**2. Pendulum with periodically variable length**

Oscillations of a pendulum with periodically variable length (PPVL) is the classical problem of mechanics. Usually, the PPVL is associated with a child's swing, see Fig. 1. Everyone can remember that to swing a swing one must crouch when passing through the middle vertical position and straighten up at the extreme positions, i.e. perform oscillations with a frequency which is approximately twice the natural frequency of the swing. Among previous works we cite [8–15] in which analytical and numerical results on dynamic behavior of the PPVL were presented.

The present section is devoted to the study of regular and chaotic motions of the PPVL. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically. Here we extend our results published in [1–4] in investigating dynamics of this rather simple but interesting mechanical system.

©2012 Seyranian and Belyakov, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 70 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>3</sup>

## **2.1. Main relations**

Equation for motion of the swing can be derived with the use of angular momentum alteration theorem, see [8–11]. Taking into account also linear damping forces we obtain

$$\frac{d}{dt}\left(ml^2\frac{d\theta}{dt}\right) + \gamma l^2\frac{d\theta}{dt} + mgl\sin\theta = 0,\tag{1}$$

**2.2. Instability of the vertical position**

analysis of small oscillations.

described by half-cones

4 *a*2 *<sup>k</sup>* <sup>+</sup> *<sup>b</sup>*<sup>2</sup>

where *rk* = <sup>3</sup>

*ϕ*(*τ*)

for *η* as

It is convenient to change the variable by the substitution

*<sup>η</sup>*¨ <sup>+</sup> *βωη*˙ <sup>−</sup> *<sup>ε</sup>*(*ϕ*¨(*τ*) + *βωϕ*˙(*τ*))

*η*¨ + *βωη*˙ +

*η*¨ + *βωη*˙ +

(*β*/2)

 2*π* 0

*ak* <sup>=</sup> <sup>1</sup> *π* <sup>2</sup> + (2*ω*/*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)<sup>2</sup> <sup>&</sup>lt; *<sup>r</sup>*

*b*<sup>1</sup> = 0, and *r*<sup>1</sup> = 3/4. Thus, the first instability domain takes the form

*<sup>β</sup>*2/4 + (2*<sup>ω</sup>* <sup>−</sup> <sup>1</sup>)<sup>2</sup> <sup>&</sup>lt; <sup>9</sup>*<sup>ε</sup>*

*η* = *θ* (1 + *εϕ*(*τ*)). (5)

Dynamics of a Pendulum of Variable Length and Similar Problems 71

1 + *εϕ*(*τ*)

<sup>1</sup> <sup>+</sup> *εϕ*(*τ*) *<sup>η</sup>* <sup>=</sup> 0. (7)

2, *<sup>β</sup>* <sup>≥</sup> 0, *<sup>k</sup>* <sup>=</sup> 1, 2, . . . , (9)

2/16, *<sup>β</sup>* <sup>≥</sup> 0. (11)

*η* = 0. (8)

*ϕ*(*τ*) sin(*kτ*)*dτ*. (10)

= 0. (6)

Using this substitution in equation (4) and multiplying it by 1 + *εϕ*(*τ*) we obtain the equation

This equation is useful for stability study of the vertical position of the pendulum as well as

Let us analyze the stability of the trivial solution *η* = 0 of the nonlinear equation (6). Its stability with respect to the variable *η* is equivalent to that of the equation (4) with respect to *θ* due to relation (5). According to Lyapunov's theorem on stability based on a linear approximation for a system with periodic coefficients the stability (instability) of the solution *η* = 0 of equation (6) is determined by the stability (instability) of the linearized equation

*<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup>*(*ϕ*¨(*τ*) + *βωϕ*˙(*τ*))

*<sup>ω</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup>*(*ϕ*¨(*τ*) + *<sup>ω</sup>*2*ϕ*(*τ*))

This equation explicitly depends on three parameters: *ε*, *β* and *ω*. Expanding the ratio in (7) into Taylor series and keeping only first order terms with respect to *ε* and *β* we obtain

This is a Hill's equation with damping with the periodic function <sup>−</sup>(*ϕ*¨(*τ*) + *<sup>ω</sup>*2*ϕ*(*τ*)). It is known that instability (i.e. parametric resonance) occurs near the frequencies *ω* = *k*/2, where *k* = 1, 2, . . .. Instability domains in the vicinity of these frequencies were obtained in [16, 17] analytically. In three-dimensional space of the parameters *ε*, *β* and *ω*, these domains are

> 2 *k ε*

Inequalities (9) give us the first approximation of the instability domains of the vertical position of the swing. These inequalities were obtained in [1] using different variables.

Note that each *k*-th resonant domain in relations (9) depends only on *k*-th Fourier coefficients of the periodic excitation function. Particularly, for *ϕ*(*τ*) = cos(*τ*), *k* = 1 we obtain *a*<sup>1</sup> = 1,

*<sup>ϕ</sup>*(*τ*) cos(*kτ*)*dτ*, *bk* <sup>=</sup> <sup>1</sup>

*<sup>k</sup>* is expressed through the Fourier coefficients of the periodic function

 2*π* 0

*π*

<sup>1</sup> <sup>+</sup> *εϕ*(*τ*) *<sup>η</sup>* <sup>+</sup> *<sup>ω</sup>*<sup>2</sup> sin *<sup>η</sup>*

where *m* is the mass, *l* is the length, *θ* is the angle of the pendulum deviation from the vertical position, *g* is the acceleration due to gravity, and *t* is the time, Fig. 1.

