1 Introduction

In self-excited nonlinear oscillators subjected to harmonic forcing, frequency-locking behavior can occur near resonances. This phenomenon results from synchronization near the resonances between the frequency of the periodic forcing of the system and the frequency of the self excitation leading to frequency-locked motions for which the response of the system follows the forcing frequency. Away from the resonances the response is quasiperiodic. Usually, the transition between quasiperiodic and synchronized motions can take place at two different specific frequencies when the forcing frequency is swept backward and forward producing hysteresis and abrupt jumps in the system response. These jumps phenomena are considered as serious physical problems that may cause the failure of systems and therefore analytical approximation of their location is of great importance from application view point. The presence of hysteresis near a resonance is associated with the coexistence of two distinct stable states (periodic and quasiperiodic) and hence, jump occurs between these two states. In this context, the control of hysteresis is an importance issue to realize a high functionality of systems and to improve their specific performance. For instance, the attenuation of hysteresis in micro-electro-mechanical systems resonators was performed by acting on the quality factor which is related to the damping [1] and the suppression of hysteresis in a forced van der Pol-Duffing oscillator was reported near resonances [2, 3].

The purpose of the present paper is to investigate analytically the frequency-locking area near the subharmonic resonance of order 3 and 4. For the 3:1 resonance, we consider a van der Pol-Duffing oscillator submitted to external harmonic excitation, whereas a van der Pol–Mathieu oscillator with a quadratic nonlinearity is considered for studying the frequency-locking near the 4:1 resonance. A perturbation method [4, 5] is applied in a first step to derive an autonomous slow flow system and analysis of equilibria of this slow flow provides analytical approximations of the amplitude-frequency response. A second perturbation method is performed in a second step on the slow flow to approximate quasiperiodic solution and its modulation domain. Numerical simulation is performed to compare with the analytical finding for validation. It is worthy noticing that the first part of the 3:1 subharmonic resonance case has been reported in previous works [6, 7] and presented here for convenience. Therefore, the present paper can be considered as a logical continuity of these previous works.

2 Frequency-locking in the 3:1 subharmonic resonance

To study the frequency-locking (or entrainment) near the 3:1 subharmonic resonance, the following forced van der Pol-Duffing oscillator given in the dimensionless form is considered

$$ \ddot{x}+\omega_{0}^{2}x-\varepsilon(\alpha -\beta x^2)\dot{x}-\varepsilon\gamma x^{3} = h\cos\omega t $$
(1)

where ω 0 is the natural frequency, α, β are the damping coefficients, γ is the nonlinear component and h, ω are the amplitude and the frequency of the external excitation, respectively, and \(\varepsilon\) is a small parameter introduced to indicate that α, β and γ are small. The purpose is to approximate the amplitude-frequency response of the 3:1 resonance, the quasiperiodic modulation domain and the locations of the frequency-locked response.

2.1 Amplitude-frequency response

To approximate the the amplitude-frequency response near the 3:1 resonance, we express the 3:1 resonance condition by introducing a detuning parameter σ according to

$$ \omega_{0}^{2}=\left(\frac{\omega}{3}\right)^{2}+\varepsilon\sigma $$
(2)

and we use the method of multiple scales [5] on Eq. (1) which can be rearranged in the form

$$ \ddot{x}+\left(\frac{\omega}{3}\right)^{2}x=h\cos\omega t+\varepsilon\{-\sigma x+(\alpha -\beta x^2)\dot{x}+\gamma x^{3}\} $$
(3)

We seek a two-scale expansion of the solution in the form

$$ x(t)=x_{0}(T_{0},T_{1})+\varepsilon x_{1}(T_{0},T_{1})+O(\varepsilon^{2}) $$
(4)

where x 0, x 1 are, respectively, the zero and the first order approximations of the solution and \(T_{i}=\varepsilon^{i}t\) (i = 0, 1, 2) are the different time scales. In terms of the variables T i , the time derivatives become \(\frac{d}{dt}=D_{0}+\varepsilon D_{1}+\varepsilon^{2}D_{2}+O(\varepsilon^3) \) and \(\frac{d^2}{dt^2}=D_{0}^2+2\varepsilon D_0 D_1+\varepsilon^2 D_{1}^2+2\varepsilon^2 D_0 D_2 +O(\varepsilon^3), \) where \(D_{i}^j=\frac{\partial j}{\partial T_{i}^j}. \) Substituting (4) into (3) and equating coefficients of the same power of \(\varepsilon, \) we obtain the following set of linear partial differential equations

