A new delay equation is introduced to describe the punctuated evolution of complex nonlinear systems. A detailed analytical and numerical investigation provides the classification of all possible types of solutions for the dynamics of a population in the four main regimes dominated respectively...

A theory of exceptional extreme events, characterized by their abnormal sizes compared with the rest of the distribution, is presented. Such outliers, called "dragon-kings'', have been reported in the distribution of financial drawdowns, city-size distributions (e.g., Paris in France and London...

We suggest a novel approach to treating symbiotic relations between biological species or social entities. The main idea is the characterisation of symbiotic relations of coexisting species through their mutual influence on their respective carrying capacities, taking into account that this...

We present a general classification of the conditions under which cognitive science, concerned e.g. with decision making, requires the use of quantum theoretical notions. The analysis is done in the frame of the mathematical approach based on the theory of quantum measurements. We stress that...

One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a quot;Quantum Decision Theoryquot; of decision making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of...

We give a mathematical definition for the notion of inconclusive quantum measurements. In physics, such measurements occur at intermediate stages of a complex measurement procedure, with the final measurement result being operationally testable. Since the mathematical structure of Quantum...

The problem of defining quantum probabilities of composite events for quantum decision theory and theory of quantum measurements is considered. These theories not merely are closely related with each other, but decision theory, actually, can be treated as a part of measurement theory. We show...

We consider a class of evolution equations describing population dynamics in the presence of a carrying capacity depending on the population with delay. In an earlier work, we presented an exhaustive classification of the logistic equation where the carrying capacity is linearly dependent on the...

The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of...

A novel approach to analyzing time series generated by complex systems, such as markets, is presented. The basic idea of the approach is the Law of Self-Similar Evolution, according to which any complex system develops self-similarly. There always exist some internal laws governing the evolution...