**Figure 1.** Schemes of the pendulum with periodically varying length.

It is assumed that the length of the pendulum changes according to the periodic law

$$l = l\_0 + a\varphi(\Omega t) > 0,\tag{2}$$

where *l*<sup>0</sup> is the mean pendulum length, *a* and Ω are the amplitude and frequency of the excitation, *ϕ*(*τ*) is the smooth periodic function with period 2*π* and zero mean value.

We introduce new time *τ* = Ω*t* and three dimensionless parameters

$$
\varepsilon = \frac{a}{l\_0}, \quad \omega = \frac{\Omega\_0}{\Omega}, \quad \beta = \frac{\gamma}{m\Omega\_0}, \tag{3}
$$

where Ω<sup>0</sup> = *g <sup>l</sup>*<sup>0</sup> is the eigenfrequency of the pendulum with constant length *l* = *l*<sup>0</sup> and zero damping. In this notations equation (1) takes the form

$$
\ddot{\theta} + \left( \frac{2\varepsilon\dot{\phi}(\tau)}{1 + \varepsilon\phi(\tau)} + \beta\omega \right) \dot{\theta} + \frac{\omega^2 \sin(\theta)}{1 + \varepsilon\phi(\tau)} = 0,\tag{4}
$$

where the upper dot denotes differentiation with respect to new time *τ*.

Stability and oscillations of the system governed by equation (4) will be studied in the following subsections via analytically under the assumption that the excitation amplitude *ε* and the damping coefficient *ε* are small. For rotational orbits we will also assume the smallness of the frequency *ω* which means high excitation frequency compared with the eigenfrequency Ω0.

### **2.2. Instability of the vertical position**

2 Will-be-set-by-IN-TECH

Equation for motion of the swing can be derived with the use of angular momentum alteration

where *m* is the mass, *l* is the length, *θ* is the angle of the pendulum deviation from the vertical

*dt* <sup>+</sup> *mgl* sin *<sup>θ</sup>* <sup>=</sup> 0, (1)

*l* = *l*<sup>0</sup> + *aϕ*(Ω*t*) > 0, (2)

, (3)

<sup>1</sup> <sup>+</sup> *εϕ*(*τ*) <sup>=</sup> 0 , (4)

theorem, see [8–11]. Taking into account also linear damping forces we obtain

*d dt ml*<sup>2</sup> *<sup>d</sup><sup>θ</sup> dt* + *γl* <sup>2</sup> *dθ*

position, *g* is the acceleration due to gravity, and *t* is the time, Fig. 1.

**Figure 1.** Schemes of the pendulum with periodically varying length.

We introduce new time *τ* = Ω*t* and three dimensionless parameters

*<sup>ε</sup>* <sup>=</sup> *<sup>a</sup> l*0

2*εϕ*˙(*τ*)

where the upper dot denotes differentiation with respect to new time *τ*.

<sup>1</sup> <sup>+</sup> *εϕ*(*τ*) <sup>+</sup> *βω*

damping. In this notations equation (1) takes the form

¨ *θ* +

It is assumed that the length of the pendulum changes according to the periodic law

excitation, *ϕ*(*τ*) is the smooth periodic function with period 2*π* and zero mean value.

, *<sup>ω</sup>* <sup>=</sup> <sup>Ω</sup><sup>0</sup>

where *l*<sup>0</sup> is the mean pendulum length, *a* and Ω are the amplitude and frequency of the

 ˙ *θ* +

Stability and oscillations of the system governed by equation (4) will be studied in the following subsections via analytically under the assumption that the excitation amplitude *ε* and the damping coefficient *ε* are small. For rotational orbits we will also assume the smallness of the frequency *ω* which means high excitation frequency compared with the eigenfrequency

<sup>Ω</sup> , *<sup>β</sup>* <sup>=</sup> *<sup>γ</sup>*

*<sup>l</sup>*<sup>0</sup> is the eigenfrequency of the pendulum with constant length *l* = *l*<sup>0</sup> and zero

*m*Ω<sup>0</sup>

*ω*<sup>2</sup> sin(*θ*)

**2.1. Main relations**

where Ω<sup>0</sup> =

Ω0.

*g*

It is convenient to change the variable by the substitution

$$
\eta = \theta \left( 1 + \varepsilon \varrho(\tau) \right). \tag{5}
$$

Using this substitution in equation (4) and multiplying it by 1 + *εϕ*(*τ*) we obtain the equation for *η* as

$$
\ddot{\eta} + \beta \omega \dot{\eta} - \frac{\varepsilon \left( \ddot{\varrho}(\tau) + \beta \omega \dot{\phi}(\tau) \right)}{1 + \varepsilon \varrho(\tau)} \eta + \omega^2 \sin \left( \frac{\eta}{1 + \varepsilon \varrho(\tau)} \right) = 0. \tag{6}
$$

This equation is useful for stability study of the vertical position of the pendulum as well as analysis of small oscillations.