$$ D_{0}^{2}x_{0}(T_{0},T_{1})+\left(\frac{\omega}{3}\right)^{2}x_{0}(T_{0},T_{1})=h\cos\omega T_{0} $$
(5)
$$ D_{0}^{2}x_{1}(T_{0},T_{1})+\left(\frac{\omega}{3}\right)^{2}x_{1}(T_{0},T_{1})=-2D_{0}D_{1}x_{0}-\sigma x_{0}+(\alpha-\beta x_{0}^2)D_{0}x_{0}+\gamma x_{0}^{3}. $$
(6)

The solution to the first order is given by

$$ x_{0}(T_{0},T_{1})=r\,(T_{1})\cos\left(\frac{\omega}{3} T_{0}+\theta(T_{1})\right)+F\cos\omega T_{0} $$
(7)

where r and θ are, respectively, the amplitude and the phase of the response and \(F=-\frac{9h}{8\omega^{2}}. \) Substituting (7) into (6), removing secular terms and using the expressions \(\frac{dr}{dt}=\varepsilon D_{1}r+O(\varepsilon^{2})\) and \(\frac{d\theta}{dt}=\varepsilon D_{1}\theta+O(\varepsilon^{2}), \) we obtain the slow flow modulation equations of amplitude and phase

$$ \begin{aligned} \frac{dr}{dt}&=Ar-Br^{3}-(H_{1}\sin 3\theta+ H_{2}\cos 3\theta)r^{2},\\ \frac{d\theta}{dt}&=S-Cr^{2}-(H_{1}\cos 3\theta- H_{2}\sin 3\theta)r \end{aligned} $$
(8)

where \(A=\frac{\alpha}{2}-\frac{\beta F^{2}}{4}, B=\frac{\beta}{8},\,C={ \frac{9\gamma}{8\omega}}, S=\frac{3\sigma}{2\omega}-\frac{9\gamma F^{2}}{4\omega},\, H_{1}=\frac{9\gamma F}{8\omega}\) and \(H_{2}=\frac{\beta F}{8}. \) Equilibria of the slow flow (8), corresponding to periodic solutions of Eq. (1), are determined by setting \(\frac{dr}{dt}=\frac{d\theta}{dt}=0. \) This leads to the amplitude-frequency response equation

$$ A_{2}r^{4}+A_{1}r^{2}+A_{0}=0 $$
(9)

where A 2 = B 2 + C 2, A 1 =  − (2AB + 2SC + H 21  + H 22 ) and A 0 = A 2 + S 2.

Figure 1 shows the variation of the amplitude-frequency response curve, as given by (9), for the given parameters α = 0.01, β = 0.05, γ = 0.1 and h = 1. The solid line denotes the stable branch and the dashed line denotes the unstable one. The stability analysis has been done using the Jacobian of the slow flow system (8). Results from numerical simulation (circles) using Runge-Kutta method are also plotted for validation.

Fig. 1
figure 1

Amplitude-frequency response curve near 3:1 resonance. Solid line for stable, dashed line for unstable and circles for numerical simulation, α = 0.01,  β = 0.05,  γ = 0.1,  h = 1

Full size image

2.2 Quasiperiodic modulation domain

To approximate quasiperiodic solutions of (1), corresponding to periodic solutions of the slow flow (8), we perform a second perturbation analysis on the following slow flow cartesian system equivalent to the polar form (8)

$$ \begin{aligned} \frac{du}{dt}&=Sv+\mu\{Au-(Bu+Cv)(u^{2}+v^{2})+2H_{1}uv-H_{2}(u^{2}-v^{2})\},\\ \frac{dv}{dt}&=-Su+\mu\{Av-(Bv-Cu)(u^{2}+v^{2})+H_{1}(u^{2}-v^{2})+2H_{2}uv\} \end{aligned} $$
(10)

where \(u=r\cos\theta\) and \(v=-r\sin\theta. \) The small parameter μ introduced in the equations specifies the scaling of the coefficients that appear therein. Notice that this double perturbation procedure has been successfully applied in previous works to capture the dynamic in quasiperiodic Mathieu equations [8, 9, 10, 11, 12]. We rewrite (10) in the form

$$ \begin{aligned} \frac{du}{dt}&=Sv+\mu f(u,v),\\ \frac{dv}{dt}&=-Su+\mu g(u,v) \end{aligned} $$
(11)

where f(u, v) and g(u, v) are given in (10).

