Related papers: On the Complexity of some Geometrical Objects
We propose a theory of chaos for discrete systems, based on their representation in a space of ``binary histories'', $ {\cal B^{\infty}} $. We show that $ {\cal B^{\infty}} $ is a metrizable Cantor set which embeds the attractor $\Lambda$,…
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of…
Volume of metric balls relates to rate-distortion theory and packing bounds on codes. In this paper, the volume of balls in complex Grassmann manifolds is evaluated for an arbitrary radius. The ball is defined as a set of hyperplanes of a…
We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow…
We show that the derivative of the (measure) transfer operator with respect to the parameter of the map is a divergence. Then, for physical measures of discrete-time hyperbolic chaotic systems, we derive an equivariant divergence formula…
We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
Mixed-monotone systems are separable via a decomposition function into increasing and decreasing components, and this decomposition function allows for embedding the system dynamics in a higher-order monotone embedding system. Embedding the…
Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less…
The framework of a new scale invariant analysis on a Cantor set $C\subset $ $% I=[0,1] $, presented originally in {\it S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009)}, is clarified and extended further. For an arbitrarily small…
We describe a numerical technique to compute the equilibrium measure, in logarithmic potential theory, living on the attractor of Iterated Function Systems composed of one-dimensional affine maps. This measure is obtained as the limit of a…
We extend the notion of estimation entropy of autonomous dynamical systems proposed by Liberzon and Mitra [1] to nonlinear dynamical systems with uncertain inputs with bounded variation. We call this new notion the {$\epsilon$}-estimation…
The coupling complexity index is an information measure introduced within the framework of ordinal symbolic dynamics. This index is used to characterize the complexity of the relationship between dynamical system components. In this work,…
A method to define the complex structure and separate the conformal mode is proposed for a surface constructed by two-dimensional dynamical triangulation. Applications are made for surfaces coupled to matter fields such as $n$ scalar fields…
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to arbitrary fine resolution and arbitrary high…
We consider the number of Bowen sets which are necessary to cover a large measure subset of the phase space. This introduce some complexity indicator characterizing different kind of (weakly) chaotic dynamics. Since in many systems its…
We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in in our revious publications where computability, in the rigorous sense of…
We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical…
We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity…