Resonances of a forced van der Pol equation with parametric damping

Abstract

This work entails an analysis of secondary resonances in the parametrically damped van der Pol equation, with and without external excitation. A potential application of this system is a vertical-axis wind-turbine blade, which can have cyclic damping, aeroelastic self-excitation, and direct excitation. We analyze the system using the method of multiple scales and numerical solutions. For the case without external excitation, the analysis reveals nonresonant phase drift (quasiperiodic responses) and subharmonic resonance with possible phase drift or phase locking (periodic responses). The case of external excitation consists of a constant load and a harmonic load with the same frequency as the parametric term. Hard excitation is treated for nonresonant conditions and secondary resonances. Subharmonic and superharmonic resonances show possible phase drift and phase locking. Primary resonance is observed but not analyzed here.

1 Introduction

In this paper, we study the responses of an oscillator with van der Pol terms, parametric damping and direct excitation. A potential application of this system is a vertical-axis wind-turbine blade, which can endure direct excitation and parametric damping [1, 2], as well as aeroelastic self-excitation, the effects of which can be loosely modeled with van der Pol nonlinearity [3, 4]. Here, the general behavior of this system is studied, rather than the specific responses of a specific model of an application system. As both parametric excitation and van der Pol nonlinearity can induce instabilities and oscillations, we seek to understand the combined effect of such terms in this system.

The nonlinear damping in the van der Pol equation was originally introduced to model electrical oscillations [5]. This nonlinearity is well known to induce limit-cycle oscillations. Holmes and Rand [6] studied the bifurcation of the variational equation of the forced van der Pol oscillator. Barbosa et al. [7] studied the modified version of the classical van der Pol oscillator containing derivatives of fractional order. They applied approximations to fractional-order operators to show the dynamics of the model through numerical simulations. Náprstek and Fischer [8] assessed the original sub- and supersynchronization effects and their dynamic stability in a generalized van der Pol oscillator. Barron and Sen [9] investigated the synchronization of four coupled van der Pol oscillators representing the synchronization of the self-excited vibrations in turbomachine blades due to elastic coupling.

Parametric excitation also induces significant behavior in dynamical systems and therefore has been of keen interest, in particular through the Mathieu equation [3, 4, 10] in the context of parametric stiffness. Parametric damping has been shown to generate instabilities [2, 11] similar to those of the Mathieu equation [3, 4] with period-1 or period-2 oscillation, and to decay with quasiperiodic dynamics when stable [2], the latter shown by combining the Floquet solution with a harmonic balance [12, 13].

There have been vast studies on related systems that consider parametric excitation and/or direct excitation with and without nonlinearity. Examples include Mathieu oscillators without forcing [14,15,16,17], with forcing [18,19,20,21,22,23,24], and a Mathieu–van der Pol system [25]. Pandey et al. [26] inquired into frequency locking in a forced Mathieu–van der Pol–Duffing system near the principal resonances with the application of optically actuated MEMS resonators. Belhaq and Fahsi [27] examined the effect of a fast harmonic parametric excitation on frequency-locking in 2:1 and 1:1 resonances in a similar system.

Szabelski and Warminski [28] analyzed systems with three sources of vibration: parametric, self-excited and external. Warminski [29] studied the nonlinear dynamics of a self-, parametric, and externally excited oscillator with time delay by applying the method of multiple scales. Similar to the current paper, Chakraborty and Sarkar [30] studied the parametrically excited van der Pol oscillator with a modified non-linear damping term. They presented resonance and anti-resonance effects using an approach for bookkeeping slowly varying amplitudes and small terms. They also illustrated that coupling between two van der Pol oscillators can be applied as a binary switch.

Parametric excitation has also been studied in the context of wind turbine blades [21, 22, 31, 32]. Luongo and Zulli [33] studied a self-excited tower under turbulent wind flow. The tower was assumed to be a nonlinear system where the mean wind led to self excitation and the turbulent part caused parametric and external excitation.

In this study, we combine parametric damping with self-excitation of a van der Pol equation. With a particular choice of scaling and excitation frequencies, the equation becomes

$$\begin{aligned}{} & {} \ddot{x}+\epsilon (c_0+c_1 \cos \omega t +\alpha x^2 ) {\dot{x}} \nonumber \\ {}{} & {} + {\omega _n}^2 x = f_{0}+f_{1}\sin (\omega t), \end{aligned}$$

(1)

where $\epsilon \ll 1$. The variables $c_0$ and $c_1$ are the scaled mean damping and amplitude of the parametric damping, respectively, and $f_0$ and $f_1$ are mean and cyclic direct excitation constants. The excitation frequency is $\omega$ and the linearized natural frequency is $\omega _n$. We will refer to this as the parametrically damped van der Pol (PDVDP) equation with external excitation. We apply the first-order method of multiple scales [3, 34] to study the unforced and externally forced cases with parametric damping at frequency $\omega$.

In Sect. 2, we set up the analysis using the method of multiple scales. In Sect. 3, we analyze the case without external excitation, where we consider the nonresonant case and subharmonic resonance of order 1/2. In Sect. 4, we study the system with both parametric and external excitation, where in addition to previous resonance cases, we also analyze the superharmonic resonance of order 2. We provide a brief comment on the strength of nonlinearity in Sect. 5 and concluding remarks in Sect. 6.

2 Perturbation analysis: method of multiple-scales

The core of this study is the approximation of the solution to Eq. (1) based on the method of multiple scale (MMS) [3, 4]. We expand the displacement as $x(T_0,T_1,\ldots ) = x_0(T_0,T_1,\ldots ) + \epsilon x_1(T_0,T_1,\ldots ) + \epsilon ^2 x_2(T_0,T_1,\ldots ) + \cdots$, where the time scales are $T_i = \epsilon ^i t$, and $\epsilon \ll 1$. By using the chain rule, we obtain the derivatives for $n\in {\mathbb {N}}$ as $\frac{d^n}{dt^n}(\cdot ) = (D_0 + \epsilon D_1 + \epsilon ^2 D_2 + \cdots )^n(\cdot )$ , where $D_i = \frac{\partial }{\partial T_i}$. Here, we carry out the analysis up to the first order by considering the two time scales $T_0 = t$ and $T_1 = \epsilon t$ and therefore expand the displacement as

$$\begin{aligned} x(T_0,T_1)\approx x_0(T_0,T_1)+\epsilon x_1(T_0,T_1) . \end{aligned}$$

(2)

By substituting the expansion (2) in Eq. (1) and using the derivatives, coefficients of similar powers of $\epsilon$ equate as

$$\begin{aligned}{} & {} \epsilon ^0:{D_0}^2 x_0 + {\omega _n}^2 x_0 = f_{0}+f_{1}\sin (\omega T_0), \end{aligned}$$

(3)

$$\begin{aligned}{} & {} \epsilon ^1: {D_0}^2 x_1 + {\omega _n}^2 x_1\nonumber \\{} & {} \quad = -2D_0 D_1 x_0 -(c_0+c_1 \cos \omega T_0+ \alpha {x_0}^2)(D_0 x_0).\nonumber \\ \end{aligned}$$

(4)

The relationship between the excitation and the natural frequencies specifies different cases of resonance.

