310 research outputs found
Visualizing the Template of a Chaotic Attractor
Chaotic attractors are solutions of deterministic processes, of which the
topology can be described by templates. We consider templates of chaotic
attractors bounded by a genus-1 torus described by a linking matrix. This
article introduces a novel and unique tool to validate a linking matrix, to
optimize the compactness of the corresponding template and to draw this
template. The article provides a detailed description of the different
validation steps and the extraction of an order of crossings from the linking
matrix leading to a template of minimal height. Finally, the drawing process of
the template corresponding to the matrix is saved in a Scalable Vector Graphics
(SVG) file.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
An analysis of chaos via contact transformation
Transition from chaotic to quasi-periodic phase in modified Lorenz model is
analyzed by performing the contact transformation such that the trajectory in
{\Vec R}^3 is projected on {\Vec R}^2. The relative torsion number and the
characteristics of the template are measured using the eigenvector of the
Jacobian instead of vectors on moving frame along the closed trajectory.
Application to the circulation of a fluid in a convection loop and
oscillation of the electric field in single-mode laser system are performed.
The time series of the eigenvalues of the Jacobian and the scatter plot of the
trajectory in the transformed coordinate plane in the former and
in the latter, allow to visualize characteristic pattern change at
the transition from quasi-periodic to chaotic. In the case of single mode
laser, we observe the correlation between the critical movement of the
eigenvalues of the Jacobian in the complex plane and intermittency.Comment: 20 pages, 24 eps figures, 2 gif figures, use elsart.cls, accepted for
publication in Chaos,Solitons & Fractals(2003
Reduction of continuous symmetries of chaotic flows by the method of slices
We study continuous symmetry reduction of dynamical systems by the method of
slices (method of moving frames) and show that a `slice' defined by minimizing
the distance to a single generic `template' intersects the group orbit of every
point in the full state space. Global symmetry reduction by a single slice is,
however, not natural for a chaotic / turbulent flow; it is better to cover the
reduced state space by a set of slices, one for each dynamically prominent
unstable pattern. Judiciously chosen, such tessellation eliminates the singular
traversals of the inflection hyperplane that comes along with each slice, an
artifact of using the template's local group linearization globally. We compute
the jump in the reduced state space induced by crossing the inflection
hyperplane. As an illustration of the method, we reduce the SO(2) symmetry of
the complex Lorenz equations.Comment: to appear in "Comm. Nonlinear Sci. and Numer. Simulat. (2011)" 12
pages, 8 figure
Cartography of high-dimensional flows: A visual guide to sections and slices
Symmetry reduction by the method of slices quotients the continuous
symmetries of chaotic flows by replacing the original state space by a set of
charts, each covering a neighborhood of a dynamically important class of
solutions, qualitatively captured by a `template'. Together these charts
provide an atlas of the symmetry-reduced `slice' of state space, charting the
regions of the manifold explored by the trajectories of interest. Within the
slice, relative equilibria reduce to equilibria and relative periodic orbits
reduce to periodic orbits. Visualizations of these solutions and their unstable
manifolds reveal their interrelations and the role they play in organizing
turbulence/chaos.Comment: 12 Pages, 12 figure
Continuous symmetry reduction and return maps for high-dimensional flows
We present two continuous symmetry reduction methods for reducing
high-dimensional dissipative flows to local return maps. In the Hilbert
polynomial basis approach, the equivariant dynamics is rewritten in terms of
invariant coordinates. In the method of moving frames (or method of slices) the
state space is sliced locally in such a way that each group orbit of
symmetry-equivalent points is represented by a single point. In either
approach, numerical computations can be performed in the original state-space
representation, and the solutions are then projected onto the symmetry-reduced
state space. The two methods are illustrated by reduction of the complex Lorenz
system, a 5-dimensional dissipative flow with rotational symmetry. While the
Hilbert polynomial basis approach appears unfeasible for high-dimensional
flows, symmetry reduction by the method of moving frames offers hope.Comment: 32 pages, 7 figure
Revealing the state space of turbulent pipe flow by symmetry reduction
Symmetry reduction by the method of slices is applied to pipe flow in order
to quotient the stream-wise translation and azimuthal rotation symmetries of
turbulent flow states. Within the symmetry-reduced state space, all travelling
wave solutions reduce to equilibria, and all relative periodic orbits reduce to
periodic orbits. Projections of these solutions and their unstable manifolds
from their -dimensional symmetry-reduced state space onto suitably
chosen 2- or 3-dimensional subspaces reveal their interrelations and the role
they play in organising turbulence in wall-bounded shear flows. Visualisations
of the flow within the slice and its linearisation at equilibria enable us to
trace out the unstable manifolds, determine close recurrences, identify
connections between different travelling wave solutions, and find, for the
first time for pipe flows, relative periodic orbits that are embedded within
the chaotic attractor, which capture turbulent dynamics at transitional
Reynolds numbers.Comment: 24 pages, 12 figure
Phase Space Analysis of Cardiac Spectra
Cardiac diseases are one of the main reasons of mortality in modern,
industrialized societies, and they cause high expenses in public health
systems. Therefore, it is important to develop analytical methods to improve
cardiac diagnostics. Electric activity of heart was first modeled by using a
set of nonlinear differential equations. Latter, variations of cardiac spectra
originated from deterministic dynamics are investigated. Analyzing the power
spectra of a normal human heart presents His-Purkinje network, possessing a
fractal like structure. Phase space trajectories are extracted from the time
series graph of ECG. Lower values of fractal dimension, D indicate dynamics
that are more coherent. If D has non-integer values greater than two when the
system becomes chaotic or strange attractor. Recently, the development of a
fast and robust method, which can be applied to multichannel physiologic
signals, was reported. This manuscript investigates two different ECG systems
produced from normal and abnormal human hearts to introduce an auxiliary phase
space method in conjunction with ECG signals for diagnoses of heart diseases.
Here, the data for each person includes two signals based on V_4 and modified
lead III (MLIII) respectively. Fractal analysis method is employed on the
trajectories constructed in phase space, from which the fractal dimension D is
obtained using the box counting method. It is observed that, MLIII signals have
larger D values than the first signals (V_4), predicting more randomness yet
more information. The lowest value of D (1.708) indicates the perfect
oscillation of the normal heart and the highest value of D (1.863) presents the
randomness of the abnormal heart. Our significant finding is that the phase
space picture presents the distribution of the peak heights from the ECG
spectra, giving valuable information about heart activities in conjunction with
ECG.Comment: 10 pages, 8 figures, 1 table. arXiv admin note: text overlap with
arXiv:2305.1045
State space geometry of the chaotic pilot-wave hydrodynamics
We consider the motion of a droplet bouncing on a vibrating bath of the same
fluid in the presence of a central potential. We formulate a rotation
symmetry-reduced description of this system, which allows for the
straightforward application of dynamical systems theory tools. As an
illustration of the utility of the symmetry reduction, we apply it to a model
of the pilot-wave system with a central harmonic force. We begin our analysis
by identifying local bifurcations and the onset of chaos. We then describe the
emergence of chaotic regions and their merging bifurcations, which lead to the
formation of a global attractor. In this final regime, the droplet's angular
momentum spontaneously changes its sign as observed in the experiments of
Perrard et al. (Phys. Rev. Lett., 113(10):104101, 2014).Comment: Accepted for publication in Chaos: An Interdisciplinary Journal of
Nonlinear Scienc
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