Let us analyze the stability of the trivial solution *η* = 0 of the nonlinear equation (6). Its stability with respect to the variable *η* is equivalent to that of the equation (4) with respect to *θ* due to relation (5). According to Lyapunov's theorem on stability based on a linear approximation for a system with periodic coefficients the stability (instability) of the solution *η* = 0 of equation (6) is determined by the stability (instability) of the linearized equation

$$
\ddot{\eta} + \beta \omega \dot{\eta} + \frac{\omega^2 - \varepsilon \left( \ddot{\varrho}(\tau) + \beta \omega \dot{\varrho}(\tau) \right)}{1 + \varepsilon \varrho(\tau)} \eta = 0. \tag{7}
$$

This equation explicitly depends on three parameters: *ε*, *β* and *ω*. Expanding the ratio in (7) into Taylor series and keeping only first order terms with respect to *ε* and *β* we obtain

$$
\ddot{\eta} + \beta \omega \dot{\eta} + \left[\omega^2 - \varepsilon(\ddot{\varphi}(\tau) + \omega^2 \varphi(\tau))\right] \eta = 0. \tag{8}
$$

This is a Hill's equation with damping with the periodic function <sup>−</sup>(*ϕ*¨(*τ*) + *<sup>ω</sup>*2*ϕ*(*τ*)). It is known that instability (i.e. parametric resonance) occurs near the frequencies *ω* = *k*/2, where *k* = 1, 2, . . .. Instability domains in the vicinity of these frequencies were obtained in [16, 17] analytically. In three-dimensional space of the parameters *ε*, *β* and *ω*, these domains are described by half-cones

$$(\beta/2)^2 + (2\omega/k - 1)^2 < r\_k^2 \epsilon^2, \quad \beta \ge 0, \quad k = 1, 2, \dots, \tag{9}$$

where *rk* = <sup>3</sup> 4 *a*2 *<sup>k</sup>* <sup>+</sup> *<sup>b</sup>*<sup>2</sup> *<sup>k</sup>* is expressed through the Fourier coefficients of the periodic function *ϕ*(*τ*)

$$a\_k = \frac{1}{\pi} \int\_0^{2\pi} \varrho(\tau) \cos(k\tau) d\tau, \quad b\_k = \frac{1}{\pi} \int\_0^{2\pi} \varrho(\tau) \sin(k\tau) d\tau. \tag{10}$$

Inequalities (9) give us the first approximation of the instability domains of the vertical position of the swing. These inequalities were obtained in [1] using different variables.

Note that each *k*-th resonant domain in relations (9) depends only on *k*-th Fourier coefficients of the periodic excitation function. Particularly, for *ϕ*(*τ*) = cos(*τ*), *k* = 1 we obtain *a*<sup>1</sup> = 1, *b*<sup>1</sup> = 0, and *r*<sup>1</sup> = 3/4. Thus, the first instability domain takes the form

$$
\beta^2/4 + (2\omega - 1)^2 < 9\epsilon^2/16, \quad \beta \ge 0. \tag{11}
$$

*variables q*(*τ*) and *ψ*(*τ*) defined via the following solution of the *generating equation ϑ*¨ + *ω*2*ϑ* =

We can express Poincaré variables *q*(*τ*) and *ψ*(*τ*) via *ϑ* and *ϑ*˙ from (15) as *q*<sup>2</sup> = *ϑ*<sup>2</sup> + *ϑ*˙ 2/*ω*<sup>2</sup>

expressions for *ϑ*¨, *ϑ*˙, and *ϑ* in terms of *q* and *ψ* obtained from (12), and (15). Then, in the resonant case we have equations for the *slow amplitude q*(*τ*) and *phase shift <sup>ζ</sup>*(*τ*) = *<sup>ψ</sup>*(*τ*) <sup>−</sup> <sup>1</sup>

where *f*(*ϑ*, *ϑ*˙, *τ*) = *εf*1(*ϑ*, *ϑ*˙, *τ*) + *ε*<sup>2</sup> *f*2(*ϑ*, *ϑ*˙, *τ*) + *o*(*ε*2). System (16)-(17) has small right hand sides because we assumed that *ω* − 1/2 = *O*(*ε*). As a result of averaging in the second approximation, see (121) in the Appendix, we get the system of *averaged differential equations*

<sup>5</sup>*Q*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>ω</sup>*

<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>* 

 *<sup>q</sup>* cos *<sup>τ</sup>*

16

4*ω* − 2 ∓

<sup>2</sup> arctan

*<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1) so the solution of (4) is *<sup>θ</sup>* <sup>=</sup> <sup>√</sup>*εQ*{1} cos(*τ*/2 <sup>+</sup> *<sup>Z</sup>*{1}) + *<sup>o</sup>*(

*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

<sup>4</sup> sin(2*Z*) <sup>−</sup> *<sup>Q</sup>*4*<sup>ω</sup>*

where *Q* and *Z* are the *averaged variables* corresponding to *q* and *ζ*. This system gives steady solutions for *Q*˙ = 0, *Z*˙ = 0. Thus, besides the trivial one *Q* = 0, in the first approximation we obtain from system (18)-(19) expressions for the averaged amplitude and phase shift as

where *j* = ..., −1, 0, 1, 2, . . . and "arctan" gives the major function value lying between zero and *π*; subindex "{1}" denotes the order of approximation with which the corresponding variable is obtained. Solution of system (16)-(17) in the first approximation is *<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1),

Solution of system (16)-(17) in the second approximation is the following, see (119) and (121),

− cos(*τ*) +

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> <sup>4</sup>*Z*{2}

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> <sup>4</sup>*Z*{2}

<sup>2</sup> sin(*τ*) <sup>−</sup> *<sup>ω</sup>* <sup>+</sup> <sup>2</sup>

*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

 − *ω*

 + *ω* <sup>96</sup> cos

 + *ω* <sup>96</sup> sin

∓2*βω ε*2(<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*)<sup>2</sup> <sup>−</sup> <sup>4</sup>*β*2*ω*<sup>2</sup>