• 1. Nonresonant: no specific relationship between $\omega$ and $\omega _n$

• 2. Primary resonance: $\omega \approx \omega _n$

• 3. Superharmonic resonance: $\omega \approx \omega _n / 2$

• 4. Subharmonic resonance: $\omega \approx 2\omega _n$

In the next sections, we elaborate on this perturbation analysis for specific cases without and with external excitation and apply other tools to examine the dynamics with emphasis on secondary resonances.

3 Parametric excitation without external excitation

We start with the case where there is no external forcing, i.e., $f_0 = f_1 = 0$. As a survey of the possible dynamics, Fig. 1 shows a frequency sweep from $\omega =0$ to beyond $\omega = 2 \omega _n$, when $\omega _n = 1$, $\epsilon = 0.1,$ $c_0 = -1$, and $c_1 = 1$ (these parameters are dimensionless). The sweep, as a bifurcation diagram, represents samples of the x variable of the nonwandering set in a Poincaré section [35] for various values of the frequency parameter. A Runge–Kutta method (Matlab ode45) is used to obtain numerical solutions at each frequency over many periods to approach steady state. As the responses are typically quasiperiodic, the plots are generated by recording 50 values of x at the downward $\dot{x} = 0$ crossing in the phase space for each excitation frequency. Thus the points in the plot represent local maxima in x(t).

The plot shows that significant quasiperiodic dynamics occur for a large range of excitation frequencies, with a periodic window around $\omega \approx 2 \omega _n$. The largest responses occur near this subharmonic range and also the low frequencies. Superharmonic and primary resonances are not apparent beyond a possible frequency interval of periodic or nearly periodic dynamics. Figure 2 shows examples of quasiperiodic responses for three different excitation frequencies.

We appeal to perturbation analysis to explain these responses. The solution to the zeroth order Eq. (3) with $f_0=0$ and $f_1=0$ is

$$\begin{aligned} x_0(T_0, T_1) = A(T_1)e^{i \omega _n T_0} + c.c., \end{aligned}$$

(5)

where c.c. stands for the corresponding complex conjugate terms. We obtain the solvability conditions by substituting Eq. (5) into the right-hand side of Eq. (4) and eliminating the secular terms, which are the terms that make the solution to $x_1$ grow without bound in time, and thus should be eliminated. By plugging Eq. (5) into (4), we obtain

$$\begin{aligned}{} & {} {D_0}^2 x_1 + {\omega _n}^2 x_1=(-2 i \omega _n A'- i c_0 \omega _n A \nonumber \\{} & {} \quad - \alpha i \omega _n A^2 {\bar{A}} )e^{i \omega _n T_0}\nonumber \\{} & {} \quad +\frac{c_1}{2}\left( i \omega _n {\bar{A}} e^{i (\omega - \omega _n) T_0} \right) +c.c.+ \text {N.S.T.} , \end{aligned}$$

(6)

where N.S.T. stands for non-secular terms, and $A' = D_1 A$. The homogeneous solution of Eq. (6) is of the form $e^{i \omega _n T_0}$. Therefore any right-hand side term that is of the same frequency will become secular and cause $x_1$ to grow without bound.

We seek the resonance cases that lead to additional secular terms. The right-hand side of Eq. (6) merely shows a subharmonic resonance case. However, as shown in Fig. 1 as well as Eq. (6), the system has significant oscillatory behavior at the nonresonant case, that is when there is no specific relationship between the excitation frequency $\omega$ and the natural frequency $\omega _n$. The solvability conditions for these two cases are as follows.

• Nonresonant:

  $$\begin{aligned} 2 A' + c_0 A + \alpha A^2 {\bar{A}} = 0 \end{aligned}$$

  (7)

• Subharmonic resonance of order 1/2 ($\omega = 2 \omega _n + \epsilon \sigma$):

  $$\begin{aligned} 2 A' + c_0 A + \alpha A^2 {\bar{A}} - \frac{c_1}{2} {\bar{A}} e^{i \sigma T_1} = 0 \end{aligned}$$

  (8)

  where $\sigma$ is a detuning parameter.

Fig. 1

PDVDP with parametric excitation only. The numerically solved response amplitude versus the excitation frequency when $f_0=f_1=0, \omega _n=1, \epsilon =0.1, c_0=-1$, $c_1=1$, and $\alpha =1$ . The embedded subplot zooms in on the strong subharmonic resonance window. The circles are amplitudes predicted by the perturbation analysis

3.1 Nonresonant case

We first consider the nonresonant case, where the solvability condition takes the form $2 A' + c_0 A + \alpha A^2 {\bar{A}} = 0$. We recall that A is a complex function of $T_1$. Writing it as $A(T_1) = \frac{1}{2} a(T_1) e^{i \beta (T_1)},$ the solvability condition becomes

$$\begin{aligned} a' + i a \beta ' + \frac{1}{2} c_0 a + \frac{1}{8} \alpha a^3= 0. \end{aligned}$$

(9)

By separating the real and imaginary parts, we obtain the following governing equations of amplitude a and phase $\beta$ as

$$\begin{aligned} a' + \frac{1}{2} c_0 a + \frac{1}{8} \alpha a^3 = 0, \quad a \beta ' = 0. \end{aligned}$$

(10)

The response amplitude has steady-state values, obtained by setting $a' = 0$, that depend on the parameter $c_0$. When $c_0<0$, there is a stable steady-state amplitude of $a=2\sqrt{\frac{- c_0}{\alpha }}$. This amplitude and the solvability condition that leads to it are the same as in the regular van der Pol equation when $c_0 = -1$. By eliminating the solvability condition, we keep the remaining terms in Eq. (6) and find the particular solution to be

$$\begin{aligned} x_1 = Q_1 e^{3i\omega _n T_0} + Q_2 e^{i(\omega +\omega _n) T_0} + Q_3 e^{i(\omega _n-\omega ) T_0},\nonumber \\ \end{aligned}$$

(11)

where

$$\begin{aligned} Q_1= & {} \dfrac{\alpha i A^3}{8 \omega _n}, \quad Q_2 = \dfrac{c_1 i \omega _n A}{2\omega (\omega + 2\omega _n)},\nonumber \\ Q_3= & {} \dfrac{c_1 i \omega _n A}{2\omega (\omega - 2\omega _n)}, \end{aligned}$$

(12)

in which $A=\dfrac{1}{2}a e^{i \beta }$. We note that Eq. (11) is valid if $\omega \ne 2 \omega _n$. Then the leading-order nonresonant solution is

$$\begin{aligned}{} & {} x = x_0 + \epsilon x_1 =a \cos (\omega _n T_0 + \beta )\nonumber \\{} & {} \qquad +\,\epsilon ( 2 Q_1 \cos (3\omega _n T_0) + 2 Q_2 \cos (\omega +\omega _n)T_0\nonumber \\{} & {} \qquad +\, 2 Q_3 \cos (\omega -\omega _n)T_0). \end{aligned}$$

(13)

This result for the nonresonant case demonstrates a uniformly present oscillation term at the amplitude a that comes from the van der Pol element, plus small oscillatory parametric terms with frequency-dependent amplitudes, and with two independent frequencies, such that in typical cases the result is quasiperiodic.

