Trajectory Scaling Functions at the Onset of Chaos: Experimental Results

Belmonte, Andrew; Vinson, Michael; Glazier, James A.; Gunaratne, Gemunu H.; Kenny, Brian G.

doi:10.1103/physrevlett.61.539

Cited by 16 publications

(11 citation statements)

References 18 publications

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“…Consequently, an orbit that approaches close to a fixed point on the Poincaré section will maintain proximity to the corresponding periodic orbit of the flow for a finite time interval. Hence, a point sufficiently close to a fixed point of a Poincaré map is likely to make a close return [14,23]. We search for all such close returns on the Poincaré section.…”

Section: A Preliminariesmentioning

confidence: 99%

“…We search for all such close returns on the Poincaré section. Typically, such points cluster into a few groups, each of which is assumed to be associated with a fixed point of the Poincaré map; the fixed point is estimated to be the centroid of the cluster [23].…”

Section: A Preliminariesmentioning

confidence: 99%

“…In this sense, irregular flows are organized around a "skeleton" of unstable cycles. Methods to extract periodic orbits from chaotic signals and spatiotemporally chaotic flows have been introduced [15,22] and used to analyze experimental time series [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

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Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

et al. 2013

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Dynamical systems analysis is performed for reacting flows stabilized behind four symmetric bluff bodies to determine the effects of shape on the nature of flame stability, acoustic coupling, and vortex shedding. The task requires separation of regular, repeatable aspects of the flow from experimental noise and highly irregular, nonrepeatable small-scale structures caused primarily by viscous-mediated energy cascading. The experimental systems are invariant under a reflection, and symmetric vortex shedding is observed throughout the parameter range. As the equivalence ratio-and, hence, acoustic coupling-is reduced, a symmetry-breaking transition to von Karman vortices is initiated. Combining principal-components analysis with a symmetry-based filtering, we construct bifurcation diagrams for the onset and growth of von Karman vortices. We also compute Lyapunov exponents for each flame holder to help quantify the transitions. Furthermore, we outline changes in the phase-space orbits that accompany the onset of von Karman vortex shedding and compute unstable periodic orbits (UPOs) embedded in the complex flows prior to and following the bifurcation. For each flame holder, we find a single UPO in flows without von Karman vortices and a pair of UPOs in flows with von Karman vortices. These periodic orbits organize the dynamics of the flow and can be used to reduce or control flow irregularities. By subtracting them from the overall flow, we are able to deduce the nature of irregular facets of the flows.

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Section: A Preliminariesmentioning

confidence: 99%

Section: A Preliminariesmentioning

confidence: 99%

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Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

et al. 2013

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“…Its restriction to the rst half of the segment, from 0 to 1=2, has a unique inverse, which we will call F 1 0 , and its restriction to the second half, from 1=2 to 1, also has a unique inverse, which we will call F 1 1 . For example, the segment labeled (2) (0; 1) in gure 3 is formed from the inverse image of the unit interval by mapping (0) , the unit interval, with F 1 1 and then F 1 0 , so that the segment (2) …”

Section: Scaling Functionsmentioning

confidence: 99%

Dynamic scaling function at the quasiperiodic transition to chaos

Mainieri

Ecke

1994

Physica D: Nonlinear Phenomena

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We obtain a ve-step approximation to the quasiperiodic dynamic scaling function for experimental Rayleigh-B enard convection data. When errors are taken into account in the experiment, the f( ) spectrum of scalings is equivalent to just two of these ve scales. To overcome this limitation, we develop a robust technique for extracting the scaling function from experimental data by reconstructing the dynamics of the experiment.

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“…3 " 5 Many experimental systems are believed to make similar transitions to chaos, with the path determined by evaluating universal features of the transition that are unique to that path. 6,7 Some examples of universal quantities that can be used for this purpose are the generalized dimensions 4 or singularity spectra 5 of the attracting set and the trajectory scaling function 3 of the orbit at the onset.Much less is known about dynamical systems beyond the onset of chaos. Chaotic experimental systems relax to regions of phase space with very complex structure and zero measure.…”

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confidence: 99%

Chaos beyond onset: A comparison of theory and experiment

1989

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The chaotic dynamics from a nonlinear electronic circuit is shown to exhibit the universal topological structure of maps on an annulus. This suggests that the corresponding universality class is large enough to include physical systems. We suggest that low-dimensional strange attractors fall into a few classes, each characterized by distinct universal topological features.PACS numbers: 05.45.+b Several routes to chaos in dynamical systems have been proposed and analyzed in detail, the best known being the period doubling 1 and the quasiperiodic routes. 2 At the onset of chaos, such dynamical systems exhibit scaling, and certain qualitative and quantitative features of the transition are universal. 3 " 5 Many experimental systems are believed to make similar transitions to chaos, with the path determined by evaluating universal features of the transition that are unique to that path. 6,7 Some examples of universal quantities that can be used for this purpose are the generalized dimensions 4 or singularity spectra 5 of the attracting set and the trajectory scaling function 3 of the orbit at the onset.Much less is known about dynamical systems beyond the onset of chaos. Chaotic experimental systems relax to regions of phase space with very complex structure and zero measure. Generally such objects are fractal, and have structure on ever smaller length scales. They are called strange attractors, and are characterized by metric invariants (which do not change under smooth transformations) such as fractal dimension and Lyapunov exponent. However, they are not universal, and hence do not yield much detailed information about the experimental system: e.g., how to model the dynamics.An alternative class of invariants that can be used to characterize a strange attractor is the set of all periodic orbits. 8 " 10 Periodic orbits are topological invariants; i.e., any change of coordinates will not change the periodicity of an orbit. Thus changing the point of observation or the variable that is being observed in an experiment will not change the cycle structure. This is important since the characterization of the attractor should be robust.The significance of the cycles is seen from the following observations. First, since the strange attractor is the set of points of the phase space visited by the orbit after the transients have settled down, motion on it is ergodic. Thus the orbit of any point P on the strange attractor will make arbitrarily close returns to P. Because of the smoothness and nonlinearity of the dynamics, one should in general be able to move P by a small amount so that the close return becomes exact: i.e., there is a periodic point arbitrarily close to P. This means that the periodic points are dense on the strange attractor. Since the motion on the attractor is chaotic, these cycles have to be unstable.The second important observation is that the structure of the strange attractor in the neighborhood of a periodic point and the motion of points in this neighborhood are determined by the tangent space of the...

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Trajectory Scaling Functions at the Onset of Chaos: Experimental Results

Abstract: We use an averaging technique to estimate the periodic points for attractors near the transition to chaos via period doubling and quasiperiodicity in two experimental systems. Using the averaged experimental data we then evaluate the trajectory scaling functions for both routes to chaos.

Cited by 16 publications

References 18 publications

Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

Dynamic scaling function at the quasiperiodic transition to chaos

Chaos beyond onset: A comparison of theory and experiment

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