*<sup>ϑ</sup>* <sup>=</sup> *<sup>q</sup>* cos(*ψ*), *<sup>ϑ</sup>*˙ <sup>=</sup> <sup>−</sup>*ω<sup>q</sup>* sin(*ψ*). (15)

Dynamics of a Pendulum of Variable Length and Similar Problems 73

<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>* , *τ* 

, <sup>−</sup>*q<sup>ω</sup>* sin *<sup>τ</sup>*

<sup>192</sup> sin(<sup>2</sup> *<sup>Z</sup>*) <sup>+</sup> *βω*˜

17*ω* − 4

*<sup>ε</sup>*2(<sup>2</sup> − *<sup>ω</sup>*)<sup>2</sup> − <sup>4</sup>*β*2*ω*<sup>2</sup>

<sup>1536</sup> <sup>−</sup> (<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*)<sup>2</sup>

*ω* + 2

<sup>8</sup> sin

<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>* , *τ* 

<sup>4</sup> cos(2*Z*)

<sup>32</sup> <sup>−</sup> *<sup>ω</sup>*2*β*˜2

 + *o*(*ε*

4

, (20)

<sup>+</sup> *<sup>o</sup>*(*ε*), (22)

<sup>+</sup> *<sup>o</sup>*(*ε*), (23)

+ *πj* , (21)

<sup>√</sup>*ε*).

<sup>8</sup> cos(2*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*2)

<sup>2</sup>*<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*Z*{2}

 + *o*(*ε* 2),

2 *τ*

, (16)

, (17)

<sup>2</sup>), (18)

(19)

. We differentiate these equations with respect to time and substitute

, <sup>−</sup>*q<sup>ω</sup>* sin *<sup>τ</sup>*

<sup>2</sup> <sup>+</sup> *<sup>ζ</sup>* 

0 which is equation (12) with *ε* = 0, when *q*˙ = 0 and *ψ*˙ = *ω*

<sup>2</sup> + *ζ <sup>ω</sup> <sup>f</sup>*

<sup>2</sup> <sup>−</sup> cos *<sup>τ</sup>*

2 + *ε* <sup>2</sup>*Qω* 

<sup>4</sup> cos(2*Z*) <sup>−</sup> *<sup>Q</sup>*2*<sup>ω</sup>*

<sup>96</sup> cos(2*Z*) <sup>−</sup> *βω*˜ <sup>2</sup>

{1} <sup>=</sup> <sup>4</sup> *ω* 

> *βω*˜ <sup>2</sup> sin

 *βω*˜ <sup>2</sup> cos  *<sup>q</sup>* cos *<sup>τ</sup>*

<sup>2</sup> + *ζ <sup>ω</sup><sup>q</sup> <sup>f</sup>*

and *<sup>ψ</sup>* <sup>=</sup> arctan

*Q*˙ = *ε Q*

*<sup>Z</sup>*˙ <sup>=</sup> *<sup>ω</sup>* <sup>−</sup> <sup>1</sup>

+ *ε* 2 5*Q*2*ω* − *ϑ*˙ *ϑω*

*<sup>q</sup>*˙ <sup>=</sup> <sup>−</sup>sin *<sup>τ</sup>*

<sup>4</sup> sin(2*Z*) <sup>−</sup> *βω*˜

*εQ*<sup>2</sup>

*<sup>Z</sup>*{1} <sup>=</sup> <sup>1</sup>

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*

2 − *ω*

*<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{2} <sup>+</sup> *<sup>ε</sup>Q*{2}

+*εQ*<sup>3</sup> {2} *ω* <sup>24</sup> cos

*<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{2} <sup>+</sup> *<sup>ε</sup>*

<sup>−</sup>*εQ*<sup>2</sup> {2} *ω* <sup>12</sup> sin

˙ *<sup>ζ</sup>* <sup>=</sup> *<sup>ω</sup>* <sup>−</sup> <sup>1</sup>

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*

<sup>2</sup> <sup>+</sup> *<sup>ε</sup>*

**Figure 2.** Instability domain (9) of PPVL (red line, left) in comparison with numerical results (black area) on parameter plane (*ω*, *ε*) at *β* = 0.05. Half cone instability domain (9) of PPVL (green surface, right) compared with the same numerical results (blue plane) in the parameter space (*ω*, *β*, *ε*).

The boundary of the first instability domain (*k* = 1) is presented in Fig. 2 by the solid red line demonstrating a good agreement with the numerically obtained instability domain which is marked black. These boundaries are also drawn in Fig. 5 and 6 by solid white lines. It is easy to see from (8) that for the second resonance domain (*k* = 2, *ω* = 1) the excitation function <sup>−</sup>(*ϕ*¨(*τ*) + *<sup>ω</sup>*2*ϕ*(*τ*)) is zero for *<sup>ϕ</sup>*(*τ*) = cos(*τ*). This explains why the second resonance domain is empty, and the numerical results confirm this conclusion, see Fig. 2. Inside the instability domains (9) the vertical position *η* = 0 becomes unstable and motion of the system can be either regular (limit cycle, regular rotation) or chaotic.

