Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

Euzébio, Rodrigo D.; Llibre, Jaume

doi:10.1080/00207160.2015.1046847

Cited by 4 publications

(2 citation statements)

References 29 publications

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“…In this study, a Van der Pol oscillator was used to show unit synchronization or desynchronization. Notably, the Van der Pol oscillator was selected for various reasons; first, it is a popular oscillator in biological and neural modeling carried out in previous studies (Nomura et al, 1993 ; Hu and Chung, 2013 ; Euzébio and Llibre, 2016 ). Second, as the Van der Pol oscillator ignores details of neuronal reactions, avoiding complexities in the model would be quite suitable.…”

Section: Methodsmentioning

confidence: 99%

A Neuro-Computational Model for Discrete-Continuous Dual-Task Process

Talarposhti

Ahmadi-Pajouh

Towhidkhah

2022

Front. Comput. Neurosci.

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Studies on dual-task (DT) procedures in human behavior are important, as they can offer great insight into the cognitive control system. Accordingly, a discrete-continuous auditory-tracking DT experiment was conducted in this study with different difficulty conditions, including a continuous mouse-tracking task concurrent with a discrete auditory task (AT). Behavioral results of 25 participants were investigated via different factors, such as response time (RT), errors, and hesitations (pauses in tracking tasks). In DT, synchronization of different target neuron units was observed in corresponding brain regions; consequently, a computational model of the stimulus process was proposed to investigate the DT interference procedure during the stimulus process. This generally relates to the bottom-up attention system that a neural resource allocates for various ongoing stimuli. We proposed a black-box model based on interactions and mesoscopic behaviors of neural units. Model structure was implemented based on neurological studies and oscillator units to represent neural activities. Each unit represents one stimulus feature of task concept. Comparing the model's output behavior with the experiment results (RT) validates the model. Evaluation of the proposed model and data on RT implies that the stimulus of the AT affects the DT procedure in the model output (84% correlation). However, the continuous task is not significantly changed (26% correlation). The continuous task simulation results were inconsistent with the experiment, suggesting that continuous interference occurs in higher cognitive processing regions and is controlled by the top-down attentional system. However, this is consistent with the psychological research finding of DT interference occurring in response preparation rather than the stimulus process stage. Furthermore, we developed the proposed model by adding qualitative interpretation and saving the model's generality to address various types of discrete continuous DT procedures. The model predicts a justification method for brain rhythm interactions by synchronization, and manipulating parameters would produce different behaviors. The decrement of coupling parameter and strength factor would predict a similar pattern as in Parkinson's disease and ADHD disorder, respectively. Also, by increasing the similarity factor among the features, the model's result shows automatic task performance in each task.

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Section: Methodsmentioning

confidence: 99%

A Neuro-Computational Model for Discrete-Continuous Dual-Task Process

Talarposhti

Ahmadi-Pajouh

Towhidkhah

2022

Front. Comput. Neurosci.

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“…Wen et al [33] investigate the dynamics of Mathieu equation with two kinds of van der Pol fractional-order terms. Euzebio and Llibre [9] discuss some aspects on the periodic solutions of the extended Duffing-van der Pol oscillator. They show that it can bifurcate one or three periodic solutions from a two-dimensional manifold filled by periodic solutions of the referred system.…”

Section: Introduction or The First Sectionmentioning

confidence: 99%

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Cai

Zhang

2017

NAMC

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In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis.

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Chaos and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation

Zhou

Chen

2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Chaotic dynamics and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and a harmonic excitation are investigated analytically in this paper. With the Fourier series, the system is expanded to the equivalent smooth system, and chaos arising from heteroclinic intersections is studied with the Melnikov method. The chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena, such as chaotic bands for some excitation frequencies, are presented. The relationship between the frequency range of chaotic bands and the amplitude of the excitation as well as the damping is obtained analytically. Particularly, for some system parameters satisfying a particular relationship, chaos cannot occur for any excitation amplitudes or frequencies. Subharmonic bifurcations are investigated with a subharmonic Melnikov method. It is analytically proved that the system may undergo chaotic motions through infinite or finite odd order subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold and revolution from subharmonic bifurcations to chaos obtained by analytical methods.

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Sufficient conditions for the existence of periodic solutions of the extended Duffing–Van der Pol oscillator

Cited by 4 publications

References 29 publications

A Neuro-Computational Model for Discrete-Continuous Dual-Task Process

A Neuro-Computational Model for Discrete-Continuous Dual-Task Process

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Chaos and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation

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