Renormalization Group as a Probe for Dynamical Systems

Sarkar, Abir De; Bhattacharjee, Jayanta K.

doi:10.1088/1742-6596/319/1/012017

Cited by 6 publications

(13 citation statements)

References 276 publications

(377 reference statements)

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“…To the authors' knowledge these higher order flow equations are reported for the first time in the literature. We remark that above flow equations match exactly with O(ε 3 ) flow equations of the Van der Pol equation [8]. Although not done explicitly, we expect that the O (ε 4 ) VdP flow equations would also have the equivalent forms.…”

Section: Computation Of Amplitude By Rg Methodsmentioning

confidence: 54%

“…We report here the RG solution upto order 3. To the authors' knowledge this seems to be the first higher order computation other than second order computations reported so far by various authors [6,8]. A comparison of the amplitude of the periodic cycle with the exact computations reveals that even the present higher order perturbative approximations fails to give accurate estimation for moderate values of ε.…”

Section: Introductionmentioning

confidence: 68%

“…Since the limit cycle represents a periodic motion, so we suppose that the initial approximation to the solution u (τ ) to (8) can be taken as u 0 (τ ) = cos τ Let, ω 0 and a 0 respectively denote the initial approximations of the frequency ω and the amplitude a.…”

Section: Computation Of Amplitude By Hammentioning

confidence: 99%

See 2 more Smart Citations

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

Palit¹,

Datta

2015

Differ Equ Dyn Syst

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A comparative study of the homotopy analysis method and an improved renormalization group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter ε for the amplitudes a(ε) of the limit cycles for both these oscillators are derived. The improvement in the renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.

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Section: Computation Of Amplitude By Rg Methodsmentioning

confidence: 54%

Section: Introductionmentioning

confidence: 68%

See 1 more Smart Citation

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

Palit¹,

Datta

2015

Differ Equ Dyn Syst

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“…Similar to the limit cycle, another type of closed orbit observed in both linear and nonlinear systems is called a center [17,84], as illustrated in Fig. 2.1.…”

Section: Limit Cycle Oscillationsmentioning

confidence: 99%

“…2.12d). The amplitude of centers in linear systems is dependent on the initial condition, while centers in conservative nonlinear systems are more robust against any perturbations [84]. Examples of centers are unforced undamped Duffing oscillator, Lotka-Volterra system, Lienard system, etc.…”

Section: Limit Cycle Oscillationsmentioning

confidence: 99%

An Introduction to Dynamical Systems Theory

Sujith

Pawar

2021

Springer Series in Synergetics

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As we discussed in Chap. 1, the onset of thermoacoustic instability is a nonlinear phenomenon resulting from the coupled interaction between the acoustic field in a combustor and the heat release rate field of the flame. A thermoacoustic system can undergo different dynamical transitions (called bifurcations), due to a change in any system (control) parameter. We observe that the characteristics of such dynamical transitions are governed primarily by the type of the underlying flow field, i.e., laminar or turbulent flows, present in a combustor. As a result, we notice the occurrence of different dynamical states during the onset of thermoacoustic instability in laminar and turbulent thermoacoustic systems. Furthermore, tools based on linear time series analysis are not sufficient to reveal many features exhibited by nonlinear thermoacoustic systems. In order to overcome the limitations of linear approaches in analyzing the data from experiments or numerical simulations, we need different methodologies based on dynamical systems theory to accurately characterize the nonlinear behaviors of thermoacoustic systems. In this chapter, we will introduce the basics of dynamical systems theory and cover different topics such as stability analysis, bifurcations, attractors, nonlinear measures, phase space reconstruction, Poincaré section, and recurrence plot. In the subsequent chapters, we will explore the application of these methodologies to perform time series analysis of the data obtained from experiments and numerical simulations performed on both laminar and turbulent thermoacoustic systems. Dynamical SystemThe properties of most natural and man-made systems vary with time. Some examples include the displacement of a swinging pendulum, the growth of a particular species in forest, the temperature of the earth's surface in a day, heartbeats, the local velocity of a turbulent flow, etc. A system whose behavior changes as

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Oscillators: Old and new perspectives

Bhattacharjee¹,

Roy²

2014

AIP Conference Proceedings

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Renormalization Group as a Probe for Dynamical Systems

Cited by 6 publications

References 276 publications

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

An Introduction to Dynamical Systems Theory

Oscillators: Old and new perspectives

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