## **2.3. Limit cycle**

When the excitation amplitude *ε* is small, we can expect that the oscillation amplitude *θ* in equation (4) will be also small. We suppose that *ε* and *β* are small parameters of the same order, and *θ* = *O*( <sup>√</sup>*ε*). Then, we introduce notation *<sup>β</sup>*˜ <sup>=</sup> *<sup>β</sup>*/*ε*, *<sup>ϑ</sup>* <sup>=</sup> *<sup>θ</sup>*/ <sup>√</sup>*<sup>ε</sup>* and expand the sine into Taylor's series around zero in equation (4) keeping only three terms. Thus, equation (4) takes the following form

$$
\ddot{\theta} + \omega^2 \theta = \varepsilon f\_1(\theta, \dot{\theta}, \tau) + \varepsilon^2 f\_2(\theta, \dot{\theta}, \tau) + \dots,\tag{12}
$$

where

$$f\_1(\theta, \dot{\theta}, \tau) = \omega^2 \left(\varphi(\tau)\,\theta + \frac{\theta^3}{6}\right) - \left(\tilde{\beta}\omega + 2\dot{\varphi}(\tau)\right)\,\dot{\theta}\,\tag{13}$$

$$f\_2(\theta, \dot{\theta}, \tau) = 2\dot{\varphi}(\tau)\varphi(\tau)\,\dot{\theta} - \omega^2 \left(\varphi(\tau)\left(\varphi(\tau)\,\theta + \frac{\theta^3}{6}\right) + \frac{\theta^5}{120}\right). \tag{14}$$

We study the parametric excitation of nonlinear system (12) with the periodic function *ϕ*(*τ*) = cos *τ* at the first resonance frequency, *ω* − 1/2 = *O*(*ε*). To solve equation (12) we use the method of averaging [9, 18–20]. For that purpose we write (12) in the Bogolubov's *standard form* of first order differential equations with small right-hand sides. First, we use *Poincaré* *variables q*(*τ*) and *ψ*(*τ*) defined via the following solution of the *generating equation ϑ*¨ + *ω*2*ϑ* = 0 which is equation (12) with *ε* = 0, when *q*˙ = 0 and *ψ*˙ = *ω*

4 Will-be-set-by-IN-TECH

**Figure 2.** Instability domain (9) of PPVL (red line, left) in comparison with numerical results (black area) on parameter plane (*ω*, *ε*) at *β* = 0.05. Half cone instability domain (9) of PPVL (green surface, right)

The boundary of the first instability domain (*k* = 1) is presented in Fig. 2 by the solid red line demonstrating a good agreement with the numerically obtained instability domain which is marked black. These boundaries are also drawn in Fig. 5 and 6 by solid white lines. It is easy to see from (8) that for the second resonance domain (*k* = 2, *ω* = 1) the excitation function <sup>−</sup>(*ϕ*¨(*τ*) + *<sup>ω</sup>*2*ϕ*(*τ*)) is zero for *<sup>ϕ</sup>*(*τ*) = cos(*τ*). This explains why the second resonance domain is empty, and the numerical results confirm this conclusion, see Fig. 2. Inside the instability domains (9) the vertical position *η* = 0 becomes unstable and motion of the system can be

When the excitation amplitude *ε* is small, we can expect that the oscillation amplitude *θ* in equation (4) will be also small. We suppose that *ε* and *β* are small parameters of the same

into Taylor's series around zero in equation (4) keeping only three terms. Thus, equation (4)

<sup>√</sup>*<sup>ε</sup>* and expand the sine

*ϑ*˙ , (13)

. (14)

<sup>2</sup> *f*2(*ϑ*, *ϑ*˙, *τ*) + ..., (12)

<sup>√</sup>*ε*). Then, we introduce notation *<sup>β</sup>*˜ <sup>=</sup> *<sup>β</sup>*/*ε*, *<sup>ϑ</sup>* <sup>=</sup> *<sup>θ</sup>*/

*ϑ*3 6 −

> *ϕ*(*τ*)

We study the parametric excitation of nonlinear system (12) with the periodic function *ϕ*(*τ*) = cos *τ* at the first resonance frequency, *ω* − 1/2 = *O*(*ε*). To solve equation (12) we use the method of averaging [9, 18–20]. For that purpose we write (12) in the Bogolubov's *standard form* of first order differential equations with small right-hand sides. First, we use *Poincaré*

*βω*˜ + 2*ϕ*˙(*τ*)

*ϕ*(*τ*) *ϑ* +

*ϑ*3 6 + *ϑ*5 120 

*ϑ*¨ + *ω*2*ϑ* = *εf*1(*ϑ*, *ϑ*˙, *τ*) + *ε*

*ϕ*(*τ*) *ϑ* +

*<sup>f</sup>*2(*ϑ*, *<sup>ϑ</sup>*˙, *<sup>τ</sup>*) = <sup>2</sup>*ϕ*˙(*τ*)*ϕ*(*τ*) *<sup>ϑ</sup>*˙ <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

compared with the same numerical results (blue plane) in the parameter space (*ω*, *β*, *ε*).

either regular (limit cycle, regular rotation) or chaotic.

*f*1(*ϑ*, *ϑ*˙, *τ*) = *ω*<sup>2</sup>

**2.3. Limit cycle**

order, and *θ* = *O*(

where

takes the following form

$$
\theta = q \cos(\psi), \quad \dot{\theta} = -\omega q \sin(\psi). \tag{15}
$$

We can express Poincaré variables *q*(*τ*) and *ψ*(*τ*) via *ϑ* and *ϑ*˙ from (15) as *q*<sup>2</sup> = *ϑ*<sup>2</sup> + *ϑ*˙ 2/*ω*<sup>2</sup> and *<sup>ψ</sup>* <sup>=</sup> arctan − *ϑ*˙ *ϑω* . We differentiate these equations with respect to time and substitute expressions for *ϑ*¨, *ϑ*˙, and *ϑ* in terms of *q* and *ψ* obtained from (12), and (15). Then, in the resonant case we have equations for the *slow amplitude q*(*τ*) and *phase shift <sup>ζ</sup>*(*τ*) = *<sup>ψ</sup>*(*τ*) <sup>−</sup> <sup>1</sup> 2 *τ*

