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

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 following form, see (122),

$$\begin{aligned} \dot{X}\_1 &= X\_2 - b\_{\prime} \\ \dot{X}\_2 &= -\frac{3\varepsilon\omega^2}{2}\sin\left(X\_1\right) - \beta\omega X\_2 \end{aligned} \tag{26}$$

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

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

*<sup>X</sup>*˙ <sup>2</sup> <sup>=</sup> <sup>−</sup>9*ε*2*ω*<sup>2</sup>

2 ,

16

from (32) that there are asymptotically stable steady solutions

9*ε*2*ω*

9*ε*2*ω*

*<sup>X</sup>*1(1) <sup>=</sup> <sup>−</sup> arcsin <sup>8</sup>*b<sup>β</sup>*

*<sup>X</sup>*1(2) <sup>=</sup> *<sup>π</sup>* <sup>+</sup> arcsin <sup>8</sup>*b<sup>β</sup>*

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

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

and unstable steady solutions

1 − *<sup>X</sup>*<sup>2</sup> <sup>−</sup> *<sup>b</sup>* 2 <sup>2</sup> + *ε*2 27

which have steady state solutions determined by the following equation

sin (*X*1) <sup>=</sup> <sup>−</sup>*<sup>b</sup>* <sup>8</sup>*<sup>β</sup>*

has the following boundary condition depicted in Fig. 5 with a bold solid line

*<sup>ω</sup>* <sup>≥</sup> <sup>8</sup>*<sup>β</sup>* 9*ε*<sup>2</sup>

1

1

equations for |*b*| = 2, see (128),

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

9*ε*2*ω*

From equation (32) we get that the domain of rotations with |*b*| = 2 in the parameter space

 1 <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2/27

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

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

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

 1 <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2/27

sin (*X*1) <sup>−</sup> *βω*

Dynamics of a Pendulum of Variable Length and Similar Problems 77

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2/27 <sup>+</sup> <sup>2</sup>*πk*, *<sup>k</sup>* <sup>=</sup> ..., <sup>−</sup>1, 0, 1, 2, . . . (34)

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2/27 <sup>+</sup> <sup>2</sup>*πk*, *<sup>k</sup>* <sup>=</sup> ..., <sup>−</sup>1, 0, 1, 2, . . . (35)

<sup>2</sup> *<sup>X</sup>*<sup>2</sup> ,

. (32)

. (33)

(31)

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

$$b\sin\left(X\_1\right) = -b\frac{2\beta}{3\epsilon\omega}.\tag{27}$$

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

$$
\omega \ge \frac{2\beta}{3\epsilon}.\tag{28}
$$

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

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 asymptotically stable steady solutions

$$X\_{1(1)} = -b\arcsin\left(\frac{2\beta}{3\omega\varepsilon}\right) + 2\pi k, \quad k = \dots, -1, 0, 1, 2, \dots \tag{29}$$

and unstable solutions

$$X\_{1(2)} = \pi + b \arcsin\left(\frac{2\beta}{3\omega\varepsilon}\right) + 2\pi k, \quad k = \dots, -1, 0, 1, 2, \dots \tag{30}$$

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