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(Color online) Small amplitude period 1 oscillation, L ¼ 5.5 cm and f ¼ 1.10 Hz. (a) Numerical time series, (b) numerical DFT, (c) experimental time series, and (d) experimental DFT.
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The sign of the largest Lyapunov exponent is the fundamental indicator of chaos in a dynamical system. However, although the extraction of Lyapunov exponents can be accomplished with (necessarily noisy) the experimental data, this is still a relatively data-intensive and sensitive endeavor. This paper presents an alternative pragmatic approach to i...
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... was simulated using a fourth-order Runge-Kutta time stepping scheme 22 and is in excellent agreement with the experimental results. Figures 6 and 7 show the numerical and experimentally obtained time series' and DFT's of a small amplitude and large amplitude period 1 (i.e., one response cycle for every forcing cycle) os- cillation, respectively. Figure 8 shows the same for a period 3 (i.e., one response cycle for every 3 forcing cycles) oscilla- tion with the addition of the phase portraits to highlight the more complicated motion. ...
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... this plot was set at 1% of the maximum peak height to eliminate false peaks (due to noise) being counted. It is clear that this method produces a useful alterna- tive bifurcation diagram as the number of peaks in the chaotic and nonchaotic regions is easy to distinguish. Note that dashed vertical lines denote the locations of the four samples in Figs. ...
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... There is also excellent agreement with the peak count plots. Figure 14 shows the numerically obtained largest LE bifurcation diagram for the same parameter range. The curve was obtained by simulating the equation of motion, and cal- culating the largest LE using the method in Ref. 5. Once again the dashed vertical lines denote the four samples in Figs. 6-9. The regions of positive largest LE agree well with the regions of chaos in Figs. 10-13. Note that for a non- chaotic trajectory, the true largest LE of a temporally forced system is zero since the time state is non-converging. How- ever, since the numerical method has direct access to each state one may remove time from the divergence ...
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Citations
... At a Reynolds number of 87 000, at which the flow parameters matched those of Tang and Dowells experimental study [27] , the lift and drag coefficient reduced-frequencies were observed to be in good agreement between the DNS simulation and experimental results. By employing the heuristic flow characterization developed by Wiebe and Virgin [35] and in agreement with existing literature [36] , each Reynolds number is associated with qualitatively different flow regimes: periodic flow, quasi-periodic flow and chaotic flow, as shown in Table 1 . Steady-state time histories and power spectra of the lift coefficient ( ) as computed at different Reynolds numbers are shown in the supplementary material. ...
... Figure 2 shows steady-state time histories and power spectra of the lift coefficient (CL) as computed at different Reynolds numbers. By employing the heuristic flow characterization developed by Wiebe and Virgin [39] and in agreement with existing literature [40], each Reynolds number is associated with qualitatively different flow regimes: periodic flow, quasi-periodic flow and chaotic flow, as shown in Table 1. All simulations were performed with a time step of 0.002 seconds. ...
Recent efforts have shown machine learning to be useful for the prediction of nonlinear fluid dynamics. Predictive accuracy is often a central motivation for employing neural networks, but the pattern recognition central to the network function is equally valuable for purposes of enhancing our dynamical insight into confounding dynamics. In this paper, convolutional neural networks (CNNs) were trained to recognize several qualitatively different subsonic buffet flows over a high-incidence airfoil, and a near-perfect accuracy was performed with only a small training dataset. The convolutional kernels and corresponding feature maps, developed by the model with no temporal information provided, identified large-scale coherent structures in agreement with those known to be associated with buffet flows. An approach named Gradient-weighted Class Activation Mapping (Grad-CAM) was then applied to the trained model to indicate the importance of these feature maps in classification. Sensitivity to hyperparameters including network architecture and convolutional kernel size was also explored, and results show that smaller kernels are better at coherent structure identification than are larger kernels. A long-short term memory CNN was subsequently used to demonstrate that with the inclusion of temporal information, the coherent structures remained qualitatively comparable to those of the conventional CNN. The coherent structures identified by these models enhance our dynamical understanding of subsonic buffet over high-incidence airfoils over a wide range of Reynolds numbers.