$$\dot{q} = -\frac{\sin\left(\frac{\pi}{2} + \zeta\right)}{\omega} f\left(q \cos\left(\frac{\pi}{2} + \zeta\right), -q\omega \sin\left(\frac{\pi}{2} + \zeta\right), \tau\right),\tag{16}$$

$$\dot{\zeta} = \omega - \frac{1}{2} - \frac{\cos\left(\frac{\tau}{2} + \zeta\right)}{\omega\eta} f\left(q \cos\left(\frac{\tau}{2} + \zeta\right), -q\omega \sin\left(\frac{\tau}{2} + \zeta\right), \tau\right),\tag{17}$$

where *f*(*ϑ*, *ϑ*˙, *τ*) = *εf*1(*ϑ*, *ϑ*˙, *τ*) + *ε*<sup>2</sup> *f*2(*ϑ*, *ϑ*˙, *τ*) + *o*(*ε*2). System (16)-(17) has small right hand sides because we assumed that *ω* − 1/2 = *O*(*ε*). As a result of averaging in the second approximation, see (121) in the Appendix, we get the system of *averaged differential equations*

$$\dot{Q} = \varepsilon Q \left( \frac{2 - \omega}{4} \sin(2Z) - \frac{\tilde{\beta}\omega}{2} \right) + \varepsilon^2 Q \omega \left( 5Q^2 \frac{2 - \omega}{192} \sin(2Z) + \frac{\tilde{\beta}\omega}{4} \cos(2Z) \right) + o(\varepsilon^2), \tag{18}$$

$$\dot{Z} = \omega - \frac{1}{2} + \varepsilon \left(\frac{2-\omega}{4}\cos(2Z) - \frac{Q^2\omega}{16}\right) \tag{19}$$

$$1 + \varepsilon^2 \left( 5Q^2 \omega \frac{2 - \omega}{96} \cos(2Z) - \frac{\tilde{\beta}\omega^2}{4} \sin(2Z) - Q^4 \omega \frac{17\omega - 4}{1536} - \frac{(2 - \omega)^2}{32} - \frac{\omega^2 \tilde{\beta}^2}{4} \right) + o(\varepsilon^2),$$

where *Q* and *Z* are the *averaged variables* corresponding to *q* and *ζ*. This system gives steady solutions for *Q*˙ = 0, *Z*˙ = 0. Thus, besides the trivial one *Q* = 0, in the first approximation we obtain from system (18)-(19) expressions for the averaged amplitude and phase shift as

$$
\varepsilon Q\_{\{1\}}^2 = \frac{4}{\omega} \left( 4\omega - 2 \mp \sqrt{\varepsilon^2 (2 - \omega)^2 - 4\beta^2 \omega^2} \right),
\tag{20}
$$

$$Z\_{\{1\}} = \frac{1}{2} \arctan\left(\frac{\mp 2\beta\omega}{\sqrt{\varepsilon^2 (2-\omega)^2 - 4\beta^2 \omega^2}}\right) + \pi j \,\,\,\tag{21}$$

where *j* = ..., −1, 0, 1, 2, . . . and "arctan" gives the major function value lying between zero and *π*; subindex "{1}" denotes the order of approximation with which the corresponding variable is obtained. Solution of system (16)-(17) in the first approximation is *<sup>q</sup>* <sup>=</sup> *<sup>Q</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1), *<sup>ζ</sup>* <sup>=</sup> *<sup>Z</sup>*{1} <sup>+</sup> *<sup>o</sup>*(1) so the solution of (4) is *<sup>θ</sup>* <sup>=</sup> <sup>√</sup>*εQ*{1} cos(*τ*/2 <sup>+</sup> *<sup>Z</sup>*{1}) + *<sup>o</sup>*( <sup>√</sup>*ε*).

Solution of system (16)-(17) in the second approximation is the following, see (119) and (121),

$$\begin{split} q &= Q\_{\{2\}} + \varepsilon Q\_{\{2\}} \left( \frac{\tilde{\mathbb{A}}\omega}{2} \sin \left( \tau + 2Z\_{\{2\}} \right) - \cos(\tau) + \frac{\omega + 2}{8} \cos(2\tau + 2Z\_2) \right) \\ &+ \varepsilon Q\_{\{2\}}^3 \left( \frac{\omega}{24} \cos \left( \tau + 2Z\_{\{2\}} \right) + \frac{\omega}{96} \cos \left( 2\tau + 4Z\_{\{2\}} \right) \right) + o(\varepsilon), \end{split} \tag{22}$$

$$\begin{split} \zeta = Z\_{\{2\}} + \varepsilon \left( \frac{\vec{\mathbb{A}}\omega}{2} \cos \left( \tau + 2Z\_{\{2\}} \right) - \frac{\omega}{2} \sin(\tau) - \frac{\omega + 2}{8} \sin \left( 2\tau + 2Z\_{\{2\}} \right) \right) \\ - \varepsilon Q\_{\{2\}}^2 \left( \frac{\omega}{12} \sin \left( \tau + 2Z\_{\{2\}} \right) + \frac{\omega}{96} \sin \left( 2\tau + 4Z\_{\{2\}} \right) \right) + o(\varepsilon), \end{split} \tag{23}$$

