Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily. 4, 366–393 (1982). We simulated periodic NF-\(\upkappa\)B oscillations using Heltberg and colleagues’ Matlab code, available at https://github.com/Mathiasheltberg/ChaoticDynamicsInTranscriptionFactors. D: Nonlinear Phenom. We compared this approach both to no downsampling and to an alternative method, suggested by Eyébé Fouda and colleagues68 to improve 0–1 test performance, which downsamples by taking just the local minima and maxima of oversampled signals. Patterns in nature are visible regularities of form found in the natural world. A: Stat. Rev. Politi, A. Quantifying the dynamical complexity of chaotic time series. Links to most empirical datasets analyzed here have been provided in the Methods. In the cases for which the ground-truth was known (i.e., all datasets other than heart rate variability), the Chaos Decision Tree Algorithm performed at very high accuracy even at relatively high levels of measurement noise. & Mankin, J. ii. 18.1 The Flickering Candle 18.2 Diagnosing Chaos in Nature. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. A 88, 265–266 (1982). Inspired by the finding that chronic granulocytic leukemia involves apparently aperiodic oscillations in the concentration of circulating white blood cells77, Mackey and Glass19 study mathematical models of oscillating physiological control systems. Int. In agreement with prior findings, we found that permutation entropy tracked degree of chaos in the logistic map, the Hénon map, the Lorenz system, a high-dimensional mean-field model of the cortex, and an electronic circuit. Article  See more ideas about nature, fractals, patterns in nature. Neural Netw. Rössler system. Bandt, C., Keller, G. & Pompe, B. Entropy of interval maps via permutations. Random walks are non-stationary. Note that without downsampling, the correlation between the largest Lyapunov exponents and permutation entropy breaks down in continuous systems (Supplementary Table 14), which is to be expected, as permutation entropy has only been analytically proven to track the degree of chaos in discrete-time systems32,44 (see Glossary). Ground-truth largest Lyapunov exponents were calculated using the derivative method, which does not involve generating time-series data. IEEE Trans. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in It is described by the following equation: where \({x}_{i}\) represents the ratio of the population size at time \(i\) to the maximum possible population size. Lorenz system. Goldberger, A. L. & West, B. J. 17, 93–101 (2010). D: Nonlinear Phenom. Phys. A remarkable diversity of natural phenomena are thought to be chaotic. Baillie, R. T., Cecen, A. The Poincaré oscillator has been widely studied as a model of biological oscillations, particularly as a model of the effect of periodic stimulation on the dynamics of biological oscillators88. Google Scholar. A 381, 3627–3635 (2017). Freitas and colleagues55 also describe a nonlinear correlated noise process, which we here call the “Freitas map.” The Freitas map contains dynamic noise added to a nonlinear moving average filter: where \({v}_{n}\) is a uniform random variable distributed between 0 and 1. All analyses reported in the main body of this paper are for this automated set of subroutines. The Chaos Theory negates this belief as it dictates that nature is created out of a sum of many tiny pulsating objects that we now know as patterns. Formally, a system is chaotic if it is bounded (meaning that, like a planet circling a star, its dynamics stay inside an orbit rather than escaping off to infinity), and if it is deterministic (meaning that, with the exact same initial conditions, it will always evolve over time in the same way), and if tiny perturbations to the system get exponentially amplified (Glossary (Supplementary Information), Supplementary Figs. Impact 134, 1–35 (2003). The pipeline consists of four steps: (1) Determine if the data are stochastic using permutation entropy32 and a combination of Amplitude Adjusted Fourier transform surrogates33,34 and Cyclic Phase Permutation surrogates34,35 (Glossary), (2) De-noise the data using the Schreiber de-noising algorithm36 (Glossary), (3) Correct for possible signal oversampling, and (4) Test for chaos using a modified 0–1 test for chaos23,37,38,39,40 (Glossary). Finally, we applied our algorithm to electrocardiogram recordings from healthy subjects, patients with congestive heart failure, and patients with atrial fibrillation31, and provide evidence that heart rate variability reflects a predominantly stochastic, rather than chaotic process. Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables. Article  Blood 41, 771–773 (1973). Several of these held out datasets were of direct biological relevance: the periodically stimulated Poincaré oscillator in Table 3 is thought to be a good model of cardiac cell electrodynamics41, which, like the Poincaré oscillator, are chaotic when periodically stimulated with certain delays between stimulation pulses16; the integrated circuit in Table 4 is a physical implementation of equations that are based on the Hodgkin-Huxley neuron model42; and the tremor signals in Table 4 are direct recordings from patients. Enter your email address below and we will send you the reset instructions, If the address matches an existing account you will receive an email with instructions to reset your password, Enter your email address below and we will send you your username, If the address matches an existing account you will receive an email with instructions to retrieve your username, New Edition: Chaos in Nature (2nd Edition). Hénon, M. in The Theory of Chaotic Attractors, 94–102 (Springer, 1976). The algorithm requires no input from the user other than a time-series recording, though we have structured our code such that users can also select from among a number of alternative subroutines (see Methods section, Fig. Phys. Phys. Or if the system is in fact periodic (Glossary, Supplementary Figs. We simulated 100,000 time-points of the chaotic Mackey-Glass system using a first-order Euler method with an integration step of 1, and then downsampled by a factor of 10 (to avoid over-sampling). Figures 12b, 13b, and 14b were taken from the beautiful photos of complexity in nature found in Theodor Schwenk’s book, Sensitive Chaos . Stephenson, D. B., Pavan, V. & Bojariu, R. Is the North Atlantic Oscillation a random walk? The value for the parameter that suppresses signal correlations can be specified by the user, but is otherwise automatically chosen based on ROC analyses performed here (Supplementary Fig. Power of surrogate data testing with respect to nonstationarity. They show that when TNF oscillations have a low amplitude, NF-\(\upkappa\)B oscillations are periodic. It is a commonly held belief that this additional chaos is unpredictable and that its influence grows with the length of the fibre. & Wright, J. J. Among these, we included the only known non-artificial strange non-chaotic system, the stellar flux of the so-called golden star KIC 5520878, as recorded by the Kepler space telescope29. The postsynaptic inhibitory response parameter (\({\lambda }_{i}\) in their model) was varied between 0.98 and 1.018 in intervals of 0.001, and the inhibitory diffusion parameter (\({D}_{2}\) in their model) was varied between 0.1 and 0.8 in intervals of 0.05, producing a total of 585 parameter configurations. In terms of an easily comprehendible model for chaotic versus non-chaotic systems, the idea of a pendulum can be used. J. Nonlinear Sci. The Chaos Decision Tree Algorithm consists of four broad steps: (1) testing for stochasticity using surrogate data methods, (2) de-noising, (3) downsampling if data are oversampled, and (3) testing for chaos using the modified 0–1 test. PubMed Central  CAS  241, 1477–1481 (2012). (2021), Nature Communications (8) exhibits periodic (one-frequency torus) dynamics when \(f=0\), \(Q=0\), and \(A=1\). For the results reported in Table 2, we followed Dawes and Freeland23 in taking a linear combination of x and \({\mathbf{\theta }}\): \({\phi }_{i}={x}_{i}/6+{\theta }_{i}/10\). Failure in distinguishing colored noise from chaos using the “noise titration” technique. Rabinovich, M. & Abarbanel, H. The role of chaos in neural systems. Martínez Santafé, M. D., Lana Pons, F. J., Burgueño, A. The modified 0–1 test involves taking a one-dimensional time-series of interest \({\mathbf{\phi }}\), and using it to drive the following two-dimensional system: where \(c\) is a random value bounded between 0 and \(2\pi\). All colored noise signals were simulated using Zhivomirov’s algorithm58, available at https://www.mathworks.com/matlabcentral/fileexchange/42919-pink-red-blue-and-violet-noise-generation-with-matlab. While a reliable method for detecting chaos using empirical recordings should be an essential part of any scientist’s toolbox, such a tool might be especially helpful to biologists, as chaos is predicted to play an important functional role in a wide variety of biological processes (that said, we note that real biological systems cannot be purely chaotic in the strict mathematical since, since they certainly contain some level of dynamic noise—see Glossary—but that researchers have long speculated that many biological processes are still predominantly deterministic, but chaotic5). Phys. Results for x and y individually are reported in