We use the multiple scales method by expanding solutions of (10) or (11) as

$$ \left.\begin{array}{ll} u(t)=u_{0}(T_{0},T_{1},T_{2})+\mu u_{1}(T_{0},T_{1},T_{2})+\mu^{2} u_{2}(T_{0},T_{1},T_{2})+O(\mu^{3}),\\ v(t)=v_{0}(T_{0},T_{1},T_{2})+\mu v_{1}(T_{0},T_{1},T_{2})+\mu^{2}v_{2}(T_{0},T_{1},T_{2})+O(\mu^{3}) \end{array}\right. $$
(12)

where T i  = μi t. Substituting (12) into (11) and collecting terms, we get

  • Order μ 0:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{0}+S^{2}u_{0}=0,\\ Sv_{0}=D_{0}u_{0} \end{array}\right. $$
    (13)

  • Order μ 1:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{1}+S^{2}u_{1}=-2D_{0}D_{1}u_{0}+D_{0}(f(u_{0},v_{0}))+Sg(u_{0},v_{0}),\\ Sv_{1}=D_{0}u_{1}+D_{1}u_{0}-f(u_{0},v_{0}) \end{array}\right. $$
    (14)

  • Order μ 2:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{2}+S^{2}u_{2}=-2D_{0}D_{2}u_{0}-D_{0}D_{1}u_{1}-SD_{1}v_{1} +\,D_{0}\left(u_{1}\displaystyle\frac{\partial f}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial f}{\partial v}(u_{0},v_{0})\right)\\ +\,S\left(u_{1}\displaystyle\frac{\partial g}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial g}{\partial v}(u_{0},v_{0})\right),\\ Sv_{2}=D_{0}u_{2}+D_{1}u_{1}+D_{2}u_{0}-\left(u_{1}\displaystyle\frac{\partial f}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial f}{\partial v}(u_{0},v_{0})\right) \end{array}\right. $$
    (15)

    where \(D_{i}^{j}=\frac{\partial^{j}}{\partial T_{i}^{j}}. \) A solution to the first order is given by

    $$ \begin{aligned} u_{0}(T_{0},T_{1})&=R(T_{1})\cos(S T_{0}+\varphi(T_{1})),\\ v_{0}(T_{0},T_{1})&=-R(T_{1})\sin(ST_{0}+\varphi(T_{1})) \end{aligned} $$
    (16)

    Substituting (16) into (14), removing secular terms and using the expressions \(\frac{dR}{dt}=\mu D_{1}R+O(\mu^{2})\) and \(\frac{d\varphi}{dt}=\mu D_{1}\varphi+O(\mu^{2}), \) we obtain the following slow slow flow system on R and \(\varphi\)

    $$ \begin{aligned} \frac{dR}{dt}&=AR-BR^{3},\\ \frac{d\varphi}{dt}&=-CR^{2} \end{aligned} $$
    (17)

    Then, the first-order approximate periodic solution of the slow flow (11) is given by

    $$ \begin{aligned} u(t)&=R\cos\nu t,\\ v(t)&=-R\sin\nu t \end{aligned} $$
    (18)

    where the amplitude R and the frequency ν are obtained by setting \(\frac{dR}{dt}=0\) and given, respectively, by

    $$ R=\sqrt{\frac{A}{B}} $$
    (19)
    $$ \nu=S-CR^{2} $$
    (20)

    Using (18), the modulated amplitude of quasiperiodic solution of Eq. (1) is approximated by

    $$ r(t)=\sqrt{u^{2}(t)+v^{2}(t)}=R $$
    (21)

    This indicates that the quasiperiodic modulation area is reduced, at this first-order approximation, to the unique curve given by

    $$ r_{min}=r_{max}=R $$
    (22)

    where r min and r max designate the maximum and the minimum of the amplitude modulation, respectively.