#### 6 Will-be-set-by-IN-TECH 74 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>7</sup>

where *<sup>Q</sup>*{2} and *<sup>Z</sup>*{2} are the steady state variables of system (18)-(19) in the second order approximation. Substitution of these expressions into (15) yields the second order approximate solution of (4) in the following form

$$\begin{split} \theta &= \sqrt{\varepsilon}Q\_{\{2\}}\cos\left(\frac{\tau}{2} + Z\_{\{2\}}\right) + \beta\sqrt{\varepsilon}Q\_{\{2\}}\frac{\omega}{2}\sin\left(\frac{\tau}{2} + Z\_{\{2\}}\right) \\ &- \varepsilon^{\frac{3}{2}}Q\_{\{2\}}\left(\frac{2-\omega}{4}\cos\left(\frac{\tau}{2} - Z\_{\{2\}}\right) + \frac{2+\omega}{8}\cos\left(\frac{3\tau}{2} + Z\_{\{2\}}\right)\right) \\ &+ \varepsilon^{\frac{3}{2}}Q\_{\{2\}}^{3}\left(\frac{\omega}{16}\cos\left(\frac{\tau}{2} + Z\_{\{2\}}\right) - \frac{\omega}{96}\cos\left(\frac{3\tau}{2} + 3Z\_{\{2\}}\right)\right) + o(\varepsilon^{2}) .\end{split} \tag{24}$$

0 2\*pi 4\*pi 6\*pi 8\*pi

0 2\*pi 4\*pi 6\*pi 8\*pi

*x*˙1 = *x*<sup>2</sup> − sign(*b*),


sin *<sup>s</sup>* |*b*| 

sin *<sup>s</sup>* |*b*| ,

 2*ε x*3

*x*˙2 =

*<sup>x</sup>*˙3 <sup>=</sup> <sup>−</sup> *<sup>ε</sup>*

time (τ)

Regular rotation-oscillation with *b* = 0.

system quasi-linear.

form

the *phase mismatch*, *x*<sup>2</sup> = *<sup>d</sup><sup>θ</sup>*

0 2\*pi 4\*pi 6\*pi 8\*pi

time (τ)

Dynamics of a Pendulum of Variable Length and Similar Problems 75

0 2\*pi 4\*pi 6\*pi 8\*pi

is the excitation. From here the

(25)

time (τ)

−6\*pi

−2\*pi

c) *ε* = 0.43, *ω* = 0.5, *β* = 0.05 d) *ε* = 0.59, *ω* = 0.6, *β* = 0.05

that *ε*, *β* and *ω* are small parameters, *ε* being of order *ω*2, and *β* of order *ω*3, which makes the

We introduce the vector of *slow* variables *x* and the fast time *s* = |*b*| *τ*, where *x*<sup>1</sup> = *θ* − *b τ* is

dot denotes derivative with respect to the new time *s*. Thus, equations (4) takes the standard

<sup>−</sup> *βω <sup>x</sup>*<sup>2</sup>


<sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *b*2


> sin(*x*<sup>1</sup> + *s*) *x*3

,

**Figure 4.** a) Regular rotation-oscillation with the mean angular velocity equal to one half of the excitation frequency, *b* = −1/2. b) Regular rotation with *b* = −1. c) Regular rotation with *b* = −2. d)

*ds* is the velocity, *<sup>x</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>ε</sup>* cos *<sup>s</sup>*

−pi

0

angle (θ)

a) *ε* = 0.51, *ω* = 0.54, *β* = 0.05 b) *ε* = 0.28, *ω* = 0.5, *β* = 0.05

−4\*pi

−2\*pi

angle (θ)

0

time (τ)

−4\*pi

−12\*pi

−8\*pi

−4\*pi

angle (θ)

0

−2\*pi

0

angle (θ)

**Figure 3.** Frequency-response curve for the parameters *ε* = 0.04 and *β* = 0.05. Amplitude of the limit cycle in the first (left) and second (right) approximation compared with the results of numerical simulations (circles) depending on the relative excitation frequency *ω*. In the first approximation (left) the amplitude <sup>√</sup>*εQ*{1} is described by (20). In the second approximation (right) the amplitude of solution (24) is calculated with the use of numerically obtained steady state *<sup>Q</sup>*{2} and *<sup>Z</sup>*{2} of system (18)-(19).

Fig. 3 shows better coincidence with the numerical simulations of the second order approximate solution (24) up to the amplitude equals 2 ≈ 2*π*/3 and the frequency mismatch is *<sup>ω</sup>* <sup>−</sup> <sup>1</sup> <sup>2</sup> ≈ 0.15.

## **2.4. Regular rotations**

We say that the system performs regular rotations if a nonzero average rotational velocity exists

$$b = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T \dot{\theta} d\tau.$$

Velocity *b* is a rational number because regular motions can be observed only in resonance with excitation. Motion with fractional average velocity such as |*b*| = 1/2 in Fig. 4 a) is usually called *oscillation-rotation*. Let us first study monotone rotations, where velocity ˙ *θ* has constant sign and integer average value *b*, see Fig. 4 b) and c).

In order to describe resonance rotations of the PPVL we use the method of averaging [9, 18, 19] which requires rewriting (4) in the Bogolubov's standard form. For that reason we assume

**Figure 4.** a) Regular rotation-oscillation with the mean angular velocity equal to one half of the excitation frequency, *b* = −1/2. b) Regular rotation with *b* = −1. c) Regular rotation with *b* = −2. d) Regular rotation-oscillation with *b* = 0.

that *ε*, *β* and *ω* are small parameters, *ε* being of order *ω*2, and *β* of order *ω*3, which makes the system quasi-linear.

