91 research outputs found
Complex statistics in Hamiltonian barred galaxy models
We use probability density functions (pdfs) of sums of orbit coordinates,
over time intervals of the order of one Hubble time, to distinguish weakly from
strongly chaotic orbits in a barred galaxy model. We find that, in the weakly
chaotic case, quasi-stationary states arise, whose pdfs are well approximated
by -Gaussian functions (with ), while strong chaos is identified by
pdfs which quickly tend to Gaussians (). Typical examples of weakly
chaotic orbits are those that "stick" to islands of ordered motion. Their
presence in rotating galaxy models has been investigated thoroughly in recent
years due of their ability to support galaxy structures for relatively long
time scales. In this paper, we demonstrate, on specific orbits of 2 and 3
degree of freedom barred galaxy models, that the proposed statistical approach
can distinguish weakly from strongly chaotic motion accurately and efficiently,
especially in cases where Lyapunov exponents and other local dynamic indicators
appear to be inconclusive.Comment: 14 pages, 9 figures, submitted for publicatio
Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
The recently introduced GALI method is used for rapidly detecting chaos,
determining the dimensionality of regular motion and predicting slow diffusion
in multi--dimensional Hamiltonian systems. We propose an efficient computation
of the GALI indices, which represent volume elements of randomly chosen
deviation vectors from a given orbit, based on the Singular Value Decomposition
(SVD) algorithm. We obtain theoretically and verify numerically asymptotic
estimates of GALIs long--time behavior in the case of regular orbits lying on
low--dimensional tori. The GALI indices are applied to rapidly detect
chaotic oscillations, identify low--dimensional tori of Fermi--Pasta--Ulam
(FPU) lattices at low energies and predict weak diffusion away from
quasiperiodic motion, long before it is actually observed in the oscillations.Comment: 10 pages, 5 figures, submitted for publication in European Physical
Journal - Special Topics. Revised version: Small explanatory additions to the
text and addition of some references. A small figure chang
Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
In this paper we consider a family of dynamical systems that we call
"arabesques", defined as closed chains of 2-element negative circuits. An
-dimensional arabesque system has 2-element circuits, but in addition,
it displays by construction, two -element circuits which are both positive
vs one positive and one negative, depending on the parity (even or odd) of the
dimension . In view of the absence of diagonal terms in their Jacobian
matrices, all these dynamical systems are conservative and consequently, they
can not possess any attractor. First, we analyze a linear variant of them which
we call "arabesque 0" or for short "A0". For increasing dimensions, the
trajectories are increasingly complex open tori. Next, we inserted a single
cubic nonlinearity that does not affect the signs of its circuits (that we call
"arabesque 1" or for short "A1"). These systems have three steady states,
whatever the dimension is, in agreement with the order of the nonlinearity. All
three are unstable, as there can not be any attractor in their state-space. The
3D variant (that we call for short "A1\_3D") has been analyzed in some detail
and found to display a complex mixed set of quasi-periodic and chaotic
trajectories. Inserting cubic nonlinearities (one per equation) in the same
way as above, we generate systems "A2\_D". A2\_3D behaves essentially as
A1\_3D, in agreement with the fact that the signs of the circuits remain
identical. A2\_4D, as well as other arabesque systems with even dimension, has
two positive -circuits and nine steady states. Finally, we investigate and
compare the complex dynamics of this family of systems in terms of their
symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao
Cause-effect relationship between model parameters and damping performance of hydraulic shock absorbers
Despite long-term research and development of modern shock absorbers, the
effect of variations of several crucial material and model parameters still
remains dubious. The goal of this work is therefore a study of the changes of
shock absorber dynamics with respect to typical parameter ranges in a realistic
model. We study the impact of shim properties, as well as geometric features
such as discharge coefficients and bleed orifice cross section. We derive
cause-effect relationships by nonlinear parameter fitting of the differential
equations of the model and show digressive and progressive quadratic damping
curves for shim number and thickness, sharp exponential curves for discharge
coefficients, and leakage width, as well as a linear decrease of damping
properties with bleed orifice area. Temperature increase affecting material
properties, such as density and viscosity of the mineral oil, is found to have
a mostly linear relationship with damping and pressure losses. Our results are
not only significant for the general understanding of shock absorber dynamics,
but also serve as a guidance for the development of specific models by
following the proposed methodology
Nonlinear dynamics and onset of chaos in a physical model of a damper pressure relief valve
Hydraulic valves, for the purpose of targeted pressure relief and damping,
are a ubiquitous part of modern suspension systems. This paper deals with the
analytical computation of fixed points of the dynamical system of a
single-stage shock absorber valve, which can be applied for the direct
computation of its system variables at equilibrium without brute-force
numerical integration. The obtained analytical expressions are given for the
original dimensional version of the model derived from continuity and motion
equations for a realistic valve setup. Furthermore, a large part of the study
is devoted to a systematic sensitivity analysis of the model towards crucial
parameter changes. Numerical investigation of a potential loss of stability and
following nonlinear oscillations is performed in multi-dimensional parameter
spaces, which reveals sustained valve vibrations at increased valve mass and
applied pretension force. The dynamical behaviour is analysed by phase space
orbits, as well as Fourier-transformed valve displacement data to identify
dominant frequencies. Chaotic indicators, such as Lyapunov exponents and the
Smaller Alignment Index (SALI), are utilized to understand the nature of the
oscillatory motion and to distinguish between order and chaos
Network inference combining mutual information rate and statistical tests
In this paper, we present a method that combines information-theoretical and statistical approaches to infer connec- tivity in complex networks using time-series data. The method is based on estimations of the Mutual Information Rate for pairs of time-series and on statistical significance tests for connectivity acceptance using the false discovery rate method for multiple hypothesis testing. We provide the mathematical background on Mutual Information Rate, discuss the statistical significance tests and the false discovery rate. Further on, we present results for corre- lated normal-variates data, coupled circle and coupled logistic maps, coupled Lorenz systems and coupled stochastic Kuramoto phase oscillators. Following up, we study the effect of noise on the presented methodology in networks of coupled stochastic Kuramoto phase oscillators and of coupling heterogeneity degree on networks of coupled circle maps. We show that the method can infer the correct number and pairs of connected nodes, by means of receiver operating characteristic curves. In the more realistic case of stochastic data, we demonstrate its ability to infer the structure of the initial connectivity matrices. The method is also shown to recover the initial connectivity matrices for dynamics on the nodes of Erd ̋os-R ́enyi and small-world networks with varying coupling heterogeneity in their connections. The highlight of the proposed methodology is its ability to infer the underlying network connectivity based solely on the recorded datasets
Labyrinth chaos: Revisiting the elegant, chaotic, and hyperchaotic walks
Labyrinth chaos was discovered by Otto Rössler and René Thomas in their endeavour to identify the necessary mathematical conditions for the appearance of chaotic and hyperchaotic motion in continuous flows. Here, we celebrate their discovery by considering a single labyrinth walks system and an array of coupled labyrinth chaos systems that exhibit complex, chaotic behaviour, reminiscent of chimera-like states, a peculiar synchronisation phenomenon. We discuss the properties of the single labyrinth walks system and review the ability of coupled labyrinth chaos systems to exhibit chimera-like states due to the unique properties of their space-filling, chaotic trajectories, what amounts to elegant, hyperchaotic walks. Finally, we discuss further implications in relation to the labyrinth walks system by showing that even though it is volume-preserving, it is not force-conservative
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