The numerical solutions in Fig. 2 have features described by the nonresonant case. When $\omega =0.12$, the response looks like classic beating that can arise with the sum of two incommensurate simple harmonics. This is consistent with the approximate solution in Eq. (13). The cases of $\omega = 1.93$ and $\omega = 2.055$ provide bookends around the subharmonic resonance formulated in the next section. While the two latter cases seem to have features of the nonresonant solution, we will see that the subharmonic resonance analysis describes them more accurately.

Fig. 2

PDVDP with parametric excitation only. The numerically solved time response (top row) and phase portrait (bottom row) at different excitation frequencies where $f_0=f_1=0, \omega _n=1, \epsilon =0.1, c_0=-1$, $c_1=1$, and $\alpha =1$

3.2 Subharmonic resonance of order 1/2

Here, we focus on the subharmonic resonance case, where the excitation frequency is tuned to be close to twice the natural frequency, i.e., $\omega = 2\omega _n + \epsilon \sigma$. In this setting, the solvability condition is comprised of an additional term and is given as $2 A' + c_0 A + \alpha A^2 {\bar{A}} - \frac{c_1}{2} {\bar{A}} e^{i \sigma T_1} = 0$. By letting $A(T_1) = \frac{1}{2} a(T_1) e^{i \beta (T_1)}$, we obtain

$$\begin{aligned} a' + i a \beta ' + \frac{1}{2} c_0 a + \frac{1}{8} \alpha a^3 -\frac{1}{4} c_1 a e^{i (\sigma T_1 -2 \beta )} = 0.\nonumber \\ \end{aligned}$$

(14)

We separate the real and imaginary parts and obtain the following governing equations

$$\begin{aligned}{} & {} a' + \frac{1}{2} c_0 a + \frac{1}{8} \alpha a^3 -\frac{1}{4} c_1 a \cos (\sigma T_1 -2 \beta ) = 0, \nonumber \\{} & {} \quad a \beta ' - \frac{1}{4} c_1 a \sin (\sigma T_1 -2 \beta )= 0. \end{aligned}$$

(15)

To investigate the dynamics of Eq. (15), we first make the system autonomous via the change of variables $\gamma = \sigma T_1 - 2 \beta$ to obtain

$$\begin{aligned}{} & {} a' + \frac{1}{2} c_0 a +\frac{1}{8} \alpha a^3 - \frac{1}{4} c_1 a \cos \gamma = 0, \nonumber \\{} & {} \quad a\gamma ' + \frac{1}{2} c_1 a \sin \gamma -\sigma a = 0. \end{aligned}$$

(16)

The response amplitude has steady-state values, obtained by setting $a'=\gamma '=0$, that depend on the parameters $c_0$ and $c_1$. By using the trigonometric identities, we remove $\gamma$ and finally obtain the response amplitude as

$$\begin{aligned}{} & {} a = 0, \quad \text {or} \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \quad \alpha a^2 = -4 c_0 \pm 4 \sqrt{\frac{c_1^2}{4} - \sigma ^2}. \end{aligned}$$

(18)

The nonzero solutions in Eq. (18) form an ellipse in $(a^2,\sigma )$ space. If $\frac{c_1^2}{4} - \sigma ^2 > 0$, and the right-hand side of Eq. (18) is positive, then there are both zero and non-zero real-valued response amplitudes. Otherwise, the only steady-state amplitude is zero.

Figure 3 shows the steady-state amplitude versus the excitation frequency $\omega = 2\omega _n + \epsilon \sigma$ for different values of $c_0$ and $c_1$ where $\epsilon =0.1$, $\omega _n=1$, and $\alpha =1$. As the frequency increases, fixed points in a (periodic system responses) emerge at a critical value of $\omega$ that depends on $c_1$ ($\omega \approx 1.95$ for $c_1=1$), and stable solutions follow the upper branch (stability is addressed below). The amplitude a is maximum at $\omega = 2$ and the solution disappears at another critical $\omega$ ($\omega \approx 2.05$ for $c_1=1$). In the left panel in Fig. 3, when $c_0 = -1$, a larger amplitude of parametric damping $c_1$ leads to a larger response amplitude. The inner and outer ovals are associated with $c_1=0.2$ and $c_1 = 1$, respectively. An increase in the mean value of damping $c_0$, however, decreases the response amplitude by pushing down the oval-shaped curve. In the right panel, at $c_0 = 0.5$, the oval disappears at the frequency of $\omega = 2$. When $c_0 \ge 0.5$, $a^2$ has only negative values and therefore no real nonzero solution exists.

Examples of analytical solution amplitudes from Eq. (18) are plotted as circles in the subharmonic insert in Fig. 1 for comparison with the numerical solutions. The pattern of the responses described here and shown in this insert are somewhat similar to those seen in [26] with parametric stiffness at subharmonic resonance, although that case has the influence of a direct excitation of half the frequency, and a cubic stiffness as well. The system of [30] also exhibits subharmonic periodic solutions surrounded by quasiperiodic behavior, but the structure of the slow flow near $\omega =2$ is rather different.

