Search results for "Chaotic dynamics"
showing 10 items of 197 documents
Numerical analysis of dynamical systems : unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimens…
2019
In this article, on the example of the known low-order dynamical models, namely Lorenz, Rössler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rössler system. Using the example of the Vallis system describing the El…
Hidden attractors in dynamical systems
2016
Complex dynamical systems, ranging from the climate, ecosystems to financial markets and engineering applications typically have many coexisting attractors. This property of the system is called multistability. The final state, i.e., the attractor on which the multistable system evolves strongly depends on the initial conditions. Additionally, such systems are very sensitive towards noise and system parameters so a sudden shift to a contrasting regime may occur. To understand the dynamics of these systems one has to identify all possible attractors and their basins of attraction. Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors whi…
On lower-bound estimates of the Lyapunov dimension and topological entropy for the Rossler systems
2019
In this paper, on the example of the Rössler systems, the application of the Pyragas time-delay feedback control technique for verification of Eden’s conjecture on the maximum of local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated. To this end, numerical experiments on computation of finite-time local Lyapunov dimensions and finite-time topological entropy on a Rössler attractor and embedded unstable periodic orbits are performed. The problem of reliable numerical computation of the mentioned dimension-like characteristics along the trajectories over large time intervals is discussed. peerReviewed
Semipredictable dynamical systems
2015
A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with $p$ possible dynamical states can be decomposed at each instant of time in a superposition of $N$ layers involving $p_{0}$, $p_{1}$,... $p_{N-1}$ dynamical states each, where the $p_{k\in \mathbb{N}}$, $k \in [0, N-1]$ are divisors of $p$. If the divisors coincide with the prime factors of $p$ this decomposition is unique. Conversely, we also prove that $N$ CA w…
"Table 1" of "Comprehensive measurements of $t$-channel single top-quark production cross sections at $\sqrt{s} = 7$ TeV with the ATLAS detector"
2014
Predicted and observed events yields for the 2-jet and 3-jet channels considered in this measurement. The multijet background is estimated using data-driven techniques (see Sec. VB); an uncertainty of $50\%$ is applied. All the other expectations are derived using theoretical cross sections and their uncertainties (see Secs. VA and VC in the paper).
Coexistence of periods in a bifurcation
2012
Abstract A particular type of order-to-chaos transition mediated by an infinite set of coexisting neutrally stable limit cycles of different periods is studied in the Varley–Gradwell–Hassell population model. We prove by an algebraic method that this kind of transition can only happen for a particular bifurcation parameter value. Previous results on the structure of the attractor at the transition point are here simplified and extended.
Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes1
1998
In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoret…
Transitions in a stratified Kolmogorov flow
2016
We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is $$Ri=10^{-5}$$ , and restrict our analysis to the range $$0<Re<30$$ . The effect of the stabilizing temperature is to shift bifurcation points and to reduce the region of existence of stable drifting states. The…
Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
2001
We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques: the frequency map analysis and renormalization-group methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverg…
Intermittent-Type Chaos in Nonsinusoidal Driven Oscillators
2000
The intermittent-type chaos occurring in rf- and dc- nonsinusoidal driven oscillators is investigated analytically and numerically. The attention is focused on a general class of oscillators in which the total potential VRP(,r) is the Remoissenet-Peyrard potential which has constant amplitude and is 2π-periodic in , and whose shape can be varied as a function of parameter r ( |r| < 1). A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. The parameter regions of chaotic behaviour predicted by the theore…