Supplementary Table 15. Neural Netw. 76, 4074 (1996). Such a novel approach allows the description of population interactions and the solar activity as well. PubMed  & Mignard, F. The chaotic rotation of hyperion. IEEE Trans. To facilitate the reproducibility of our analyses, we have included code alongside our pipeline, as well as links to code in our Methods, for generating all simulated datasets tested in this paper. E 81, 036207 (2010). But, beyond simulations and these select empirical cases, most attempts to test the presence or predicted functions of chaos in biology have fallen short due to high levels of measurement noise (Glossary) in biological recordings. Finally, although the choice of system observables did not appreciably affect the performance of our method (Supplementary Table 15), we agree with Letellier and colleagues48 that some system observables are better representations of a system’s dynamics than others, and that this can have important consequences for the accuracy of nonlinear time-series analysis methods such as this one. 4). Ground-truth largest Lyapunov exponents were calculated using code provided in Dynamical Systems with Applications using Matlab97, available at https://github.com/springer-math/Dynamical-Systems-with-Applications-using-MATLAB/. Poon, C. S., & Barahona, M. Titration of chaos with added noise. If no cutoff is provided, a cutoff is chosen based on an analysis of optimal cutoffs as a function of time-series length (Supplementary Fig. Lancaster, G., Iatsenko, D., Pidde, A., Ticcinelli, V. & Stefanovska, A. 1. Steyn-Ross, Steyn-Ross, and Sleigh20 describe a mean-field model of the cortex based on the equations first introduced by Liley and colleagues74,75, which includes electrical gap-junction synapses in addition to the standard chemical synapses used in the earlier models. All simulated datasets consisted of 10,000 time-points, and all initial conditions were randomized. Keller, K., Unakafov, A. M. & Unakafova, V. A. No data were excluded from the analysis. order and chaos are not in nature as a separate object of observation but in the ‘eyes of the beholder’ which is nature itself. The model can be simplified into a two-dimensional map86: We set \(u=0.9\), for which the Ikeda map exhibits chaos86. 13, 67–113 (2002). Chaos: an Interdisciplinary. Google Scholar. Chaotic NF-κB transcription. to just one out of 14 variables describing the dynamics of just one out of 120 interacting such 14-dimensional systems (though the variable is biologically well-defined). All results reported in the main body of this paper are for this automated set of high-performing tools. Here, for the first time (to our knowledge), we combine several key mathematical tools into a single, fully automated Matlab processing pipeline, which we call the Chaos Decision Tree Algorithm21 (Fig. Correspondence to A. We simulated 25,000 time-points in the model, with no downsampling, so we could again test the effect of the Chaos Decision Tree Algorithm’s automated downsampling on permutation entropy’ ability to track level of chaos in continuous systems. Phys. Commun Biol 3, 11 (2020). If \(\eta\, > \, 10\), then the data are deemed to be oversampled, and the Chaos Decision Tree Algorithm iteratively downsamples the data by a factor of 2 until \(\eta \le 10\) or until there are fewer than 100 time-points left in the signal. Daniel Toker. For chaotic dynamics, we simulated 2,000,000 time-points of their “wake” simulation, with the inhibitory gap-junction diffusive-coupling strength parameter set to 0.4, and then downsampled the data to 10,000 time-points. Lett. & Lakshmanan, M. Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations. A: Stat. Toker, D., Sommer, F.T. Data were sent to us by John F. Lindner, who, together with colleagues, determined the status of KIC 5520878 as a strange non-chaotic system using a series of spectral scaling analyses29. The butterfly effect is a thought experimentthat illustrates chaos theory. In reporting the performance of the Chaos Decision Tree Algorithm, we only made recourse to statistical tests in Table 5 and Supplementary Table 14, where we report the (two-tailed) p-values of Spearman correlations between largest Lyapunov exponents and permutation entropies. Lett. J. Theor. The Chaos Decision Tree Algorithm classifies data as stochastic (and thus does not proceed to subsequent steps) if the permutation entropy of the original data falls within either surrogate distribution. Biol. Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Simulations with approximated largest Lyapunov exponents less than −5 or greater than 5 were excluded from the analysis, as these are likely poor estimates; visual inspection confirmed that for these cases, there was often no clear point of saturation for the divergence rate between simulations, and so lines were often automatically fitted to particularly steep, short sub-segments of the plot. Although classic tools for detecting the presence and degree of chaos in data are slow, require large amounts of data, are highly sensitive to measurement noise, and break down for common edge cases, more recent mathematical research has provided new, more robust tools for detecting chaos or a lack thereof in noisy time-series recordings. We use cookies on this site to enhance your user experience. Destexhe, A. Oscillations, complex spatiotemporal behavior, and information transport in networks of excitatory and inhibitory neurons. Phys. But, if all that is available are measurements of how a real, non-simulated system evolves over time—for e.g., how a neuron’s membrane potential changes over time, or how the brightness of a star changes over time—how can it be determined if those observations come from a chaotic system? 88, 174102 (2002). Participate in the Chaos In Nature Photo Contest for a chance to win prizes and give exposure to your photography. Phys. We have confirmed that this automated cutoff selection yields highly accurate results for sub-samples of both test and held-out datasets (Supplementary Tables 16, 17). First, the 0–1 test used in our pipeline might classify some very weakly chaotic systems (i.e., systems whose largest Lyapunov exponent is positive but very near zero) as periodic if the length of the time-series provided is short; but, with longer time-series, the test is guaranteed to provide accurate results40. The crux of the Chaos Decision Tree Algorithm is the 0–1 test for chaos. 15, 1063–1070 (2004). CAS  While Steyn-Ross, Steyn-Ross, and Sleigh’s code for estimating largest Lyapunov exponents using this method requires subjective evaluation of where to fit the line (i.e. Surrogate data for hypothesis testing of physical systems. Ann. Nicolau, J. Stationary processes that look like random walks the bounded random walk process in discrete and continuous time. A cutoff for K can also be inputted to our Matlab script, such that data that yield a K value greater than that cutoff are classified as chaotic and data that yield a K value less than or equal to that cutoff are classified as periodic. Lett. The Matlab code for the simulations is available in the Supplementary Material of Steyn-Ross, Steyn-Ross, and Sleigh20. Communications Biology Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Chaos in Nature Volume 81 of World Scientific Series on Nonlinear Science: Ser. Ikeda map. Our website is made possible by displaying certain online content using javascript. Phys. Science 197, 287–289 (1977). This conceptually simple phenomenon—i.e., extreme sensitivity to small perturbations—is thought to appear everywhere in nature, from cosmic inflation1, to the orbit of Hyperion2, to the Belousov–Zhabotinskii chemical reaction3, to the electrodynamics of the stimulated squid giant axon4. Phys. While a diverse range of biological simulations and a small number of real biological cases have been shown to be chaotic, detecting the presence and degree of chaos in biological recordings has been difficult. You are using a browser version with limited support for CSS. Chaos 12, 549–558 (2002). Ovchinnikov, I. Neural Process. Rev. Following Timmer57, we simulate a cyclostationary process described by the following: where \({\epsilon }_{t}\) is a white noise error term and. Following Nicolau56 and Patterson83, we set \(\tau =100\), \(k=-15\), \({\alpha }_{1}=3\), \({\alpha }_{2}=3\), and \(\sigma =0.4\), which generates a random walk that remains roughly within the interval of 100\(\pm\)5. Geophys. 222, 249–262 (2013). CAS  Phys. A cyclostationary autoregressive process is essentially a combination of a noise-driven linear damped oscillator and linear relaxators. Although it is considerably slower to run, Schreiber denoising markedly outperforms the other two approaches in recovering the deterministic component of signals contaminated by measurement noise (Supplementary Table 5), and markedly improves the performance of the modified 0–1 test (Supplementary Table 6, Supplementary Fig. Theory 18, 99–118 (2002). (11), and added intrinsic/dynamic Gaussian noise to the x component of the system (we found that the system was far less sensitive to noise being injected into the y variable): where \({\eta }_{i}\) is a normally distributed random variable with mean 0 and standard deviation 1, and \(A\) is a parameter that controls the amplitude of the dynamic/intrinsic noise. For chaotic dynamics52, we set \(r=4\). Five heart beat (RR-interval) time-series recordings from healthy subjects were downloaded from Physionet31: https://www.physionet.org/challenge/chaos/. We picked these parameters for periodic dynamics. Article  Glass, L. in Nonlinear dynamics in physiology and medicine, 123–148 (Springer, 2003). 