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The multifractal critical phase (MCP) fundamentally differs from extended and localized phases, exhibiting delocalized distributions in both position and momentum spaces. The investigation on the MCP has largely focused on one-dimensional…
Many-variable differential equations with random coefficients provide powerful models for the dynamics of many interacting species in ecology. These models are known to exhibit a dynamical phase transition from a phase where population…
Following seminal work by J. Fr\"ohlich and T. Spencer on the critical exponent $\alpha=2$, we give a proof via contours of phase transition in the one-dimensional long-range ferromagnetic Ising model in the entire region of decay, where…
In this topical review, we present a brief overview of the different methods and measures to detect the occurrence of critical transitions in complex systems. We start by introducing the mechanisms that trigger critical transitions, and how…
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both…
Complex systems, such as the power grid, are essential for our daily lives. Many complex systems display (multi-)fractal behavior, correlated fluctuations and power laws. Whether the power-grid frequency, an indicator about the balance on…
Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal…
We propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wavefunction…
Current methods for determining whether a time series exhibits fractal structure (FS) rely on subjective assessments on estimators of the Hurst exponent (H). Here, I introduce the Bayesian Assessment of Scaling, an analytical framework for…
The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary…
We analyze the time series of the power loads of the 35 separated countries publicly sharing hourly data through ENTSO-E platform for more than 5 years. We apply the Multifractal Detrended Fluctuation Analysis for the demonstration of the…
Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks…
A generic saddle-node bifurcation is proposed to modelize fast transitions of finite amplitude arising in geophysical (and perhaps other) contexts, when they result from the intrinsic dynamics of the system. The fast transition is…
A variety of real world and experimental systems can display a drastic regime shift, as the evolution in one its paramaters crosses a threshold value. Assimilation of such a transition with a bifurcation has allowed to identify so called…
We present a comparative analysis of multifractal properties of financial time series built on stock indices from developing (WIG) and developed (S&P500) financial markets. It is shown how the multifractal image of the market is altered…
In the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest. Ageing phenomena are observed during this ever-lasting non-equilibrium evolution. A simple instance of such a behaviour is…
We first apply the WT-MFDFA, MFDFA, and WTMM multifractal methods to binomial multifractal time series of three different binomial parameters and find that the WTMM method indicates an enhanced difference between the fractal components than…
In many complex systems studied in statistical physics, inter-arrival times between events such as solar flares, trades and neuron voltages follow a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time…
Fault diagnosis in multimode processes plays a critical role in ensuring the safe operation of industrial systems across multiple modes. It faces a great challenge yet to be addressed - that is, the significant distributional differences…
Multivariate time series forecasting is widely used in various fields. Reasonable prediction results can assist people in planning and decision-making, generate benefits and avoid risks. Normally, there are two characteristics of time…