rock. We develop a series of fracture models with various degrees of roughness characterized by fractal dimensions that are based on the Weierstrass-Mandelbrot fractal function. The Lattice Boltzmann Method (LBM), a discrete numerical algorithm, is employed for characterizing the complex unsteady non-Darcy flow through the single rough fractures and validated by experimental observations under the same conditions. Comparison indicates that the LBM effectively characterizes the unsteady non-Darcy flow in single rough fractures. Our LBM model predicts experimental measurements of unsteady fluid flow through single rough fractures with great satisfactory, but significant deviation is obtained from the conventional cubic law, showing the superiority of LBM models of single rough fractures