Fig. 3

PDVDP with parametric excitation only: nonzero steady-state response amplitude versus the excitation frequency in the case of subharmonic resonance where $\omega _n=1$, $\epsilon =0.1$, and $\alpha =1$, from Eq. (17). Left: $c_0 = -1$ and $c_1=\{0.2,0.4,0.6,0.8,1\}$. Right: $c_1 = 1$ and $c_0=\{-1,-0.7,-0.4,-0.1,0.2\}$. Solid and dotted curves are stable and unstable branches, respectively

3.2.1 Stability analysis

Stability of the steady-state solutions in Eq. (18) is determined from the Jacobian of Eqs. (16), given by

$$\begin{aligned} \begin{bmatrix} -\frac{1}{4} \alpha a^2 &{} - \frac{1}{2} a \sigma \\ 0 &{} -(c_0 + \frac{1}{4} \alpha a^2) \end{bmatrix} \end{aligned}$$

(19)

which has trace $T=-c_0 - \alpha \frac{a^2}{2}$ and determinant $D = \frac{1}{4}\alpha a^2(c_0 + \frac{\alpha a^2}{4})$. Here, a represents the equilibrium amplitude. A fixed point (steady-state amplitude a) is stable if $D>0$, i.e., $\alpha a^2 + 4c_0 > 0$ ($\alpha$ is a positive number), and $T<0$, i.e., $\alpha a^2 + 2c_0 > 0$. Thus, based on Eq. (18), if $\frac{c_1^2}{4} - \sigma ^2 > 0$, then the nonzero fixed point exists and the $D>0$ criterion implies that the positive (upper) branch is stable, while the lower branch is unstable. The $a=0$ solution is neutrally stable if $c_0 > 0$. If $\frac{c_1^2}{4} - \sigma ^2 > 0$, then fixed point defined by $\alpha a^2 = -4 c_0 + 4 \sqrt{\frac{c_1^2}{4} - \sigma ^2}$ is stable.

3.2.2 Dynamics of the amplitude and phase

Figure 4 shows the stable and unstable branches (schematic) of the steady-state response amplitude (top panel), and the amplitude-phase trajectories (bottom panel) at the set of frequencies $\omega = \{ 1.93, 1.98, 2, 2.04, 2.05, 2.055\}$. The parameters are set to be $\epsilon = 0.1$, $\omega _n = 1$, $c_0 = -1$, $c_1 = 1$, and $\alpha =1$. The upper panel shows that between frequencies of $\omega = 1.95$ and $\omega = 2.05$, there exist two steady-state response amplitudes a, one on each branch of the oval-shaped curve. For $\omega <1.95$ and $\omega >2.05$, there are no nonzero fixed points. However, note that Eqs. (16) exist on a cylindrical state space, and for these frequency ranges they admit a “whirling” solution, whose mean is approximated in the top panel. Examples of whirling are depicted in the the lower panels for $\omega = 1.93$ for which the whirling solution travels from right to left, and for $\omega = 2.055$, for which the whirling solution travels from left to right. As such, the amplitude a of the leading order solution has a periodic fluctuation, and the oscillator is quasiperiodic, as labeled in the top panel of Fig. 4.

Fig. 4

PDVDP with parametric excitation only. Top panel: steady-state response amplitude versus excitation frequency close to the subharmonic resonance. Bottom panel: amplitude-phase trajectories at different corresponding frequencies $\omega = \{ 1.93, 1.98, 2, 2.04, 2.05, 2.055\}$. The simulation parameters are $\epsilon = 0.1$, $\omega _n = 1$, $c_0 = -1$, $c_1 = 1$, and $\alpha =1$. The range of the vertical axes is 0 to 4, and the range of the horizontal axes is $-\pi$ to $\pi$

As $\omega$ increases, two fixed points in $(a,\gamma )$ are created in a saddle-node bifurcation at $\omega = 1.95$. Likewise, when the frequency reaches $\omega =2.05$, the two fixed points in $(a,\gamma )$ collide and disappear in another saddle-node bifurcation. Panel 2 shows the two fixed points after the first saddle-node bifurcation. With increasing frequency, the saddle moves to the left, while the stable node moves to the right. In panel 3, at $\omega = 2.0$, the stable node has $\gamma = 0$ while the saddle has $\gamma = \pm \pi /2$ and is visible at both sides of the slice of the cylindrical phase space. In panel 4, the saddle and node have continued their trek to the left and right, respectively, and they are now approaching each other. In panel 5, at $\omega = 2.05$, the second saddle-node bifurcation has been reached, and the two fixed points coalesce. Panel 6 then shows whirling trajectories after the second saddle-node bifurcation.

Some information about the whirling orbits can be gleaned from Eqs. (16) when $\alpha =1$. These equations have a rectangular “trapping region” [35] bounded by $a^2_{\textrm{min}} = -4c_0 - 2c_1$ and $a^2_{\textrm{max}} = -4c_0 + 2c_1$. These bounds are shown as the gray shaded area in the top panel. When $\omega = 1.93$ and $\omega = 2.055$ (before and after the saddle-node bifurcations), there are no fixed points in the trapping region, and therefore the trapping region contains a whirling orbit bounded by $a_{\textrm{min}}$ and $a_{\textrm{max}}$. This is consistent with Figs. 2 and 4, where the response amplitudes are trapped in the range $[\sqrt{2}, \sqrt{6}]$.

Let us examine the temporal characteristics of the whirling orbit. The second of Eqs. (16) is a differential equation for $\gamma$ which is independent of a. Depending on $c_1$ and $\sigma$, there can be a range of $\gamma$ for which its phase flow is fast, and a range that is slow. In the lower part of Fig. 4, in panel 1 (negative $\sigma$) the slow interval is $-\pi< \gamma < 0$, while in panel 6 (positive $\sigma$) the slow interval is $0< \gamma < \pi$. In both cases, the $\gamma$ flow is fast to $a_{\textrm{max}}$ and slow to $a_{\textrm{min}}$, somewhat like a relaxation oscillation. This effect can be seen in Fig. 2, for the cases of $\omega = 1.93$ and $\omega =2.055$, where the response amplitudes decrease slowly to $a=\sqrt{2}$, and increase quickly to the maximum amplitude near $a=\sqrt{6}$.

The second of Eq. (16) is independent of a and is separable. It can be integrated to obtain $T_1$ as a function of $\gamma$. For example, when $\sigma >0$, a cycle is completed from $\gamma = -\pi$ to $\gamma = \pi$. These conditions can be used to determine the integration constants. As a result, and accounting for $T_1 = \epsilon t$, the period of whirling is

$$\begin{aligned} T_{\textrm{est}} = \dfrac{2 \pi \epsilon }{\sigma \sqrt{1-(c_1/2\sigma )^2}}, \end{aligned}$$

(20)

which is an estimate, since the slow flow of Eq. (16) was obtained through an asymptotic perturbation expansion. Using this expression, the period of beating of the solutions depicted in Fig. 2 is $T_{\textrm{est}} = 128$ (nondimensional) compared to the observed $T = 135$ for the case of $\omega = 1.93$, and $T_{\textrm{est}} = 225$ compared to the observed $T = 274$ for the case of $\omega = 2.055.$

Since $2 \beta = \sigma T_1 - \gamma$, the leading-order solution is

$$\begin{aligned} x_0= & {} a(T_1)\cos (\omega _n T_0 + \beta (T_1)) \nonumber \\ {}= & {} a(T_1)\cos \left( \dfrac{1}{2}(\omega T_0 -\gamma (T_1)\right) . \end{aligned}$$

(21)

Thus, at subharmonic resonance, the response has a reference frequency at half the forcing frequency, with amplitude and phase fluctuations. When $1.95< \omega < 2.05$ and there is a stable fixed point in $(a,\gamma )$, the steady-state solution has a fixed amplitude and phase and is phase locked. Otherwise the oscillator is in phase drift: when $\omega < 1.95$ the decreasing phase whirl increases the mean frequency, and when $\omega > 2.05$ the increasing phase whirl decreases the mean frequency, by an estimated amount $2 \pi /T_{\textrm{est}}$, while the amplitude fluctuates.

