310 research outputs found

    Visualizing the Template of a Chaotic Attractor

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    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

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    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 X−ZX-Z in the former and ∣X∣−∣Z∣|X|-|Z| 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

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    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

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    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

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    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

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    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 ∞\infty-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

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    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

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    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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