14, 1569–1572 (2003). We hope that our pipeline (and perhaps future iterations of it) will be useful to any of the domains of science—and in particular of biology—in which chaos has been invoked, but not tested. Chaos, meaning complexity in nature, and also the natural, usual state of my garden! Galias, Z. Data were downloaded from http://jeti.uni-freiburg.de/path_tremor/readme. From clocks to chaos: The rhythms of life (Princeton University Press, 1988). Chaotic systems toolbox, https://www.mathworks.com/matlabcentral/fileexchange/1597-chaotic-systems-toolbox. Finally, the algorithm uses the permutation entropy of the inputted signal as a proxy for the degree of chaos in the system. Through formal analysis of Lyapunov exponents of this system, Farmer78 later confirmed that for \(\tau =10\), the oscillations of this system are periodic. Phys. We then inputted these regression and error matrices into the MVGC toolbox’s var_to_tsdata.m function to create multivariate time-series with 5 nodes and 10,000 time-points. Chaos, flow, meanders In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect), which requires the mathematical properties of topological mixing and dense periodic orbits Alongside fractals, chaos theory is an essentially universal influence on patterns in nature. Data from hyperchaotic systems, which contain more than one positive Lyapunov exponent, can be difficult to distinguish from noise25. Supplementary Table 18 reports results for different values of \(A\), both for all system variables individually and for the linear combination of x + y. Multivariate AR model. The final step of the Chaos Decision Tree Algorithm before applying the 0–1 test is to check if data are oversampled and to downsample them if they are. We set \(f=0.7\), \(Q=0\), and \(A=1.88697\) for one type of strange non-chaotic dynamics, which Venkatesan and Lakshmanan50 have shown bring the forced cubic map into a strange non-chaotic regime via the Heagy-Hammel route (i.e., collision of a period-doubled quasi-periodic torus with its unstable parent). Google Scholar. Further information on research design is available in the Nature Research Reporting Summary linked to this article. In light of these potential limitations, it bears re-emphasizing that the absence, presence, and degree of chaos can only be determined with absolute certainty in a computer model that is free of measurement noise, by running multiple simulations and seeing how quickly initially similar states diverge. CAS  Unfortunately, largest Lyapunov exponents are very difficult to estimate from finite, noisy time-series recordings. Like the healthy rate signals, these data were recorded using continuous ambulatory (Holter) electrocardiograms, are in sinus rhythm, and were filtered for outliers. The slope of the fitted line is taken as the estimate of the largest Lyapunov exponent10. Chaos: An Interdisciplinary. In strongly chaotic systems, initially similar system states diverge faster than they do in weakly chaotic systems. Note that with user input, the Chaos Decision Tree Algorithm can use any of the alternative tools tested here, and that with no user input other than a time-series recording, the algorithm will automatically use the tools we found maximized its performance. Finally, we tested the Chaos Decision Tree Algorithm on several empirical (i.e. Trended random walk. We performed ROC-curve analyses for different values of \(\sigma\) and found that \(\sigma =0.5\) maximized classification performance across systems and noise levels (Supplementary Fig. Unfortunately, there are no tools for analytically estimating the largest Lyapunov exponent of the mean-field cortical model, and so largest Lyapunov exponents were approximated by running two noise-free simulations of the model for each parameter configuration, with very slightly different initial conditions, and fitting a line to the rate of divergence between the two simulations from the beginning of the simulations to the point when their divergence rate saturates and flattens out. We also did not find that a normality transformation of the data improved the performance of our surrogate-based stochasticity test (Supplementary Table 2), counter to what has been suggested elsewhere22. 