3.3 Comments

In the perturbation analysis of the system with no external forcing, we have only detected a subharmonic resonance. The first-order perturbation analysis does not reveal a primary or superharmonic resonance. Similarly, the frequency sweep (Fig. 1) does not indicate such resonant activity for the simulated parameters.

4 Parametric and external excitation

In this case, the external forcing terms $f_0$ and $f_1$ are nonzero. Similar to the previous case, as a survey of the possible dynamics, Fig. 5 shows a frequency sweep from $\omega =0$ to beyond $\omega = 3 \omega _n$, with parameters $\omega _n = 1$, $c_0 = -1$, $c_1 = 1$, $f_0 = 0.2$, and $f_1=1$ when $\alpha =1$. The sweep is based on numerical simulations and the steady-state response amplitudes are plotted. The plot shows that significant quasiperiodic dynamics occur for a large range of excitation frequencies with periodic windows around $\omega \approx \omega _n$, $\omega \approx 2 \omega _n$, and $\omega \approx 3 \omega _n$. The largest responses occur near the primary resonance range. The expanded windows show details of responses within subharmonic and superharmonic resonances.

Fig. 5

PDVDP with parametric and external excitation. The numerically solved response amplitude versus the excitation frequency $\omega$ where $f_0=0.2, f_1=1, \omega _n=1, \epsilon =0.1, c_0=-1$, $c_1=1$, and $\alpha =1$. Small circles in the insert are predictions from the perturbation analysis

In this case, the particular solution to the leading order Eq. (3) is

$$\begin{aligned}&x_0(T_0, T_1) = \Gamma + \, i \Lambda \, e^{i \omega T_0}\nonumber \\&\quad + A(T_1)e^{i \omega _n T_0} - \, i \Lambda \, e^{-i \omega T_0}+ {\bar{A}}(T_1)e^{-i \omega _n T_0}, \end{aligned}$$

(22)

where $\Gamma = \frac{f_0}{\omega _n^2}$ and $\Lambda = \frac{f_1}{2 (\omega ^2- \omega _n^2)}$. By plugging Eq. (22) into (4), we obtain

$$\begin{aligned}{} & {} {D_0}^2 x_1 + {\omega _n}^2 x_1 =( -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A\nonumber \\{} & {} \quad -\,i \alpha \omega _n A^2 {\bar{A}} -2 \alpha i \omega _n \Lambda ^2 A ) e^{i \omega _n T_0} \nonumber \\{} & {} \quad +\,( c_0 \omega \Lambda + \omega \alpha \Gamma ^2\Lambda + \omega \alpha \Lambda ^3 +2 \alpha \omega A {\bar{A}}\Lambda ) e^{i \omega T_0}\nonumber \\{} & {} \quad +\, \left( \frac{c_1}{2} \omega \Lambda + 2i \alpha \omega \Gamma \Lambda ^2\right) e^{2i \omega T_0} \nonumber \\{} & {} \quad -\, \alpha \omega \Lambda ^3 \, e^{3 i \omega T_0} +(i \frac{c_1}{2} \omega _n {\bar{A}} -2 \omega _n \alpha \Lambda \Gamma {\bar{A}}\nonumber \\{} & {} \quad +\,2 \omega \alpha \Lambda \Gamma {\bar{A}}) e^{i (\omega -\omega _n) T_0} \nonumber \\{} & {} \quad +\,i \alpha ( 2\omega - \omega _n) \Lambda ^2 \, {\bar{A}} e^{i (2\omega -\omega _n) T_0}\nonumber \\{} & {} \quad +\, \alpha (\omega - 2\omega _n ) \Lambda {\bar{A}}^2 e^{i (\omega - 2\omega _n) T_0} +c.c. + \text {N.S.T.} \end{aligned}$$

(23)

The right-hand side of Eq. (23) shows different cases of resonance; each produces different secular terms. Among all of the resonances in Eq. (23), we obtain the following solvability conditions.

• 1. Nonresonant:

  $$\begin{aligned}{} & {} -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A -i \alpha \omega _n A^2 {\bar{A}}\nonumber \\{} & {} \quad -2 \alpha i \omega _n \Lambda ^2 A =0 \end{aligned}$$

  (24)

• 2. Primary resonance ($\omega = \omega _n + \epsilon \sigma$):

  $$\begin{aligned}{} & {} \quad -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A \nonumber \\ {}{} & {} \quad -i \alpha \omega _n A^2 {\bar{A}} -2 \alpha i \omega _n \Lambda ^2 A \nonumber \\{} & {} \quad +(c_0 \omega \Lambda + \omega \alpha \Gamma ^2\Lambda + \omega \alpha \Lambda ^3 \nonumber \\ {}{} & {} \quad +2 \alpha \omega A {\bar{A}}\Lambda +i ( 2\omega \nonumber \\{} & {} \quad - \omega _n) \alpha \Lambda ^2 \, {\bar{A}}) e^{i \sigma T_1}=0 \end{aligned}$$

  (25)

• 3. Subharmonic resonance of order 1/2 ($\omega = 2 \omega _n + \epsilon \sigma$):

  $$\begin{aligned}{} & {} -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A -i \alpha \omega _n A^2 {\bar{A}} \nonumber \\{} & {} \quad -2 \alpha i \omega _n \Lambda ^2 A +(i \frac{c_1}{2} \omega _n {\bar{A}}\nonumber \\{} & {} \quad +2 (\omega -\omega _n) \alpha \Lambda \Gamma {\bar{A}})e^{i \sigma T_1}=0 \end{aligned}$$

  (26)

• 4. Subharmonic resonance of order 1/3 ($\omega = 3 \omega _n + \epsilon \sigma$):

  $$\begin{aligned}{} & {} -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A -i \alpha \omega _n A^2 {\bar{A}}\nonumber \\ {}{} & {} -2 \alpha i \omega _n \Lambda ^2 A +(\omega - 2\omega _n ) \alpha \Lambda {\bar{A}}^2 e^{i \sigma T_1}=0 \end{aligned}$$

  (27)

• 5. Superharmonic resonance of order 2 ($2\omega = \omega _n + \epsilon \sigma$):

  $$\begin{aligned}{} & {} -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A -i \alpha \omega _n A^2 {\bar{A}}\nonumber \\{} & {} -2 \alpha i \omega _n \Lambda ^2 A +(\frac{c_1}{2} \omega \Lambda +2i \omega \alpha \Gamma \Lambda ^2) e^{i \sigma T_1} =0\nonumber \\ \end{aligned}$$

  (28)

• 6. superharmonic Resonance of Order 3 ($3\omega = \omega _n + \epsilon \sigma$):

  $$\begin{aligned}{} & {} -2 i \omega _n A' - i c_0 \omega _n A -i \alpha \omega _n \Gamma ^2 A -i \alpha \omega _n A^2 {\bar{A}}\nonumber \\{} & {} -2 \alpha i \omega _n \Lambda ^2 A - \omega \alpha \Lambda ^3 e^{i \sigma T_1} =0 \end{aligned}$$

  (29)

Here we study cases 1, 3 and 5. Case 4 does not involve the parametric term, and case 6 is of minimal significance. However, the subharmonic of order 1/2 and the superharmonic of order 2 involve both van der Pol and parametric damping terms together. (Note that for the nonresonant case we include $x_1$, which is affected by $c_1$, discussed below.)