... The Poincare section is computed using 1000 cycles of the repetitive movements, with the first 30% of motions being ignored since they are considered to be a transient response. The heuristic approach proposed by Wiebe and Virgin [37], which works by counting the number of peaks in the Discrete Fourier Transform (DFT), is applied to strengthen the observation. For this heuristic approach, . ...
... Using the parame-ters: µ = 1, ω 0 = 0.2, α = −3, F 1 = 2, F 2 = 0.1, γ = 2, ω 1 = 1, ω 2 = √ 5, and d = 0.25, the Poincare section of the coupled system indicates the chaotic orbit, as shown in Figure 5c. The chaotic behavior can be further observed using the heuristic approach proposed by Wiebe and Virgin [37], where there are many number peaks in the DFT, as shown in Figure 5d. rameters: = 0.1, 0 = 0.2, = -3, F1 = 2, F2 = 3, = 2, 1 = 1, 2 = 2, with coupling parameter d = 0.25, exhibits the period-2 solution. ...
... Using the parameters: = 1, 0 = 0.2, = −3, F1 = 2, F2 = 0.1, = 2, 1 = 1, 2 = √5, and d = 0.25, the Poincare section of the coupled system indicates the chaotic orbit, as shown in Figure 5c. The chaotic behavior can be further observed using the heuristic approach proposed by Wiebe and Virgin [37], where there are many number peaks in the DFT, as shown in Figure 5d. ...
Nonlinear dynamics have become a new perspective on model human movement variability; however, it is still a debate whether chaotic behavior is indeed possible to present during a rhythmic movement. This paper reports on the nonlinear dynamical behavior of coupled and synchronization models of a planar rhythmic arm movement. Two coupling schemes between a planar arm and an extended Duffing-Van der Pol (DVP) oscillator are investigated. Chaos tools, namely phase space, Poincare section, Lyapunov Exponent (LE), and heuristic approach are applied to observe the dynamical behavior of orbit solutions. For the synchronization, an orientation angle is modeled as a single well DVP oscillator implementing a Proportional Derivative (PD)-scheme. The extended DVP oscillator is used as a drive system, while the orientation angle of the planar arm is a response system. The results show that the coupled system exhibits very rich dynamical behavior where a variety of solutions from periodic, quasi-periodic, to chaotic orbits exist. An advanced coupling scheme is necessary to yield the route to chaos. By modeling the orientation angle as the single well DVP oscillator, which can synchronize with other dynamical systems, the synchronization can be achieved through the PD-scheme approach.
... Even if a chaotic relaxation might not be excluded in these cases (see, for example, Refs. [18,19] and references therein) from our numerical analysis this appears not to have a major impact on the significance of the critical damping condition and on the main results outlined in the present paper. ...
We study the dynamics of a comagnetometer based on an alkali-metal—noble-gas hybrid tri-spin system by numerically solving coupled Bloch equations. A thorough analysis of the response dynamics is carried out in the standard experimental regime where the spin system can be mapped to a damped harmonic oscillator. The results show that a linear increasing response of the comagnetometer signal is found when the noble-gas nuclear spin magnetization and the alkali-metal spin lifetime parameters satisfy an overdamping condition. We find that an upper limit for the signal amplitude of the comagnetometer is imposed by the inherent dynamics of the hybrid tri-spin system. The results agree with currently available experimental data and provide useful guiding for future experiments.
... This approach can be easily extended to work with irregularly sampled data [25,38,45]. Kulp [24] modifies a method originally proposed by Wiebe and Virgin [48]; the method employs series spectrum. In Ref. [18], authors utilize multigraph to represent information about time series. ...