In Fig. 2 the amplitude-frequency response curve as well as the quasiperiodic modulation domain obtained numerically (double circles connected with a vertical line) are shown. The horizontal line lying in the middle of circles corresponds to the first-order approximation of the quasiperiodic modulation domain as given by Eq. (22). Notice that the first-order approximation fails to capture the quasiperiodic modulation domain (double circles).

Fig. 2
figure 2

Amplitude-frequency response curve and modulation amplitude of slow flow limit cycle. Numeric: double circles, analytic to the first order: horizontal line. Values of parameters are fixed as in Fig. 1

Full size image

At the second-order approximation, a particular solution of system (14) is given by

$$ \begin{aligned} u_{1}(T_{0},T_{1},T_{2})&=\frac{R^{2}(T_{1},T_{2})}{3S}(H_{1}\cos(2S T_{0}+2\varphi(T_{1},T_{2}))-H_{2}\sin(2S T_{0}+2\varphi(T_{1},T_{2}))),\\ v_{1}(T_{0},T_{1},T_{2})&=\frac{R^{2}(T_{1},T_{2})}{3S}(H_{1}\sin(2S T_{0}+2\varphi(T_{1},T_{2}))+H_{2}\cos(2S T_{0}+2\varphi(T_{1},T_{2}))) \end{aligned} $$
(23)

Substituting (16), (19) and (23) into (15) and removing secular terms gives the following partial differential equations on R and \(\varphi\)

$$ \left.\begin{array}{ll} D_{2}R=0,\\ D_{2}\varphi=-\displaystyle\frac{2R^{2}}{3S}(H_{1}^{2}+H_{2}^{2}) \end{array}\right. $$
(24)

Then, the second-order approximate periodic solution of the slow flow (11) is given now by

$$ \begin{aligned} u(t)&=R\cos\nu t+\frac{R^{2}}{3S}(H_{1}\cos(2\nu t)-H_{2}\sin(2\nu t)),\\ v(t)&=-R\sin\nu t+\frac{R^{2}}{3S}(H_{1}\sin(2\nu t)+H_{2}\cos(2\nu t)) \end{aligned} $$
(25)

where the amplitude R and the frequency ν are obtained by setting \(\frac{dR}{dt}=D_{1}R+D_{2}R=0\) and given, respectively, by

$$ R=\sqrt{\frac{A}{B}} $$
(26)
$$ \nu = S - \left( {C + \frac{{2R^{2} }}{{3S}}\left( {H_{1}^{2} + H_{2}^{2} } \right)} \right)R^{2}$$
(27)

The modulated amplitude of quasiperiodic solution is now approximated by

$$ r(t)=\sqrt{R^{2}+\frac{R^{4}}{9S^{2}}(H_{1}^{2}+H_{2}^{2})+\frac{2R^{3}}{3S}(H_{1}\cos(3\nu t)-H_{2}\sin(3\nu t))} $$
(28)

and the envelope of this modulation is delimited by r min and r max given by

$$ r_{min}=\sqrt{R^{2}+\frac{R^{4}}{9S^{2}}(H_{1}^{2}+H_{2}^{2}) -\frac{2R^{3}}{3S}\sqrt{H_{1}^{2}+H_{2}^{2}}} $$
(29)
$$ r_{max}=\sqrt{R^{2}+\frac{R^{4}}{9S^{2}}(H_{1}^{2}+H_{2}^{2}) +\frac{2R^{3}}{3S}\sqrt{H_{1}^{2}+H_{2}^{2}}} $$
(30)

In Fig. 3 we show the modulated amplitude of the quasiperiodic solution (solid lines), as given by Eqs. (29), (30). The comparison between this analytical prediction and the numerical simulations (circles) indicates that the second-order analytical approximation can capture the quasiperiodic modulation domain. It can be seen from this figure that outside the synchronization area quasiperiodic behavior takes place. When approaching the synchronization area, the upper limit of the modulation amplitude collides with the unstable branch producing a frequency-locking.