Although Fig. 5 indicates the primary resonance as a prominent case when $\omega \approx \omega _n$, the coefficient $\Lambda$ becomes singular and would contradict the multiple-scales bookkeeping strategy. The analysis of primary resonance case requires weak excitation, as well as a second-order perturbation analysis to capture the parametric term, as in [24]. This will be analyzed in a separate study.

4.1 Nonresonant case

The solvability condition in Eq. (24) is not affected by the parametric damping term and hence the behavior is similar to the forced van der Pol equation [3, 4]. In this case, the phase equation becomes $\beta ^\prime = 0$, and hence the phase $\beta$ is constant and does not influence the oscillation frequency. Equation (24) yields the following steady-state solutions

$$\begin{aligned} a = 0, \qquad a=\frac{2}{\sqrt{\alpha }}\sqrt{-c_0 - \alpha \Gamma ^2 -2 \alpha \Lambda ^2}, \end{aligned}$$

(30)

where the zero solution is unstable and the nonzero solution exists and is stable when $\alpha \Gamma ^2 + 2 \alpha \Lambda ^2 < -c_0$. Since $\alpha \Gamma ^2 + 2 \alpha \Lambda ^2 > 0$, a negative value of $c_0$ is necessary (but not sufficient) for nonzero a. If the above condition is not satisfied, then the trivial solution $a=0$ is stable.

Since the leading-order solution has the form

$$\begin{aligned} x_0 = \Gamma -2 \Lambda \sin \omega T_0 + a \cos (\omega _n T_0 + \beta ) \end{aligned}$$

(31)

when the condition $\alpha \Gamma ^2 + 2 \alpha \Lambda ^2 < -c_0$ is satisfied, $a\ne 0$ and the response becomes quasiperiodic. Otherwise, with sufficient increase in the excitation ($\Lambda$ and $\Gamma$), a is suppressed and the response becomes periodic, known as quenching [3, 4].

The parametric terms affect the first-order correction, $x_1$, in the approximate solution $x(t) = x_0(t_0, T_1) + \epsilon x_1(T_0,T_1)$, similar to what was seen in the unforced case. In eliminating the secular terms, there are several contributions of different frequency components, including $2\omega , \omega - \omega _n, \omega + \omega _n,$ from parametric excitation and van der Pol terms, and $2\omega _n, 3\omega _n, 3\omega , 2\omega - \omega _n, 2\omega + \omega _n, \omega - 2\omega _n,$ and $\omega + 2 \omega _n$, from the van der Pol terms. Thus the first-order correction, $x_1$, can contribute additional two-frequency quasiperiodic effects which depend on parametric damping $c_1$, and the content of the total response has linear combinations of two independent frequencies.

The numerical solutions in Fig. 6 demonstrate that when $\omega = 0.12$, the response is quasiperiodic. This is consistent with the leading-order solution presented in Eq. (31). Also shown in the figure are solutions near subharmonic resonance, which is discussed in the next section.

Fig. 6

PDVDP with parametric and direct excitation: numerically solved time response (top row) and phase portrait (bottom row) at different excitation frequencies where $f_0=0.2$ and $f_1=1, \omega _n=1, \epsilon =0.1, c_0=-1$, $c_1=1$, and $\alpha =1$

4.2 Subharmonic resonance of order 1/2

In this case, the excitation and natural frequencies are related as $\omega = 2 \omega _n + \epsilon \sigma$. We see from the solvability condition in Eq. (26) that in addition to the nonresonant secular terms in Eq. (24), the parametric damping and forcing appear. We substitute $A(T_1) = \frac{1}{2} a(T_1) e^{i \beta (T_1)}$ into the equation and let $\gamma =\sigma T_1-2\beta$. Then, the autonomous coupled system of governing equations of the amplitude a and phase $\gamma$ becomes

$$\begin{aligned}{} & {} a' + \frac{1}{8} \alpha a^3 + \Big ( \frac{c_0}{2} + \frac{\alpha \Gamma ^2 }{2}\nonumber \\{} & {} + \alpha \Lambda ^2 - \frac{c_1}{4} \cos \gamma + \alpha \Gamma \Lambda (1-\frac{\omega }{\omega _n}) \sin \gamma \Big )a= 0,\nonumber \\{} & {} a \gamma ' -a \sigma \!+\! \frac{c_1}{2} a \sin \gamma \!+\! 2 a \alpha \Gamma \Lambda (1\!-\!\frac{\omega }{\omega _n}) \cos \gamma \!=\! 0.\nonumber \\ \end{aligned}$$

(32)

The fixed points of Eqs. (32) are obtained in the steady-state case when $a'=\gamma '=0$, which admits $a=0$ and a nontrivial solution. The equations for the nontrivial solution take the form

$$\begin{aligned} A_1 \sin \gamma + B_1 \cos \gamma = C_1, \quad A_2 \sin \gamma + B_2 \cos \gamma = C_2, \end{aligned}$$

where the coefficients $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, and $C_2$ are functions of the parameters and the amplitude a (increasing $\alpha$ reduces the steady-state amplitudes, a). By solving for $\sin \gamma$ and $\cos \gamma$, and using the trigonometric identities, we can first obtain the phase, and then we remove the variable $\gamma$ and form an algebraic equation to obtain the steady-state amplitude a as,

$$\begin{aligned}{} & {} \alpha ^2 a^4+8 a^2 ( \alpha {c_0}+2 \alpha ^2 \Lambda ^2 + \alpha ^2 \Gamma ^2)\nonumber \\{} & {} \quad +4 \left( 4 {c_0}^2+16 \alpha {c_0} \Lambda ^2- {c_1}^2+16 \alpha ^2 \Lambda ^4+4 \sigma ^2\right) \nonumber \\{} & {} \quad + 16 \Gamma ^2 \left( 2 \alpha {c_0} -4 \alpha ^2 \Lambda ^2 \frac{\omega }{\omega _n} (\frac{\omega }{\omega _n} -2 ) + \alpha ^2 \Gamma ^2 \right) =0.\nonumber \\ \end{aligned}$$

