Oscillators comprising of both cubic and quadratic nonlinear terms are termed in Kovacic and Gatti [1] as Dufing-Helmholtz oscillators. In case of a linear damped system with displacement, Q(t), we may represent Dufing-Helmholtz system as:

where and refer the linear damping coeficient, the natural frequency, strength of quadratic and cubic non-linearities of the system, respectively. It may be noted that, for the presence of only quadratic non-linearity, i.e. , equation (1) represents a Helmholtz asymmetric oscillator as in Kovacic and Brennan [2] while in the absence of the quadratic nonlinear term i.e., , equation (1) refers to the Dufing type oscillator used earlier to study pendulum dynamics with cubic nonlinear restoring term (Kovacic and Gatti [1]). The application of Cardano’s transformation i.e., allows one to write the Dufing-Helmholtz system (1.1) as a Dufing type oscillator,

with defining the transformed natural frequency of the system, is a constant and may be considered as a steady bias for the oscillator as in Das [3] and f is the time dependent excitation that controls the dynamics externally (Mallik [4]). Various instances of the Dufing type oscillator and synchronization of coupled oscillators have been investigated earlier, both analytically and numerically in Bhardwaj and Das [5], Mickens [6] and Nayfeh and Mook [7] and, in recent years in Brennan and Kovacic [8], Chen, et.al. [9], Das and Bhardwaj [10], Das [3], Kalm´ar-Nagy and Balachandran [11], Lie, et.al. [12], Mickens [13], Ramlan, et.al. [14], WaWrzynski [15], WaWrzynski [16]. Among the commonly used tools of nonlinear system analysis (Chaudhary, et.al. [17] and Kovacic and Brennan [18]) viz., frequency response curve, backbone curve, time histories, phase portraits, bifurcation analysis, Poincare section, basin of attraction, etc., the frequency response curve is commonly used in engineering mechanics to estimate the resonance frequency, jump-up/down frequencies etc. In particular, recently in WaWrzynski [16], the idea of origin point is introduced, based on bistability region and unstable solution domain of the periodically excited Dufing system, to understand its dynamics. Earlier, in Brennan, et.al. [19], Nayfeh and Mook [7], using Harmonic balance method (Mickens [6]) wherein a periodic solution is expressed in terms of superposition of sinusoidal functions and multi-scale perturbation analysis involving two time scales, a linearly damped harmonically excited Dufing type system is known to exhibit jump-up frequency as dependent mainly on the nonlinearity parameter, , whereas the jump-down frequency is ob served to be dependent on both and the damping parameter, respectively. The primary aim of the present work is to recast the usual small perturbation method (Mickens [13], Nayfeh and Mook [7]) of multiple time scales perturbation method [MTPM] (applicable to only weakly nonlinear forced oscillatory system) to be amenable to investigate the strongly nonlinear oscillator, , larger values of the nonlinear parameter , as well. Such a procedure involves (i) defining a new expansion parameter, α as a function of weak nonlinear parameter, ϵ and (ii) the introduction of a frequency de-tuning parameter, in the squared frequency, of the forcing term as in Burton and Rahman [20]. Based on the improved/modified method, the efect of more general form of excitation, i.e., amplitude modulated excitation (AME) on the primary and secondary resonance of a linearly damped

2

Dufing type system, , equation (2), has been investigated. Using the improved procedure, the amplitude-frequency response curves of Dufing type system when excited by AME, for diferent parameter settings are obtained in case of primary resonance.

In section II, we briefly outline the method of MTPM applied to weakly nonlinear Dufing type oscillator and its extension for the strongly nonlinear case. The numerical results regarding the efect of AME on the primary resonance both for (i) weakly and (ii) strongly nonlinear oscillator, for diferent values of parameters of the system are discussed. In section III, we present the results of numerical simulation of the response of the system. Appendix further provides some insight into the stability of the system.

In this section, we consider the forcing function, , of the externally excited Dufing system, equation (2), to be amplitude modulated. Therefore, we write, as (WaWrzynski [16]):

where and are amplitudes of the external harmonic excitation and modulation with main excitation frequency or carrier frequency, , and modulating frequency, , respectively. Since, the AME excitation of the Dufing type system therefore comprises of two additional frequencies, namely, (1) and besides the main excitation frequency , Ω. Therefore, with increased complexity of excitation, we may write the reduced form of the Dufingtype equation as,

The dynamical response of the AME Dufing type system, , equation (4), is investigated, in the following, using the multi time scale expansion method (Mickens [13], Mickens [6], Nayfeh and Mook [7]) for both instances of weak and strong nonlinearity.

In this paper, the problem of a DufingHelmholtz oscillator excited by an amplitude modulated signal is considered. Using Cardano’s transformation, the problem is shown to get reduced to a Dufing -type system excited by an amplitude modulated (AME) signal and a constant bias signal that depends on the parameters of the original system. Assuming the excitation amplitudes , modulation , damping and nonlinearity parameter, to be a perturbation on the system, a multi time scale perturbation method (MTPM) involving an expansion parameter, has been used to understand the dynamical complexities involved in getting the amplitudefrequency relationship, exhibiting bi-stability on varying the modulation parameter, , and the damping parameter, of the Dufing-type system (Figs.2.1-2.2) near primary-resonance. Phase plane trajectories exhibiting the stable and unstable manifolds for specific values of the parameters involved in the system is further illustrated in Fig.2.3.

A modified perturbation scheme (MMTPM) extending the MTPM scheme is further developed, incorporating the Lindstedt-Poincare transformation and a transformation , so as to obtain the response of the AME excited Dufing -type system even for very high values of the parameter, ϵ i.e, strongly nonlinear system. Simulation results showing the amplitude-frequency response around the back bone curve for and , as obtained in Fig.2.4 further reveal the bi-stable behavior of the system. For the parameter of the Dufing-type system as given in Fig.2.4, the peak amplitude, for is observed to be larger than that when

Appendix A.

The linear stability analysis of the amplitude and phase response for theDuffing-type system can be analyzed by considering equations (16)-(17) and writingits Jacobian matrix as,

Where

The characteristic roots, λ1,2, may therefore be written as,

Alternatively, it is observed that, the determinant, ∆, and the trace of J , are given by,

It is to be noted that, sum of the eigenvalues, i.e., Tr(J ), is negative. There-fore, for the stability of the fixed point of the amplitude and phase response, boththe eigenvalues viz., λ1,2should have negative real part. However, if one of theeigenvalue say λ1is having negative real part, then the fixed point of the systemwould be saddle only if the real part of the other eigenvalue (λ2), is positive. Incase, one of the eigenvalue, say λ2, is zero, then the system undergoes a bifurcation,i.e., saddle node/ pitch-fork bifurcation and the following condition holds,

A typical frequency-response curve is shown in Fig.3.5 where the point P2andP3refers to the jump-up and jump-down points respectively.The curve between thepoint P2and P3corresponds to the unstable domain, whereas the curve joiningthe points P1, X1and P3along with the curve joining P2, X3and P4refers to thestable region (Kalḿar-Nagy and Balachandran [11]).