*2.4.2. Rotations with relative velocity* |*b*| = 2

8 Will-be-set-by-IN-TECH

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

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

It is the third order approximation of averaged equation where regular rotations with |*b*| = 1 can be observed, see Fig. 4 b). In the third order approximation averaged equations take the

where *X*<sup>1</sup> and *X*<sup>2</sup> are the averaged slow variables *x*<sup>1</sup> and *x*2. Auxiliary variable *x*<sup>3</sup> = 1 + *ε* cos(*s*/*b*) has unit average *X*<sup>3</sup> = 1 and is excluded from the consideration. Excluding variable *X*<sup>2</sup> from the steady state conditions *X*˙ <sup>1</sup> = 0 and *X*˙ <sup>2</sup> = 0 in (26) we obtain the equation for the

sin (*X*1) <sup>=</sup> <sup>−</sup>*<sup>b</sup>* <sup>2</sup>*<sup>β</sup>*

*ω* ≥ 2*β* 3*ε*

Inequality (28) determines the domain in parameter space, where rotations with |*b*| = 1 can exist. The boundary of this domain is depicted with a bold dashed line in Fig. 5 on the

Stability of the solutions obtained from (27) was studied in [2]. There was found the condition for asymptotic stability cos(*X*1) > 0. Hence, if inequality (28) is strict, then there are

Thus, we conclude that if the parameters satisfy strict inequality (28) there are two stable regular rotations *θ* = *b τ* + *X*1(1) + *o*(1) in opposite directions (*b* = ±1) and two unstable

<sup>2</sup> sin (*X*1) <sup>−</sup> *βωX*<sup>2</sup> ,

points are well bounded below by analytically obtained curves for corresponding |*b*|.

*<sup>X</sup>*˙ <sup>1</sup>=*X*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*,

*<sup>X</sup>*˙ <sup>2</sup>=−3*εω*<sup>2</sup>

Thus, it is clear from (27) that equation (26) has a steady state solution only if

 2*β* 3*ωε*

> 2*β* 3*ωε*

*θ*(0) = 0.05. Domains of these

<sup>3</sup>*εω* . (27)

. (28)

+ 2*πk*, *k* = ..., −1, 0, 1, 2, . . . (29)

+ 2*πk*, *k* = ..., −1, 0, 1, 2, . . . (30)

(26)

space (*ω*,*ε*) with *β* = 0.05 and initial conditions *θ*(0) = *π*, ˙

approximations of equations (25).

*2.4.1. Rotations with relative velocity* |*b*| = 1

following form, see (122),

averaged phase mismatch *X*<sup>1</sup>

parameter plane (*ω*,*ε*) for *β* = 0.05.

asymptotically stable steady solutions

and unstable solutions

*X*1(1) = −*b* arcsin

*X*1(2) = *π* + *b* arcsin

rotations *θ* = *b τ* + *X*1(2) + *o*(1) in opposite directions.

Rotations with higher averaged velocities |*b*| = 2, . . . correspond to higher excitation amplitudes *ε*. That is why we consider the coefficient *ω* being of order *ε*, and *β* being of order *ε*3. With this new ordering we obtain the sixth order approximation of the averaged equations for |*b*| = 2, see (128),

$$\begin{split} \dot{X}\_1 &= X\_2 - \frac{b}{2}, \\ \dot{X}\_2 &= -\frac{9\varepsilon^2 \omega^2}{16} \left( 1 - \left( X\_2 - \frac{b}{2} \right)^2 + \frac{\varepsilon^2}{27} \right) \sin \left( X\_1 \right) - \frac{\beta \omega}{2} X\_2. \end{split} \tag{31}$$

which have steady state solutions determined by the following equation

$$\sin\left(X\_1\right) = -b \frac{8\beta}{9\epsilon^2 \omega} \left(\frac{1}{1 + \epsilon^2/27}\right). \tag{32}$$

From equation (32) we get that the domain of rotations with |*b*| = 2 in the parameter space has the following boundary condition depicted in Fig. 5 with a bold solid line

$$
\omega \ge \frac{8\beta}{9\epsilon^2} \left(\frac{1}{1 + \epsilon^2/27}\right). \tag{33}
$$

System (31) has similar structure to system (26). That is why stability condition for its steady state solutions appears to be the same: cos(*X*1) > 0. Hence, if inequality (33) is strict, we find from (32) that there are asymptotically stable steady solutions

$$X\_{1(1)} = -\arcsin\left(\frac{8b\beta}{9\varepsilon^2\omega}\left(\frac{1}{1+\varepsilon^2/27}\right)\right) + 2\pi k, \quad k = \dots, -1, 0, 1, 2, \dots \tag{34}$$

and unstable steady solutions

$$X\_{1(2)} = \pi + \arcsin\left(\frac{8b\beta}{9\epsilon^2\omega}\left(\frac{1}{1+\epsilon^2/27}\right)\right) + 2\pi k, \quad k = \dots, -1, 0, 1, 2, \dots \tag{35}$$

Thus, as in the previous case, if the parameters satisfy strict inequality (33) there are two stable regular rotations *θ* = *b τ* + *X*1(1) + *o*(1) in opposite directions (*b* = ±2) and two unstable rotations *θ* = *b τ* + *X*1(2) + *o*(1) in opposite directions.