Fig. 3
figure 3

Modulation amplitude of slow flow limit cycle. Numerical simulation: double circles, analytical approximation to the second order: double lines; r s : saddles amplitude. Values of parameters are fixed as in Fig. 1

Full size image

Figure 4 illustrates examples of phase portraits of the slow flow (10) for some values of ω picked from Fig. 3. For small values of ω, a slow flow limit cycle born by Hopf bifurcation exists and attracts all initial conditions. The related phase portraits are shown in subfigure (a) for ω = 2.72. The subfigure (b) for ω = 2.80 indicates the coexistence of a stable periodic orbit (stable equilibrium born by saddle-node bifurcation) and quasiperiodic (stable limit cycle) responses. As the forcing frequency varies, the stable slow flow limit cycle approaches the saddles and disappears via a heteroclinic bifurcation. The corresponding phase portrait is shown in subfigure (c) for the value ω = 2.86256. This mechanism gives rise to frequency-locking, in which the response of the system follows the 3:1 subharmonic frequency (see subfigure (d) for ω = 2.90). We point out that in this study, we have approximated the analytically quasiperiodic modulated domain in the case where the limit cycle disappears leaving the cycles located outside the heteroclinic connection, as shown in Fig. 4b,c. The other case of heteroclinic bifurcation leaving the cycles inside the connection is not considered here. The phase portraits corresponding to this bifurcation are illustrated in Fig. 4e,f.

Fig. 4
figure 4

Examples of phase portraits of the slow flow at different frequencies picked from Fig. 3. Values of parameters are fixed as in Fig. 1

Full size image

3 Frequency-locking in the 4:1 subharmonic resonance

To investigate the frequency-locking near the 4:1 resonance, we consider a van der Pol–Mathieu oscillator with a quadratic nonlinearity given in the dimensionless form as

$$ \ddot{x}+\omega_{0}^{2}(1+\varepsilon h\cos\omega t)x-(\varepsilon^{2}\alpha -\varepsilon\beta x)\dot{x}-\varepsilon c x^{2}=0 $$
(31)

where ω0 is the natural frequency, α, β are the damping coefficients, c is the quadratic component, h, ω are, respectively, the amplitude and the frequency of the parametric excitation, and \(\varepsilon\) is introduced for scaling the parameters. It is worth noticing that while the dynamic and the local bifurcations of Eq. (31) near the 4:1 resonances has been extensively studied [13, 14], the investigation of the global bifurcations corresponding to the frequency-locking threshold is still unexplored from analytical view point.

Our purpose here is to approximate the amplitude-frequency response near the 4:1 resonance and the modulation domain of the quasiperiodic response of Eq. (31) as well.

3.1 Amplitude-frequency response

We impose the 4:1 resonance condition by introducing a detuning parameter σ according to

$$ \omega_{0}^{2}=\left(\frac{\omega}{4}\right)^{2}+\,\varepsilon^{2}\sigma $$
(32)

We use the method of Bogoliubov-Mitropolsky [4] on Eq. (31) which can be rearranged in the form

$$ \ddot{x}+\left(\frac{\omega}{4}\right)^{2}x=\varepsilon\{c x^{2}-\beta x\dot{x}-h\left(\frac{\omega}{4}\right)^{2}x\cos\omega t\}+\varepsilon^{2}\{-\sigma x+\alpha\dot{x}\} $$
(33)

Following previous works [15], the expressions of the 4:1 subharmonic solution up to the third order are given by

$$ \begin{aligned} x(t)&=r\cos\left(\frac{\omega}{4}t+\theta\right)+\frac{8cr^{2}}{\omega^{2}}-\frac{8cr^{2}}{3\omega^{2}}\cos\left(\frac{\omega}{2}t+2\theta\right)-\frac{2\beta r^{2}}{3\omega}\sin\left(\frac{\omega}{2}t+2\theta\right)\\ &\quad+\frac{hr}{48}\cos\left(\frac{5\omega}{4}t+\theta\right)+\frac{hr}{16}\cos\left(\frac{3\omega}{4}t-\theta\right) +\left(\frac{16c^{2}}{3\omega^{4}}-\frac{\beta^2}{2\omega^{2}}\right)r^{3}\cos\left(\frac{3\omega}{4}t+3\theta\right)\\ &\quad+\frac{10\beta cr^{3}}{3\omega^{3}}\sin\left(\frac{3\omega}{4}t+3\theta\right) +\frac{4hcr^{2}}{9\omega^{2}}\cos(\omega t)-\frac{2h\beta r^{2}}{45\omega}\sin(\omega t)\\ &\quad-\frac{7hcr^{2}}{9\omega^{2}}\cos\left(\frac{\omega}{2}t-2\theta\right) +\frac{h\beta r^{2}}{36\omega}\sin\left(\frac{\omega}{2}t-2\theta\right)-\frac{hcr^{2}}{21\omega^{2}}\cos\left(\frac{3\omega}{2}t+2\theta\right)\\ &\quad-\frac{h\beta r^{2}}{60\omega}\sin\left(\frac{3\omega}{2}t+2\theta\right) +\frac{h^{2}r}{1536}\cos\left(\frac{7\omega}{4}t-\theta\right)+\frac{h^{2}r}{7680}\cos\left(\frac{9\omega}{4}t+\theta\right) \end{aligned} $$
(34)