(33)

Solving for $a^2$ yields

$$\begin{aligned}{} & {} a^2= -4 \Big (\frac{c_0}{\alpha } + \Gamma ^2 +2 \Lambda ^2 \Big )\nonumber \\{} & {} \quad \pm 2 \Gamma \Lambda \sqrt{\frac{ ({c_1}^2- 4 \sigma ^2)}{\alpha ^2 \Gamma ^2 \Lambda ^2}+\frac{16 (\omega - {\omega _n})^2}{{\omega _n}^2}}\Big ), \end{aligned}$$

(34)

which is valid if the square root in the solution is real, and if $a^2 \ge 0$. The first criterion reduces to $4 \sigma ^2 < c_1^2 + 16 \alpha ^2 \Gamma ^2 \Lambda ^2$, when using $\omega -\omega _n \approx \omega _n.$ Thus the frequency range of fixed amplitude solutions increases with $c_1$, $f_0$, and $f_1$.

Based on Eq. (31) and the definition of $\gamma$, the leading order solution takes the form

$$\begin{aligned} x_0 = \Gamma - 2 \Lambda \sin (\omega T_0) + a \cos \left( \dfrac{\omega }{2}T_0 - \dfrac{\gamma }{2}\right) . \end{aligned}$$

(35)

For the case when a and $\gamma$ are fixed and stable, there is a periodic (phase locked) response of fundamental frequency $\omega /2$. When a steady-state solution a does not exist, the response is in phase drift and is quasiperiodic.

Fig. 7

PDVDP with parametric and external excitation in the case of subharmonic resonance where $\omega _n=1$, $\alpha =1$, $\epsilon =0.1$, $c_0 =-1$, and $f_1=1$. The three plotted curves correspond to $c_1 = \{0.2, 0.5,1\}$, and the panels are for $f_0=\{0.2, 0.4, 0.6\}$. Upper and lower curves are stable and unstable branches, respectively. The larger values of $c_1$ produce the larger ovals

Fig. 8

PDVDP with parametric and direct excitation. Amplitude-phase trajectories at different corresponding frequencies $\omega = \{ 1.93, 1.98, 2, 2.04, 2.05, 2.055\}$. The simulation parameters are $\epsilon = 0.1$, $\omega _n = 1$, $c_0 = -1$, $c_1 = 1$, and $\alpha =1$. The range of the vertical axes is 0 to 4, and the range of the horizontal axes is $-\pi$ to $\pi$

Figure 7 shows the steady-state response amplitude versus the excitation frequency for small values of detuning parameter, when $\epsilon =0.1$ and $-\sigma _{0}<\sigma <\sigma _{0}$, where $\sigma _0=0.5$. Note that these figures show the amplitude a of one term in Eq. (35). The phase $\gamma$ would affect peak-to-peak amplitudes. The mean damping and periodic forcings are set to be constant, $c_0=-1$ and $f_1=1$, while different values of $c_1 = \{0.2, 0.5,1\}$ produce different closed curves. The larger values of $c_1$ are associated with the larger ovals. We see that as the constant forcing term $f_0$ is increasing between $\{0.2,0.4,0.6\}$, the ovals are expanding and the limit cycle amplitude decreases, both very slightly. The upper curves represent the stable responses and the lower curves are the unstable responses. The stable upper curves represent responses with constant a and $\gamma$, and considering the leading order solution of Eq. (35), responses are phase locked. Outside the interval of the closed curve, a and $\gamma$ are not fixed, and we have phase drift.

Figure 6 shows two numerical phase-drift solutions near subharmonic resonance. Like the unforced case, $\omega = 1.93$ and $\omega = 2.055$ are outside the interval of the subharmonic phase locking, as seen in the insert of Fig. 5. The analytical estimates of amplitude, and phase as well, for the phase-locked case, can be inserted into Eq. (35) to estimate the total response amplitude. Samples within $1.95< \omega < 2.05$ are plotted as circles in the insert of Fig. 5.

Figure 8 shows the dynamics of the amplitude and phase for the forced excitation, represented by Eqs. (32). The behavior is similar to the unforced case. For $\sigma <\sigma _0$, a whirling solution travels from right to left, and for $\sigma >\sigma _0$ the whirling solution travels from left to right. Therefore, in the leading order solution the amplitude a has a periodic fluctuation, and the oscillator is quasiperiodic. If $4 \sigma ^2 < c_1^2 + 16 \alpha ^2 \Gamma ^2 \Lambda ^2$ there exists a nonzero response amplitude. For all $\sigma$ there is a trapping region $[a_{\textrm{min}},a_{\textrm{max}}]$, obtained from the first of Eqs. (32) as $a_{\textrm{min}}=1.27$ and $a_{\textrm{max}}=2.38$ for the given parameters. These values are consistent with Figs. 8 and  6. The second of Eqs. (32) is used to find the time characteristics of the whirling orbit of the system with the external excitation. For a complete cycle, from $\gamma = -\pi$ to $\gamma = \pi$, $T_{\textrm{est}}$ is calculated. Considering $T_1 = \epsilon t$, the estimated period of whirling for the case of panel 6 in Fig. 8 is $T_{\textrm{est}}=286.1$ compared to the observed $T=241$ in Fig. 6.

By considering Eqs. (32), we can argue that if there are no fixed points in $(a,\gamma )$, we can expect whirling, where the leading order solution (Eq. 35) includes a time varying amplitude a, limited by $[a_{\textrm{min}},a_{\textrm{max}}]$ of the trapping region, and phase $\gamma$, and as such the response will be quasiperiodic. In contrast, when there is a stable fixed point in $(a,\gamma )$, the response becomes phase locked and periodic. The subharmonic behavior of the parametric plus direct excitation is thus similar to that of the parametric excitation only, although the solutions for the steady amplitudes are more complicated and distorted by the direct excitation terms $f_0$ and $f_1$.

