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Vakakis, A. F.; Azeez, Mohammed F. Abdul

doi:10.1023/a:1008202529152

Cited by 34 publications

(4 citation statements)

References 14 publications

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“…Moreover, the biggest challenge comes in obtaining analytic expressions for the global manifolds (the expressions for the homoclinic or heteroclinic orbits). Vakakis and Azeez [21] developed an iterative technique to approximate certain homoclinic orbits. Mikhlin [18] and Feng et al [13] used the Padé and quasi-Padé approximation to construct these types of orbits.…”

Section: (Communicated By Hirokazu Ninomiya)mentioning

confidence: 99%

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Zhang¹,

Wang²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Section: (Communicated By Hirokazu Ninomiya)mentioning

confidence: 99%

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Zhang¹,

Wang²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…The starting point of the renormalization group is the removal of divergences (or resonance terms) from the perturbation series such that stable characteristics of system structure and dynamics are extracted which are insensitive to details, so it can be regarded as an altermative formulation of asymptotic analysis [9,11,12]. Usually asymptotic analysis techniques such as multiple scale method (MS), boundary layer method (BL) and WKB approximation are sophisticated and quite daunting to use because of their complexity and limitations [13][14][15]. On the contrary, the renormalization group method does naive perturbation expansion and requires little prior knowledge and hence is very convenient to apply in practice.…”

Section: Introductionmentioning

confidence: 99%

Koopman analysis in oscillator synchronization

Hu,

Lan

2020

Preprint

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Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems while a global statistical approach is overly cursory. Koopman operator technique seems to well balance the two approaches. In this paper, we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighbouring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.

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“…Mikhlin [16] used a quasi-Padé approximant method to construct homoclinic orbits of the Lorenz equation. Vakakis and Azeez [17] expressed the solution in power series such as t → ±0 and introduced global approximants to match the local solutions by means of a Padé-like procedure. According to the characteristics of a n-dimensional quadratic system, Li and Zhu.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

Feng¹,

Zhang²,

Wang³

et al. 2013

Chinese Phys. B

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In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

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Cited by 34 publications

References 14 publications

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

Koopman analysis in oscillator synchronization

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

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