On the Renormalization Group Techniques for the Cubic-Quintic Duffing Equation

We apply the renormalization group techniques for solving the nonlinear cubic-quintic Duffing equation in the presence of an external periodic, non-autonomous force with an additional damping term. We also make a comparative study with the multiple-time scale approach and show that the correction to the frequency is the same. Introduction In recent times the method of renormalization group (RG) has been employed [1]–[4] through the introduction of a set of modified variables to arrive at the elimination of secular terms. The theory of RG has rich connections with quantum field theory and is considered to be a very powerful tool to handle the so-called ’divergences’ of quantum electrodynamics [5]. It has several applications in the areas of phase transitions and critical phenomena [6, 7] and asymptotic analysis of a variety of perturbed ordinary and partial differential equations [1, 2, 8]. RG argument has also been used to study jump phenomena and stability in nonlinear oscillators [3]. In this article we discuss the RGmethod for the cubic-quintic Duffing oscillator, proposed by Chua [9], in the presence of an external periodic (non-autonomous) force with an additional damping term moving in a sextic potential ẍ+ αẋ+ ω 0x+ νx 3 + σx = Ω cosωt. (1) In [9] perturbative analytical techniques were proposed to derive approximate periodic solutions and period-amplitude relations. Duffing oscillator with odd nonlinearities has been studied in the literature to model the nonlinear dynamics of various systems including that of a slender elastica, the compound KdV, the propagation of a short electromagnetic pulse in a nonlinear medium (see for instance, [9]–[14] and references therein) and position or momentum-dependent mass schemes [15, 16]. In particular, Linstedt-Poincaré techniques were applied for the specific case of quintic Duffing equation by Ramos [11] by an artificial parameter method. The extended scheme (1) describes a classical particle in a triple-well potential for appropriate choices of parameters. In the phase portrait at most five equilibrium points exist for it revealing a wide variety of interesting dynamical behaviour. A modified variable τ is defined by [17] τ = ω̄t+ θ ⇒ x(t) → z(τ) ≡ z(ω̄t+ θ). (2) In the following we set ν = λε, σ = ηε and Ω = fε where an intention is to carry out a perturbation analysis in terms of the infinitesimal quantity ε << 1. We thus express equation (1) in the form ω̄2z′′ + αω̄z′ + ω 0z + λεz 3 + ηεz = fε cos( ω ω̄ (τ − θ)), z′ = dz dτ (3) We look for an expansion of both z and ω̄. To first order in ε we can write z(τ) = z0(τ) + εz1(τ) +O(ε ), ω̄ = ω̄0 + εω̄1 +O(ε ). (4) International Frontier Science Letters Submitted: 2016-03-14 ISSN: 2349-4484, Vol. 10, pp 1-7 Revised: 2016-05-28 doi:10.18052/www.scipress.com/IFSL.10.1 Accepted: 2016-09-02 2016 SciPress Ltd., Switzerland Online: 2016-12-12 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ Taking ω̄0 = ω0 so that ω̄ = ω0 + εω̄1 +O(ε), we substitute (4) into equation (3) and then collecting the terms of like powers in the perturbation parameter ε (up to order ε) yields the flow of equations ε : z′′ 0 + α ω0 z′ 0 + z0 = 0 , (5) ε : z′′ 1 + α ω0 z′ 1 + z1 = − 2ω̄1 ω0 z′ 0 − αω̄1 ω 0 z′ 0 − λ ω 0 z 0 − η ω 0 z 0 + f ω 0 cos( ω ω̄ (τ − θ)) . (6) A natural assumption is that the coefficient α of the damping term is non-negative and hence we discuss the following two cases.