The paper deals with a generalized relational tensor, a novel discrete structure to store information about a time series, and algorithms (1) to fill the structure, (2) to generate a time series from the structure, and (3) to predict a time series, for both regularly and irregularly sampled time series. The algorithms combine the concept of generalized z-vectors with ant colony optimization techniques. In order to estimate quality of the storing/re-generating procedure, a difference between characteristics of the initial and regenerated time series is used. The structure allows working with a multivariate time series, with an irregularly sampled time series, and with a number of series as well. For chaotic time series, a difference between characteristics of the initial time series (the highest Lyapunov exponent, the auto-correlation function) and those of the time series re-generated from a structure is used to assess the effectiveness of the algorithms in question. The approach has shown fairly good results for periodic and benchmark chaotic time series and satisfactory results for real-world chaotic data.
... As has been shown previously in [8], a multiple peaks spectral structure is produced in a certain condition [3]. Here we want to point out that, a multiple frequency spectra is only a necessary condition for the presence of chaos [18], but not enough. Practically, although super long, the nuclear spins still have finite lifetimes. ...
With numerical calculation of coupled Bloch equations, we have simulated the spin dynamics of nuclear magnetic resonance gyroscope based on alkali metal-noble gas hybrid trispin system. From the perspective of damping harmonic oscillator, a thorough analysis of the response dynamics is demonstrated. The simulation results shows a linear increasing response of gyroscope signal while the noblge gas nuclear spin magnetization and alkali atomic spin lifetime parameters are at the over damping condition. An upper limit of response is imposed on the NMR gyroscope signal due to the inherent dynamics of the hybrid trispin system. The results agrees with present available experimental results and provide useful guidings for future experiments.
... C haos has been widely studied for decades in physics, mathematics, and engineering [1][2][3][4] . Since chaos is generally defined as aperiodic dynamical behavior of deterministic systems that exhibit a high sensitivity to initial conditions 1 , it could be considered predictable in a mathematical sense, assuming all relevant information about the system is known. ...
Advances in machine learning have revolutionized capabilities in applications ranging from natural language processing to marketing to health care. Recently, machine learning techniques have also been employed to learn physics, but one of the formidable challenges is to predict complex dynamics, particularly chaos. Here, we demonstrate the efficacy of quasi-recurrent neural networks in predicting extremely chaotic behavior in multistable origami structures. While machine learning is often viewed as a “black box”, we conduct hidden layer analysis to understand how the neural network can process not only periodic, but also chaotic data in an accurate manner. Our approach shows its effectiveness in characterizing and predicting chaotic dynamics in a noisy environment of vibrations without relying on a mathematical model of origami systems. Therefore, our method is fully data-driven and has the potential to be used for complex scenarios, such as the nonlinear dynamics of thin-walled structures and biological membrane systems.
... Indeed, changes in power spectra of some dynamical systems as they bifurcate to chaos have been described [2][3][4]. Although several measures were proposed to characterize these spectral changes [2,[5][6][7], a close relation between these measures and the attractor topology is unclear and it is not supported by numerical simulations [2,7]. ...
... It is generally accepted that time series observed from chaotic systems exhibit some characteristic signatures such as fractal geometry and broadband frequency content. Indeed, time series of chaotic trajectories often display an exponential decay in their power spectrum at high frequencies, different from the classical power-law of stochastic colored noises [5,8,9]. However, some studies have proved that the spectra of colored noises cannot be related to the dynamical route to chaos [10,11]. ...
Although classical spectral analysis is a natural approach to characterise linear systems, it cannot describe a chaotic dynamics. Here, we propose the ordinal spectrum, a method based on a spectral transformation of symbolic sequences, to characterise the complexity of a time series. In contrasts with other nonlinear mapping functions (e.g. the state-space reconstruction) the proposed representation is a natural approach to distinguish, in a frequency domain, a chaotic behavior. We test the method in different synthetic and real-world data. Our results suggest that the proposed approach may provide new insights into the non-linear oscillations observed in different real data.
... Pour finir l'étude dynamique des bistables, nous présentons une deuxième méthode pour révéler le comportement chaotique de la structure sans faire recours aux sections de Poincaré. La méthode de comptage de pics [137] consiste à analyser le spectre de la réponse temporelle. La première étape est de procéder à une FFT de la réponse. ...
