Related papers: Rigorous numerics for critical orbits in the quadr…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically…
We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which the diffeomorphisms have arbitrarily large finite numbers of…
We consider the family of piecewise linear maps $F(x,y)=\left(|x| - y + a, x - |y| + b\right),$ where $(a,b)\in \R^2$. In previous work, we identified a novel phenomenon: certain maps of this class possess one-dimensional invariant sets,…
Periodic orbit theory allows calculations of long time properties of chaotic systems from traces, dynamical zeta functions and spectral determinants of deterministic evolution operators, which are in turn evaluated in terms of periodic…
The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones…
An alternative numerical method is developed to find stable and unstable periodic orbits of nonlinear dynamical systems. The method exploits the high-efficiency of the Levenberg-Marquardt algorithm for medium-sized problems and has the…
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long-term behavior. It is shown that a very general class of decision problems regarding…
By varying a parameter of a one-dimensional piecewise smooth map, stable periodic orbits are observed. In this paper, complete analytic characterization of these stable periodic orbits is obtained. An interesting relationship between the…
We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in…
We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(\lambda)$ for $\lambda$ in $\mathbb{C}$, defined by the condition that each $f\in…
This paper deals with the period function of the reversible quadratic centers \begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y, \end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to $\Sc^2$, the…
Considering the problem of risk-sensitive parameter estimation, we propose a fairly wide family of lower bounds on the exponential moments of the quadratic error, both in the Bayesian and the non--Bayesian regime. This family of bounds,…
An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular…
We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance…
We present a detailed algorithm to construct symbolic encodings for chaotic attractors of three-dimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed…
The distribution of orbital period ratios between adjacent observed exoplanets is approximately uniform, but exhibits a strong falloff toward close orbital separations. We show that this falloff can be explained through past dynamical…
This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safety-critical software, as these…
Bifurcation analysis collects techniques for characterizing the dependence of certain classes of solutions of a dynamical system on variations in problem parameters. Common solution classes of interest include equilibria and periodic…
The planetary dynamics of $4/3$, $3/2$, $5/2$, $3/1$ and $4/1$ mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance.…