where r and θ are, respectively, the amplitude and the phase of the response which are solutions of the following slow flow modulation system

$$ \begin{aligned} \frac{dr}{dt}&=Ar-Br^{3}-(H_{1}\sin 4\theta+ H_{2}\cos 4\theta)r^{3}\\ \frac{d\theta}{dt}&=S-Cr^{2}-(H_{1}\cos 4\theta- H_{2}\sin 4\theta)r^{2} \end{aligned} $$
(35)

Here \(A=\frac{\alpha}{2},\,B=\frac{16\beta c}{8\omega^{2}},\,C={ \frac{80c^{2}}{3\omega^{3}}}+{ \frac{\beta{2}}{6\omega}},\,S=\frac{2\sigma}{\omega}+\frac{h^{2}\omega}{192}, \,H_{1}= - \frac{20hc^{2}}{9\omega^{3}}+{ \frac{h\beta^{2}}{72\omega}}\) and \(H_{2}=\frac{h\beta c}{9\omega^{2}}. \) The detailed calculation to obtain the expression of the solution (34) is given in [15].

Equilibria of this slow flow correspond to periodic solutions of Eq. (31) which are determined by setting \(\frac{dr}{dt}=\frac{d\theta}{dt}=0. \) This leads to the amplitude-frequency response equation

$$ A_{2}r^{4}+A_{1}r^{2}+A_{0}=0 $$
(36)

where A 2 = B 2 + C 2 − H 21  − H 22 , A 1 =  − 2(AB + SC) and A 0 = A 2 + S 2.

Figure 5 illustrates the variation of the amplitude-frequency response curve, as given by (36), for the given parameters α,  = 0.01,  β = 0.2, c = 1 and h = 0.2. The solid line denotes the stable branch and the dashed line denotes the unstable one. The validity of the analytical approximations (34) is tested by performing numerical simulations on Eq. (31) using the fourth–order Runge–Kutta method as shown in Fig. 6. This figure shows the result in the phase portrait of (31) for the parameter values given above. The solid line is obtained from the analytical expression (34), while the dotted line is obtained by direct numerical simulations of (31).

Fig. 5
figure 5

Amplitude−frequency response curve near the 4:1 resonance. Solid line for stable and dashed line for unstable, α = 0.01, β = 0.2, c = 1, h = 0.2

Full size image
Fig. 6
figure 6

Comparison between the analytical approximation, (34), (solid line) and the numerical integration of (31) (dotted line). Values of parameters are fixed as in Fig. 5

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3.2 Quasiperiodic modulation domain

Next we explore the periodic motions of the slow flow (35) corresponding to quasiperiodic solution of the original system (31). Such a periodic solution can be approximated by performing a perturbation analysis on the slow flow (35) or on the corresponding Cartesian one

$$ \begin{aligned} \frac{du}{dt}&=Sv+\mu (Au-(Bu+Cv)(u^{2}+v^{2})-H_{1}v(v^{2}-3u^{2})-H_{2}u(u^{2}-3v^{2}))\\ \frac{dv}{dt}&=-Su+\mu (Av-(Bv-Cu)(u^{2}+v^{2})+H_{1}u(u^{2}-3v^{2})-H_{2}v(v^{2}-3u^{2})) \end{aligned} $$
(37)