Introduction
In recent times the method of renormalization group (RG) has been employed [1]- [4] through the introduction of a set of modified variables to arrive at the elimination of secular terms. The theory of RG has rich connections with quantum field theory and is considered to be a very powerful tool to handle the so-called 'divergences' of quantum electrodynamics [5]. It has several applications in the areas of phase transitions and critical phenomena [6,7] and asymptotic analysis of a variety of perturbed ordinary and partial differential equations [1,2,8]. RG argument has also been used to study jump phenomena and stability in nonlinear oscillators [3].
In this article we discuss the RG method for the cubic-quintic Duffing oscillator, proposed by Chua [9], in the presence of an external periodic (non-autonomous) force with an additional damping term moving in a sextic potentialẍ + αẋ + ω 2 0 x + νx 3 + σx 5 = Ω cos ωt. (1) In [9] perturbative analytical techniques were proposed to derive approximate periodic solutions and period-amplitude relations. Duffing oscillator with odd nonlinearities has been studied in the literature to model the nonlinear dynamics of various systems including that of a slender elastica, the compound KdV, the propagation of a short electromagnetic pulse in a nonlinear medium (see for instance, [9]- [14] and references therein) and position or momentum-dependent mass schemes [15,16].
In particular, Linstedt-Poincaré techniques were applied for the specific case of quintic Duffing equation by Ramos [11] by an artificial parameter method. The extended scheme (1) describes a classical particle in a triple-well potential for appropriate choices of parameters. In the phase portrait at most five equilibrium points exist for it revealing a wide variety of interesting dynamical behaviour. A modified variable τ is defined by [17] In the following we set ν = λϵ, σ = ηϵ and Ω = f ϵ where an intention is to carry out a perturbation analysis in terms of the infinitesimal quantity ϵ << 1. We thus express equation (1) in the form We look for an expansion of both z andω. To first order in ϵ we can write Takingω 0 = ω 0 so thatω = ω 0 + ϵω 1 + O(ϵ 2 ), we substitute (4) into equation (3) and then collecting the terms of like powers in the perturbation parameter ϵ (up to order ϵ) yields the flow of equations A natural assumption is that the coefficient α of the damping term is non-negative and hence we discuss the following two cases.

Case-I : α = 0
In the absence of damping term, we set α = 0 and in this case the solution of equation (5) reads where a is a constant and using this solution equation (6) can be reduced to The solution of equation (8) would be a cosine function as equation (7) if the right hand side of (8) were zero. The particular solution of (8) can be obtained as implying that z(τ ) is given by where the last term is the secular or the growth term. For ω 1 = 6λa 2 +5ηa 4 16ω 0 the secular term vanishes. In the Lindstedt approach, elimination of the secular terms is done in each step of the power series by recursively fixingω 1 ,ω 2 and so on. However, as is well known, there are some difficulties with the convergence of the Lindstedt expansion although such a disadvantage is not always serious for a physical problem [18]. In the following we adopt instead the RG approach that introduces an arbitrary time scale µ and the RG constants are adjusted to eliminate terms like τ − µ, τ 2 − µ 2 so that we dealt with a finite form for z.  − µ) i.e. a singularityfree time. Towards this end we introduce renormalization parameters Z 1 (µ) and Z 2 (µ) in a perturbative manner such that

Renormalization group (RG) analysis
We utilizeĀ 1 ,B 1 in such a way that the secular terms are made to vanish.
we get from (10) up to order ϵ Inspection reveals that the divergent term vanish for the conditions For this choices ofĀ 1 andB 1 solution z becomes Since the dynamics needs to be independent of the renormalization scale i.e. ∂z ∂µ = 0 which implies and this gives

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From (24) and using τ 1 = ϵ(ωt 0 + θ) we obtain up to order ϵ, The final expression for z according to equation (19), takes the form Note that the correction to the frequency of this solution is same as the solution (18).

Case-II : α > 0
In this case we solve equation (3) in the presence of damping term i.e. for α > 0. The solution of equation (5) reads where a is a constant and D is given by When the solution (28) is used, equation (6) takes the form Particular integral of equation (30) can be determined as

International Frontier Science Letters Vol. 10
which indicates that the solution z(τ ) of equation (3) up to order ϵ can be put as Clearly this solution is free from any divergent term.

Summary
In this work we have employed the RG approach to investigate the dynamical behaviour of a cubicquintic Duffing oscillator endowed with an external periodic non-autonomous force. The RG approach ensures a divergence free result. A comparative study with the multiple-time scale approach shows that the correction to the frequency is the same. We also obtained a perturbative solution of the same equation with an additional damping term.