Related papers: Measuring Complexity in Cantor Dynamics
We consider Markov decision processes (MDPs) which are a standard model for probabilistic systems. We focus on qualitative properties for MDPs that can express that desired behaviors of the system arise almost-surely (with probability 1) or…
Dynamic Complexity is a phenomenon exhibited by a nonlinearly interacting system within which multitudes of different sizes of large scale coherent structures emerge, resulting in a globally nonlinear stochastic behavior vastly different…
We consider the basic features of complex dynamic and control systems, including systems having hierarchical structure. Special attention is paid to the problems of design and synthesis of complex systems and control models, and to the…
In this paper, we present some results on information, complexity and entropy as defined below and we discuss their relations with the Kolmogorov-Sinai entropy which is the most important invariant of a dynamical system. These results have…
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…
The problem of defining and studying complexity of a time series has interested people for years. In the context of dynamical systems, Grassberger has suggested that a slow approach of the entropy to its extensive asymptotic limit is a sign…
Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the…
We study the build up of complexity on the example of 1 kg matter in different forms. We start on the simplest example of ideal gases, and then continue with more complex chemical, biological, life and social and technical structures. We…
For any dynamical system $T:X\rightarrow X$ of a compact metric space $X$ with $g-$almost product property and uniform separation property, under the assumptions that the periodic points are dense in $X$ and the periodic measures are dense…
We consider models with topological sectors, and difficulties with their Monte Carlo simulation. In particular we are concerned with the situation where a simulation has an extremely long auto-correlation time with respect to the…
Some aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a…
We propose a measure of learning efficiency for non-finite state spaces. We characterize the complexity of a learning problem by the metric entropy of its state space. We then describe how learning efficiency is determined by this measure…
We present exact results for two complementary measures of spatial structure generated by 1D spin systems with finite-range interactions. The first, excess entropy, measures the apparent spatial memory stored in configurations. The second,…
We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence:…
Complex systems are characterized by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behavior. Examples arise both in the physical and non-physical…
We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a…
Hamiltonian systems that are either open, leaking, or contain holes in the phase space possess solutions that eventually escape the system's domain. The motion described by such escape orbits before crossing the escape threshold can be…
Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with…
We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions--approximability and distinguishability. Built upon…
This chapter presents a brief review of complexity research in mathematics education. We argue how research on complexity, as it pertains to mathematics education, can be viewed as an epistemological discourse, an historical discourse, a…