where \(u=r\cos\theta\) and \(v=-r\sin\theta. \) The small parameter μ introduced in the equations specify the scaling of the coefficients that appear therein. The system (37) can be written in the compacted form

$$ \left\{\begin{array}{ll} \displaystyle\frac{du}{dt}=Sv+\mu f(u,v),\\ \displaystyle\frac{dv}{dt}=-Su+\mu g(u,v) \end{array}\right. $$
(38)

where f(u, v) and g(u, v) are given explicitly in (37). Using the multiple scales method, the solutions of (38) are expanded as

$$ \left.\begin{array}{ll} u(t)=u_{0}(T_{0},T_{1},T_{2})+\mu u_{1}(T_{0},T_{1},T_{2})+\mu^{2} u_{2}(T_{0},T_{1},T_{2})+O(\mu^{3}),\\ v(t)=v_{0}(T_{0},T_{1},T_{2})+\mu v_{1}(T_{0},T_{1},T_{2})+\mu^{2}v_{2}(T_{0},T_{1},T_{2})+O(\mu^{3}) \end{array}\right. $$
(39)

where T i  = μ i t. Substituting (39) into (38) and collecting terms, we get

  • Order μ 0:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{0}+S^{2}u_{0}=0,\\ Sv_{0}=D_{0}u_{0} \end{array}\right. $$
    (40)

  • Order μ 1:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{1}+S^{2}u_{1}=-2D_{0}D_{1}u_{0}+D_{0}(f(u_{0},v_{0}))+Sg(u_{0},v_{0}),\\ Sv_{1}=D_{0}u_{1}+D_{1}u_{0}-f(u_{0},v_{0}) \end{array}\right. $$
    (41)

  • Order μ 2:

    $$ \left\{\begin{array}{ll} D_{0}^{2}u_{2}+S^{2}u_{2}=-2D_{0}D_{2}u_{0}-D_{0}D_{1}u_{1}-SD_{1}v_{1}+D_{0}\left(u_{1}\displaystyle\frac{\partial f}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial f}{\partial v}(u_{0},v_{0})\right)\\ +S\left(u_{1}\displaystyle\frac{\partial g}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial g}{\partial v}(u_{0},v_{0})\right),\\ Sv_{2}=D_{0}u_{2}+D_{1}u_{1}+D_{2}u_{0}-\left(u_{1}\displaystyle\frac{\partial f}{\partial u}(u_{0},v_{0})+v_{1}\frac{\partial f}{\partial v}(u_{0},v_{0})\right) \end{array}\right. $$
    (42)

    where \(D_{i}^{j}=\frac{\partial^{j}}{\partial T_{i}^{j}}. \) A solution to the first order of system (40) is given by

    $$ \begin{aligned} u_{0}(T_{0},T_{1})&=R(T_{1})\cos(S T_{0}+\varphi(T_{1})),\\ v_{0}(T_{0},T_{1})&=-R(T_{1})\sin(ST_{0}+\varphi(T_{1})) \end{aligned} $$
    (43)

    Substituting (43) into (41), removing secular terms and using the expressions \(\frac{dR}{dt}=\mu D_{1}R+O(\mu^{2})\) and \(\frac{d\varphi}{dt}=\mu D_{1}\varphi+O(\mu^{2}), \) we obtain a slow slow flow system on R and \(\varphi\) similar to (17).

At the second-order approximation, a particular solution of system (41) is given by

$$ \begin{aligned} u_{1}(T_{0},T_{1},T_{2})&=\frac{R^{3}(T_{1},T_{2})}{4S}(H_{1}\cos(3S T_{0}+3\varphi(T_{1},T_{2}))-H_{2}\sin(3S T_{0}+3\varphi(T_{1},T_{2})))\\ v_{1}(T_{0},T_{1},T_{2})&=\frac{R^{3}(T_{1},T_{2})}{4S}(H_{1}\sin(3S T_{0}+3\varphi(T_{1},T_{2}))+H_{2}\cos(3S T_{0}+3\varphi(T_{1},T_{2}))) \end{aligned} $$
(44)

Substituting (43) and (44) into (42) and removing secular terms gives the following partial differential system on R and \(\varphi: \)

$$ \left.\begin{array}{ll} D_{2}R=0,\\ D_{2}\varphi=-\displaystyle\frac{3R^{4}}{4S}(H_{1}^{2}+H_{2}^{2}) \end{array}\right. $$
(45)