### **2.5. Basins of attractions and transitions to chaos**

In order to determine domains of chaos we calculate maximal Lyapunov exponents presented in Fig. 6. We recall that positive Lyapunov exponents correspond to chaotic motions. Note that chaotic motion includes passing through the upper vertical position, i.e. irregular oscillations-rotations. This is usually called tumbling chaos. We have observed two types of transition to chaos. The first type is when the system goes through the cascade of period doubling (PD) bifurcations occurring within the instability domain of the vertical position when the excitation amplitude *ε* increases, for example at *ω* = 0.5 in Fig. 7(a). The second type is when chaos immediately appears after subcritical Andronov-Hopf (AH) bifurcation when the system enters the instability domain of the lower vertical position of PPVL, for

**Figure 5.** Absolute values |*b*| of relative rotational velocities are shown with different colors on the plane of parameters *ε* and *ω* at the damping *β* = 0.05. The correspondence between the colors and values is shown by the color bar on the right. Approximate boundaries for rotations are drawn with bold dashed line (for |*b*| = 1) and bold solid line (for |*b*| = 2).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −pi

excitation amplitude ε

AH

Dynamics of a Pendulum of Variable Length and Similar Problems 79

−pi/2

0

angle θ

(a) *ω* = 0.5 (b) *ω* = 0.67

**Figure 7.** The bifurcation diagrams for different frequencies *ω* and the same damping *β* = 0.05 show two different types of transition to chaos. (a) After Andronov-Hopf bifurcation (AH) a limit cycle appears which experiences the saddle-node bifurcation (SN) and then the cascade of period-doubling bifurcations (PD). Regular rotations with relative mean angular velocity |*b*| = 1 are denoted by orange points and |*b*| = 2 by red. (b) After subcritical AH bifurcation of the vertical equilibrium the chaotic

small which means that at *ω* = 0.67 the transition to chaos through subcritical AH bifurcation is the most typical. In the middle of Fig. 8(d) the manifold of dark blue points reveals a typical strange attractor structure. The strange attractor inherits the basin of attraction from

Elliptically excited pendulum (EEP) is a mathematical pendulum in the vertical plane whose pivot oscillates not only vertically but also horizontally with *π*/2 phase shift, so that the pivot has elliptical trajectory, see Fig. 9. EEP is a natural generalization of pendulum with vertically vibrating pivot that is one of the most studied classical systems with parametric excitation. It is often referred to simply as *parametric pendulum*, see e.g. [9, 22–27] and references therein. Stability and dynamics of EEP have been studied analytically and numerically in [28–30]. Approximate oscillatory and rotational solutions for EEP are the common examples in literature [31–34] on asymptotic methods. Sometimes EEP is presented in a slightly more general model of unbalanced rotor [31–33], where the phase shift between vertical and horizontal oscillations of the pivot can differ from *π*/2. EEP is also a special case of generally excited pendulum in [35]. The usual assumption for approximate solution in the literature is the smallness of dimensionless damping and pivot oscillation amplitudes in the EEP's equation of motion. We could find only one paper [36], where oscillations of EEP with high

In this section we study rotations of EEP with not small excitation amplitudes and with both small and not small linear damping. Our analysis uses the exact solutions for EEP with the absence of gravity and with equal excitation amplitudes, when elliptical trajectory of the pivot

motion occurs immediately.

disappeared stationary attractor.

**3. Elliptically excited pendulum**

damping and yet small relative excitation were studied.

pi/2

pi

**Figure 6.** Maximal Lyapunov's exponents are shown on the plane of parameters *ε* and *ω* at the damping *β* = 0.05. The correspondence between the colors and values is shown by the color bar on the right, where white color distinguishes zero maximal Lyapunov's exponent which corresponds to regular regime. Positive Lyapunovs' exponents characterize chaotic motions.

example at *ω* = 0.67; see Fig. 7(b). We can see the change of the system dynamics in its route to chaos along *ω* = 0.5 in the bifurcation diagram shown in Fig. 7(a), where red points denote rotations with mean angular velocity equal to one excitation frequency (|*b*| = 1) and green points denote those equal to two excitation frequencies (|*b*| = 2). The domain with the most complex regular dynamics is surrounded by the red rectangle, where the system can have coexisting oscillations, rotations and rotations-oscillations.

Basins of attractions in Fig. 8 have been plotted using program Dynamics [21]. These basins track the changes of the system dynamics in its route to chaos along *ω* = 0.67. In Fig. 8(a) the oscillatory attractor (limit cycle) coexists with stationary attractor (lower vertical position of PPVL). In Fig. 8(b) we can see the first emergence of two rotational attractors with counterrotations. This picture is in a good agreement with condition (18) for existence of rotational solutions |*b*| = 1, see Fig. 5. Closer to the boundary of chaotic region in Fig. 8(c) only stationary and rotational attractors remain. Note that the basins of rotational attractors are

**Figure 7.** The bifurcation diagrams for different frequencies *ω* and the same damping *β* = 0.05 show two different types of transition to chaos. (a) After Andronov-Hopf bifurcation (AH) a limit cycle appears which experiences the saddle-node bifurcation (SN) and then the cascade of period-doubling bifurcations (PD). Regular rotations with relative mean angular velocity |*b*| = 1 are denoted by orange points and |*b*| = 2 by red. (b) After subcritical AH bifurcation of the vertical equilibrium the chaotic motion occurs immediately.

small which means that at *ω* = 0.67 the transition to chaos through subcritical AH bifurcation is the most typical. In the middle of Fig. 8(d) the manifold of dark blue points reveals a typical strange attractor structure. The strange attractor inherits the basin of attraction from disappeared stationary attractor.
