Stable Oscillations and Devil's Staircase  in the Van Der Pol Oscillator

Gilbert, Thomas; Gammon, R.G.

doi:10.1142/s0218127400000098

Cited by 12 publications

(6 citation statements)

References 23 publications

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“…In the van der Pol oscillator, both routes – period-adding [83], [100] and period-doubling cascades – occur [91], [101]. On the other hand, our results concerning the route to chaos are in line with findings in a periodically stimulated excitable neural relaxation oscillator [27] and a simple model of the Belousov-Zhabotinsky reaction [102].…”

Section: Discussionsupporting

confidence: 90%

Modeling Brain Resonance Phenomena Using a Neural Mass Model

et al. 2011

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Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.

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Section: Discussionsupporting

confidence: 90%

Modeling Brain Resonance Phenomena Using a Neural Mass Model

et al. 2011

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“…(1) is promoted to three dimensions the relaxation oscillation dynamics becomes much richer to include oscillations of multiple periodicities, almost periodic solutions, coexisting or multistable periodic solutions as well as regions of chaos. This is achieved, as mentioned above, by introducing in time a periodic forcing term to the right-hand side of van der Pol's equation, the object of many studies up to the present day [Gilbert & Gammon, 2000]. Arneodo et al [1985] demonstrated self-generated multiple periodicities, and chaos is displayed by a third-order differential equation linear in the velocity but supplemented by a quadratic nonlinearity in the x-coordinate.…”

Section: Discussionmentioning

confidence: 99%

Dynamics of Relaxation Oscillations

Phillipson

Schuster

2001

Int. J. Bifurcation Chaos

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Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker-Haag piecewiselinear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker-Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to threedimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos. *

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“…The coexistence of OMO and SOM mechanisms adds extra degrees of freedom to the dynamic space of system and results in increased susceptibility to destabilization (detailed in Supplementary Note 2 ) 16 18 21 .When the drive power is between the SOM and OMO thresholds, TPA-associated amplitude modulations disrupt the OMO rhythm, breaking the closed OMO limit cycles and creating the non-repeating chaotic oscillations. On the other hand, if the frequency ratio between OMO and SOM is close to a rational value, they will lock each other based on the harmonic frequency locking phenomena 39 40 . Consequently, different sub-harmonic f omo states are also observed in Fig.…”

Section: Resultsmentioning

confidence: 99%

Mesoscopic chaos mediated by Drude electron-hole plasma in silicon optomechanical oscillators

Huang

et al. 2017

Nat Commun

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Chaos has revolutionized the field of nonlinear science and stimulated foundational studies from neural networks, extreme event statistics, to physics of electron transport. Recent studies in cavity optomechanics provide a new platform to uncover quintessential architectures of chaos generation and the underlying physics. Here, we report the generation of dynamical chaos in silicon-based monolithic optomechanical oscillators, enabled by the strong and coupled nonlinearities of two-photon absorption induced Drude electron–hole plasma. Deterministic chaotic oscillation is achieved, and statistical and entropic characterization quantifies the chaos complexity at 60 fJ intracavity energies. The correlation dimension D2 is determined at 1.67 for the chaotic attractor, along with a maximal Lyapunov exponent rate of about 2.94 times the fundamental optomechanical oscillation for fast adjacent trajectory divergence. Nonlinear dynamical maps demonstrate the subharmonics, bifurcations and stable regimes, along with distinct transitional routes into chaos. This provides a CMOS-compatible and scalable architecture for understanding complex dynamics on the mesoscopic scale.

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Stable Oscillations and Devil's Staircase in the Van Der Pol Oscillator

Cited by 12 publications

References 23 publications

Modeling Brain Resonance Phenomena Using a Neural Mass Model

Modeling Brain Resonance Phenomena Using a Neural Mass Model

Dynamics of Relaxation Oscillations

Mesoscopic chaos mediated by Drude electron-hole plasma in silicon optomechanical oscillators

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