Hence, the second-order approximate periodic solution of the slow flow (11) is given by

$$ \begin{aligned} u(t)&=R\cos\nu t+\frac{R^{3}}{4S}(H_{1}\cos(3\nu t)-H_{2}\sin(3\nu t))\\ v(t)&=-R\sin\nu t+\frac{R^{3}}{4S}(H_{1}\sin(3\nu t)+H_{2}\cos(3\nu t) \end{aligned} $$
(46)

where the amplitude R and the frequency ν are now given, respectively, by

$$ R=\sqrt{\frac{A}{B}} $$
(47)
$$\nu = S - \left( {C + \frac{{3R^{2} }}{{4S}}\left( {H_{1}^{2} + H_{2}^{2} } \right)} \right)R^{2}$$
(48)

Using (46), the modulated amplitude of quasiperiodic solutions is approximated by

$$ r(t)=\sqrt{R^{2}+\frac{R^{6}}{16S^{2}}(H_{1}^{2}+H_{2}^{2})+\frac{2R^{4}}{4S}(H_{1}\cos(4\nu t)-H_{2}\sin(4\nu t))} $$
(49)

and the envelope of this modulated amplitude is delimited by r min and r max given by

$$ r_{min}=\sqrt{R^{2}+\frac{R^{6}}{16S^{2}}(H_{1}^{2}+H_{2}^{2}) -\frac{2R^{4}}{4S}\sqrt{H_{1}^{2}+H_{2}^{2}}} $$
(50)
$$ r_{max}=\sqrt{R^{2}+\frac{R^{6}}{16S^{2}}(H_{1}^{2}+H_{2}^{2}) +\frac{2R^{4}}{4S}\sqrt{H_{1}^{2}+H_{2}^{2}}} $$
(51)

Figure 7 depicts, in addition to the amplitude-frequency response, the modulated amplitudes of the slow flow limit cycle r min and r max as given by Eqs. (50) and (51). The comparaison between these analytical predictions (solid lines) and the numerical simulations (double circles) reveals that our analytical approach captures the modulation amplitude of the limit cycle close to the frequency-locking domain.

Fig. 7
figure 7

Amplitude-frequency response and the modulation amplitude of slow flow limit cycle. Numerical simulation: double circles, analytical approximation: double lines on the left between circles; r s : amplitude of saddles. Values of parameters are fixed as in Fig. 5

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Figure 8 illustrates examples of phase portraits of the slow flow (37) for some values of ω picked from Fig. 7. For small values of ω, a slow flow limit cycle born by Hopf bifurcation exists and attracts all initial conditions. The related phase portrait is shown in subfigure (a) for ω = 3.45. The subfigure (b) for ω = 3.55 indicates the coexistence of two stable states, a stable periodic orbit and quasiperiodic response. As the forcing frequency varies, the stable limit cycle approaches the saddles and synchronizes as shown in subfigure (c) for the value ω = 3.6048. Here the response of the system follows the 4:1 subharmonic frequency (see subfigure (d) for ω = 3.65).

Fig. 8
figure 8

Examples of phase portraits of the slow flow at different frequencies picked from Fig. 7. Values of parameters are fixed as in Fig. 6

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4 Conclusion

In this work, we have investigated the frequency-locking area in self-excited nonlinear oscillators subjected to harmonic excitation near the 3:1 and 4:1 subharmonic resonances. Analytical method based on a double averaging procedure is performed to approximate the quasiperiodic modulation domain and the frequency-locking area in the vicinity of the two resonances. We have focused our attention on the frequency-locking threshold in the case where the limit cycle disappears leaving the cycles outside the connection (see Figs. 4c, 8c). This corresponds to the left thresholds of the frequency-locking occurrence (see solid lines in Figs. 3, 7). It was shown that even though the synchronization domain near these secondary resonances is very small, the analytical treatment proposed here is efficient to capture the left threshold in which frequency-locking occurs.

To capture the right frequency-locking threshold in the case when the limit cycle disappears leaving the cycles inside the connection (see Figs. 4e, 8f), more efforts are required in term of approximating the quasiperiodic modulation domain.