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We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that…
Computer assisted procedures of Lyapunov functions defined in given neighborhoods of fixed points for flows and maps are discussed. We provide a systematic methodology for constructing explicit ranges where quadratic Lyapunov functions…
Today's most popular CPU can operate in two different computational modes for double precision computations. This fact is not very widely recognized among scientific computer users. The present paper reports the differences the modes bring…
The logistic map is a paradigmatic dynamical system originally conceived to model the discrete-time demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics.…
Physical systems behave according to their underlying dynamical equations which, in turn, can be identified from experimental data. Explaining data requires selecting mathematical models that best capture the data regularities. Identifying…
We describe algorithms and data structures to extend a neural network library with automatic precision estimation for floating point computations. We also discuss conditions to make estimations exact and preserve high computation…
Future architectures designed to deliver exascale performance motivate the need for novel algorithmic changes in order to fully exploit their capabilities. In this paper, the performance of several numerical algorithms, characterised by…
It is widely known that numerically integrated orbits are more precise than analytical theories for celestial bodies. However, calculation of the positions of celestial bodies via numerical integration at time $t$ requires the amount of…
We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour…
A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…
We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on…
We propose a flexible gradient tracking approach with adjustable computation and communication steps for solving distributed stochastic optimization problem over networks. The proposed method allows each node to perform multiple local…
This study evaluates data-driven models from a dynamical system perspective, such as unstable fixed points, periodic orbits, chaotic saddle, Lyapunov exponents, manifold structures, and statistical values. We find that these dynamical…
In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss…
In this paper, a computationally efficient data-driven hybrid automaton model is proposed to capture unknown complex dynamical system behaviors using multiple neural networks. The sampled data of the system is divided by valid partitions…
In this work, we investigate scale invariance in the temporal evolution and chaotic regime of discrete dynamical systems. By exploiting the close interrelation between scaling and inversion transformations, we formulate scale symmetry in…
Many problems in computer science and applied mathematics require rounding a vector $\mathbf{w}$ of fractional values lying in the interval $[0,1]$ to a binary vector $\mathbf{x}$ so that, for a given matrix $\mathbf{A}$,…
A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the…
In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…