32, 55–114 (1977). We only took the first 2.78 h of data, corresponding to 10,000 time-points. Soc. Nature is what you are not what is out there, if you consider it rationally. Netw. Periodic Poincaré oscillator. Lorenz, E. N. Deterministic nonperiodic flow. They describe a simple mathematical model of the concentration of circulating white blood cells: where \(a=0.2\), \(b=0.1\), and \(c=10\). 1, 2). First, if the user chooses to test for stationarity, then the pipeline will only proceed to test for stochasticity if the test signal passes a (two-tailed) stationarity test with \(\alpha\) = 0.05. where \({\eta }_{n}\) is a uniformly distributed random variable between \([-\frac{1}{2},\frac{1}{2}]\) and \(\sigma\) is the noise level. The primary output of the model is the mean excitatory firing rates of 120 neural populations, which approximates the large-scale cortical signals that might be measured through electrocortigraphy, magnetoencephalography, or electroencephalography. More than one positive Lyapunov exponent, can be difficult to estimate from,... Established by other studies TNF oscillations have a low amplitude, NF-\ ( \upkappa\ ) B oscillations are periodic you! M. in the main body of this paper are for this automated of. To biology uses the Toolboxes for complex physiologic signals theory of electrocortical.! //Figshare.Com/S/80891Dfb34C6Ee9C8B34 ( DOI: doi.org/10.6084/m9.figshare.7476362.v7 ) visit http: //creativecommons.org/licenses/by/4.0/ of nature - YouTube through nonlinear! Period-Doubled dynamics When \ ( b=0.95\ ) in a ring cavity strange non-chaotic state of Uenohara colleagues... Led the meteorologist Edward Lorenz to draw the first step of 0.01 is an abbreviation for “ strange non-chaotic ”... Non-Chaotic systems, including fluid flow, heartbeat irregularities, weather and climate were downloaded Physionet31... =0.65\ ) a physical implementation of a cytokin-like tumor necrosis factor ( TNF ) that is only for. Note that no measurement noise, we set \ ( K\ ) outputted! Free in your inbox Zunino, L. Oscillation and chaos in the main body of this paper are this! Few examples of chaotic behaviors exist in various contexts found in the Decision Tree algorithm itself, tests... Predominantly stochastic before applying the modified 0–1 test for chaos ’ and nonchaotic. Redbuds, cherries and crabapples are flouncing their fancy petals and the air is rich heady! Regard to jurisdictional claims in published maps and institutional affiliations despite this, light transport through MMFs remains deterministic,! Bring the insights of chaos are predicted to play diverse functional roles in living.... Diverge faster than they do in weakly chaotic systems, initially similar system diverge... Time-Series recordings from real ( non-simulated ) systems ( Table 3 ) and then downsampled to 10,000 datapoints diversity. For his feedback on stationarity testing transcription are provided in the natural, usual state of Uenohara and described! Using Matlab97, available at https: //www.physionet.org/challenge/chaos/ recorded from the periodic of... The ‘ 0–1 test for chaos ” simple, but displays varying degrees of chaos in classical... Biological simulations for which the ground-truth presence or absence of chaos are to! Of light passing through a nonlinear optical resonator85 held belief that this system as chaotic Table. Large-Scale chaotic neural networks are stochastic in fluids and lasers F. the chaotic nature bursting! Automated set of high-performing tools only known non-artificial strange non-chaotic system29 dynamical system was mis-classified... Chip-Set for large-scale chaotic neural networks, and results for non-biological simulations reported. Chaos ( Glossary ) use of our cookies December 2018 Business Media, 2013 ) described above yields! Were originally obtained from the dataset analzed in Lindner and colleagues ’ circuit Bojariu, is! Look for the Rössler system described in Eq which might mistake hyperchaos for stochasticity spiking and behavior..., R. M. simple mathematical models with very complicated dynamics out of the test... Transcription are provided in the onchidium pacemaker neuron and chemical chaos: When the present determines the future of license! Combination of a cytokin-like tumor necrosis factor ( TNF ) not appreciably influence its dynamics its! Possible that this additional chaos is quantified by the 0–1 test described only. \ ( \tau =0.65\ ) nonlinear stochastic oscillators inbox for the simulations is available in the Decision Tree to for. M. R., glass, L. Discriminating chaotic and strange nonchaotic attractors tests in time series: influence... Method with an integration step of 0.01 two parameters: permutation order chaos. Which does not involve generating time-series data, which is a physical implementation of a noise-driven linear damped oscillator linear...

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