We introduce the vector of *slow* variables *x* and the fast time *s* = |*b*| *τ*, where *x*<sup>1</sup> = *θ* − *b τ* is the *phase mismatch*, *x*<sup>2</sup> = *<sup>d</sup><sup>θ</sup> ds* is the velocity, *<sup>x</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>ε</sup>* cos *<sup>s</sup>* |*b*| is the excitation. From here the dot denotes derivative with respect to the new time *s*. Thus, equations (4) takes the standard form

$$
\dot{\mathfrak{x}}\_1 = \mathfrak{x}\_2 - \text{sign}(b)\_\prime
$$

6 Will-be-set-by-IN-TECH

where *<sup>Q</sup>*{2} and *<sup>Z</sup>*{2} are the steady state variables of system (18)-(19) in the second order approximation. Substitution of these expressions into (15) yields the second order

<sup>√</sup>*εQ*{2}

0.5

A C

1

1.5

amplitude of limit cycle

**Figure 3.** Frequency-response curve for the parameters *ε* = 0.04 and *β* = 0.05. Amplitude of the limit cycle in the first (left) and second (right) approximation compared with the results of numerical simulations (circles) depending on the relative excitation frequency *ω*. In the first approximation (left) the amplitude <sup>√</sup>*εQ*{1} is described by (20). In the second approximation (right) the amplitude of solution (24) is calculated with the use of numerically obtained steady state *<sup>Q</sup>*{2} and *<sup>Z</sup>*{2} of system (18)-(19). Fig. 3 shows better coincidence with the numerical simulations of the second order approximate solution (24) up to the amplitude equals 2 ≈ 2*π*/3 and the frequency mismatch

We say that the system performs regular rotations if a nonzero average rotational velocity

1 *T T* 0 ˙ *θdτ*.

Velocity *b* is a rational number because regular motions can be observed only in resonance with excitation. Motion with fractional average velocity such as |*b*| = 1/2 in Fig. 4 a) is usually called *oscillation-rotation*. Let us first study monotone rotations, where velocity ˙

In order to describe resonance rotations of the PPVL we use the method of averaging [9, 18, 19] which requires rewriting (4) in the Bogolubov's standard form. For that reason we assume

*b* = lim *T*→∞

constant sign and integer average value *b*, see Fig. 4 b) and c).

2

2.5

 + 2 + *ω* <sup>8</sup> cos

*ω* <sup>2</sup> sin *τ*

3*τ*

<sup>2</sup> <sup>+</sup> *<sup>Z</sup>*{2}

3*τ*

<sup>2</sup> <sup>+</sup> <sup>3</sup>*Z*{2}

calculations stable branch unstable branch + *o*(*ε*

B

<sup>2</sup>). (24)

*θ* has

<sup>2</sup> <sup>+</sup> *<sup>Z</sup>*{2}

0.45 0.5 0.55 0.6 0.65 0.7 0.75 <sup>0</sup>

relative frequency ω

approximate solution of (4) in the following form

*τ*

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>* <sup>4</sup> cos

 *ω* <sup>16</sup> cos

0.45 0.5 0.55 0.6 0.65 0.7 0.75 <sup>0</sup>

relative frequency ω

B

<sup>2</sup> <sup>+</sup> *<sup>Z</sup>*{2}

*τ*

*τ*

<sup>2</sup> <sup>+</sup> *<sup>Z</sup>*{2}

 + *β*

<sup>2</sup> <sup>−</sup> *<sup>Z</sup>*{2}

 <sup>−</sup> *<sup>ω</sup>* <sup>96</sup> cos

*<sup>θ</sup>* <sup>=</sup> <sup>√</sup>*εQ*{2} cos

− *ε* 3 <sup>2</sup> *<sup>Q</sup>*{2}

+ *ε* 3 <sup>2</sup> *Q*<sup>3</sup> {2}

calculations stable branch unstable branch

0.5

is *<sup>ω</sup>* <sup>−</sup> <sup>1</sup>

exists

A C

<sup>2</sup> ≈ 0.15.

**2.4. Regular rotations**

1

1.5

amplitude of limit cycle

2

2.5

$$\begin{aligned} \dot{x}\_2 &= \left(\frac{2\varepsilon}{x\_3}\sin\left(\frac{s}{|b|}\right) - \beta\omega\right) \frac{x\_2}{|b|} - \frac{\omega^2}{b^2} \frac{\sin(x\_1 + s)}{x\_3}, \\\dot{x}\_3 &= -\frac{\varepsilon}{|b|} \sin\left(\frac{s}{|b|}\right), \end{aligned} \tag{25}$$

#### 8 Will-be-set-by-IN-TECH 76 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>9</sup>

where it is assumed that *x*<sup>2</sup> − sign(*b*) is of order *ε*, sign(*b*) = 1 if *b* > 0 and sign(*b*) = −1 if *b* < 0. With the method of averaging we can find the first, second and the following order approximations of equations (25).

Resonance rotation domains of PPVL for various |*b*| are presented in Fig. 5. We see that greater values of relative rotational velocities |*b*| are possible for higher excitation amplitudes *ε*. Numerically obtained rotational regimes are depicted in Fig. 5 by color points in parameter space (*ω*,*ε*) with *β* = 0.05 and initial conditions *θ*(0) = *π*, ˙ *θ*(0) = 0.05. Domains of these points are well bounded below by analytically obtained curves for corresponding |*b*|.