Fig. 9

PDVDP with parametric and external excitation. Steady-state response amplitude versus the excitation frequency in the case of superharmonic resonance where $\omega _n=1$, $\alpha =1$, $\epsilon =0.1$, $c_0=-1$, and $f_1=1$. The three plotted curves correspond to $c_1 = \{0.2, 0.5,1\}$, and the panels are for $f_0=\{0, 0.4, 0.8\}$

4.3 Superharmonic resonance of order 2

In this case, the excitation and natural frequency are related as $2\omega = \omega _n + \epsilon \sigma$. Similar to the subharmonic resonance, we see from the solvability condition in Eq. (28) that in addition to the nonresonant secular terms in Eq. (24), the parametric damping and forcing terms are present. $A(T_1) = \frac{1}{2} a(T_1) e^{i \beta (T_1)}$ is substituted into the equation. Letting $\gamma =\sigma T_1-\beta$, the coupled system of governing equations of the amplitude a and phase $\gamma$ becomes

$$\begin{aligned}{} & {} a'+\frac{1}{8}\alpha a^3+ \Big ( \frac{ {c_0}}{2}+\frac{1}{2} \alpha \Gamma ^2 + \alpha \Lambda ^2 \Big ) a \nonumber \\{} & {} \quad -2 \alpha \Gamma \Lambda ^2 \frac{\omega }{\omega _n} \cos \gamma -\frac{1}{2} {c_1} \Lambda \frac{\omega }{\omega _n} \sin \gamma = 0,\nonumber \\{} & {} \quad a \gamma '-a \sigma +2 \alpha \Gamma \Lambda ^2 \frac{\omega }{\omega _n} \sin \gamma \nonumber \\{} & {} \quad -\frac{1}{2} {c_1} \Lambda \frac{\omega }{\omega _n} \cos \gamma = 0. \end{aligned}$$

(36)

For the superharmonic case, the fixed points are found by setting $a'=\gamma '=0$ in Eqs. (36). By taking a similar approach as in the subharmonic case, an algebraic expression relating the steady-state amplitude a to parameters can be obtained as

$$\begin{aligned}{} & {} \frac{a^2 {\omega _n}^2 \left( \left( 4 c_0+ \alpha (a^2+4 \Gamma ^2+8 \Lambda ^2)\right) ^2 +64 \sigma ^2 \right) }{16 \Lambda ^2 \omega ^2 \left( 16 \alpha ^2 \Gamma ^2 \Lambda ^2+ {c_1}^2\right) }=1.\nonumber \\ \end{aligned}$$

(37)

Figure 9 demonstrates the steady-state response amplitude for varying values of excitation frequencies where $\omega _n=1$, $\alpha =1$, $\epsilon =0.1$, $c_0=-1$, $c_1 = \{0.2, 0.5,1\}$, and $f_0=\{0, 0.4, 0.8\}$. Marching from the upper curves to the lower curves the value of $c_1$ is descending. Increasing $f_0$ increases the amplitude of the response such that the curves with lower values of $c_1$ are expanded more.

As seen in Fig. 5, the superharmonic window is slender and yields to dominant primary resonance curve. We looked at numerical solutions in a magnified superharmonic sweep window (not shown here) and observed that the superharmonic phase locking occurring at $\omega \approx 0.5$ has very weak (slow) stability, reducing its observation in the full system. Slightly below the superharmonic condition, it is likely that the perturbation of the slow flow overcomes its weak asymptotic solution, and apparently quasiperiodic behavior is observed. The mechanism for how the primary resonance curve dominates above the superharmonic will be examined when primary resonance is studied later.

5 Brief examination of strong nonlinearity

The response of the system in Eq. (1) varies significantly when the nonlinearity becomes stronger, i.e., when $\epsilon$ increases. If $\epsilon$ is large enough, the perturbation analysis is no longer valid. We resort to numerical simulations by using the method explained in Sect. 3. Figure 10 shows a frequency sweep with $\omega \in [0,3.5]$ for different values of $\epsilon \in [0.01,3]$, where $c_0 = -1$, $c_1 = 1$, and $\alpha =1$. At each frequency, the simulation begins at the same initial conditions. While it is possible that multiple steady-state solutions coexist, only one solution is obtained at each frequency. Significant changes in the response can be observed at different frequencies as $\epsilon$ takes larger values. This figure suggests the possibility of chaotic behavior in the presence of strong nonlinearity. In particular, Fig. 11 illustrates time responses and phase portraits of the system with large nonlinearity and suggests intermittency when $\epsilon = 2$ and $\omega =3.2$.

Fig. 10

PDVDP with parametric and direct excitation: amplitude versus excitation frequency for different values of $\epsilon$, where $f_0=0.2$ and $f_1=1, \omega _n=1, c_0=-1$, $c_1=1$, and $\alpha =1$

Fig. 11

PDVDP with parametric and direct excitation: time response (top row) and phase portrait (bottom row) at different excitation frequencies where $f_0=0.2$ and $f_1=1$, $\omega _n=1$, $\epsilon =2$, $\alpha =1$, $c_0=-1$ and $c_1=1$

6 Summary and conclusion

In this paper, we studied the responses of a forced and unforced van der Pol equation with parametric damping. The first-order method of multiple scales and numerical solutions were used. Applications can include vertical-axis wind turbine blade vibration, which can have parametric damping and van-der-Pol type terms in simplified models.

The parametric damping with no external excitation was analyzed in nonresonant and subharmonic resonance cases. The system shows an oscillatory quasiperiodic behavior in the former case. In the latter resonance case, we found the steady-state amplitude versus the excitation frequency for different damping parameters. When $c_0=-1$ (negative linear damping as with the van der Pol oscillator), the resonant response amplitude increases with the parametric damping $c_1$. An increase in the mean value of damping $c_0$ (i.e., making $c_0$ less negative) decreases the response amplitude. The dynamics of the amplitude and phase showed saddle-node bifurcations coinciding with the onset of phase locking, in which periodically whirling amplitude and phase (quasiperiodic oscillations) were replaced with a fixed steady-state amplitude and phase (periodic oscillations). In the phase drift case, the period of amplitude fluctuations could be estimated.

We then studied the parametrically damped van der Pol equation with direct excitation. In the nonresonant case, the parametric damping term does not contribute in the solvability condition and therefore it showed similar behavior as the forced van der Pol oscillator. The nonresonant system can exhibit the quenching phenomenon when the excitation through the mean or cyclic direct forcing is sufficiently large.

The subharmonic resonance behavior was similar to that of the parametric excitation without direct excitation, except that the direct excitation terms complicate and distort the steady solutions. Saddle-node bifurcations in the slow flow corresponded to transitions to phase locking. The phase-drift amplitude-fluctuation period was estimated. Increasing the parametric damping parameter, $c_1$, increases the steady-state amplitude, and the direct forcing, $f_0$ and $f_1$ perturbed the amplitude curve.

In addition to the nonresonant and subharmonic resonance, the parametrically damped van der Pol oscillator with direct excitation experienced superharmonic resonance. It was shown that the amplitude of the response increases with $c_1$, $f_0$, and $f_1$.

Our numerical studies showed the primary resonance as a dominant forced response case. The analysis of this case requires further investigation that will be done as a subsequent study with weak excitation. Based on previous studies on the cases with forcing and cyclic stiffness [24, 36], we expect that a second-order multiple-scales analysis should be considered to correctly pull out the contribution of the parametric damping to the different resonance cases.

Numerical studies with strong nonlinearity suggested responses with many harmonics and chaos are possible.