Le travail effectué pendant cette thèse a porté sur l'étude de comportement d'une poutre flambée bistable soumise à plusieurs types d'actionnements. Les aspects statique et dynamique du comportement du bistable ont été analysés de manière détaillée. Le modèle cinématique exploité est le modèle Elastica extensibleLa partie statique a été consacrée à l'étude des diagrammes de bifurcation pour trois types d'actionnements : force ponctuelle, éléments piézoélectriques et forces de Laplace. Pour chacune de ces technologies, une procédure d'optimisation a été mise en œuvre. À la fin de la partie statique, des essais expérimentaux ont été réalisés. Ils ont permis de valider le modèle dans le cas de l'actionnement via les forces de Laplace.Dans la partie dynamique, un nouveau modèle permettant de simuler le comportement dynamique de la poutre a été développé. Cette approche prenant en compte trois modes de flambage a mis en évidence le phénomène de croisement de modes. Nous avons excité la poutre via une force ponctuelle sinusoïdale. L'influence de plusieurs paramètres sur la réponse de la poutre a été discutée. De plus, l'analyse des diagrammes de sections de Poincaré nous a permis de comprendre les origines du comportement chaotique. Des essais expérimentaux ont été réalisés à la fin de la partie dynamique.Les résultats des études menées ont été exploités pour la conception d'une nouvelle technologie d'actionnement des dispositifs Braille fondée sur les poutres bistables. La possibilité d'actionner ces structures via les forces électromagnétiques et via les éléments piézoélectriques a été discutée.
... The nonlinear dynamics of the flow beyond its first Hopf bifurcation, in a range of Reynolds numbers less often studied, will be characterized using both Poincaré maps and analysis of the power spectrum. Quasi-periodicity and Eulerian chaos will be identified as was done by Gollub and Benson [5] and others [18,19] . A novel concept is introduced, frequency shredding, which can explain the behavior observed in the power spectrum across the tested span of Reynolds numbers. ...
... The second assumption, while in line with existing observations of quasi-periodic fluid flows [5,32] , can be disqualified by the algorithm itself if power spectra observed to be quasi-periodic are not reduced to three dominant frequencies. In line with the chaos identification method introduced by Wiebe and Virgin [19] , only power spectrum peaks above a certain threshold, in this case S min = 10 −5 , were passed through this algorithm at each Reynolds number. ...
... The present study similarly characterizes one path to chaotic flow in the 2D regularized lid-driven cavity by changing the Reynolds number; chaotic flow will be characterized in the same way: by the presence of a broadband pattern in the frequency domain. This characterization of chaos was explored by Wiebe and Virgin [19] with a low dimensional nonlinear mechanical oscillator. They demonstrated that "peak counting" above a threshold as a system progresses from periodicity to chaos yields conclusions in line with those based on the largest Lyapunov exponent. ...
Computational simulations of a two-dimensional incompressible regularized lid-driven cavity flow were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in Eulerian chaotic flow. Semi-implicit, pseudo-spectral numerical simulations were performed at Reynolds numbers from 1000 to 25,000. Poincaré maps were used to identify transitions as the Reynolds number increased from stable-laminar flow to periodic flow, periodic flow to quasi-periodic flow, quasi-periodic flow to chaotic flow, and a sudden, brief return from chaotic flow to periodic flow. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. A power spectrum analysis, in which the novel concepts of frequency shredding and power capacity are introduced, was performed with the conclusion that no further bifurcations occurred at Reynolds numbers above 15,500 even though Eulerian chaos was not formally observed until Reynolds numbers above 18,000. While qualitative changes in the fluid flow's attractor were apparent from trajectories in the phase space, mechanisms by which such changes can occur were apparent from power spectra of flow time histories. The novel power spectrum analysis may also serve as a new approach for characterizing multi-scale nonlinear dynamical systems.
