**Appendix. The standard method of averaging**

The standard averaging method is a straightforward procedure and in the literature [9, 18] usually only two first approximations are written down. We use the averaging method down to the fifth approximation and feel obliged to present the resulting formulas.

The standard averaging method finds the transformation of the equation system with small and 2*π* time periodic right hand side

$$\dot{\mathbf{x}} = f\_1(\mathbf{x}, t) + f\_2(\mathbf{x}, t) + \dots + f\_6(\mathbf{x}, t) \tag{117}$$

into autonomous system (also called averaged system) which could be solved analytically

$$\dot{X} = F\_1(X) + F\_2(X) + \dots + F\_6(X) + \dots \tag{118}$$

where the lower index denotes the order of smallness with respect to one. A new variable *X* is presented in the following series

$$\mathbf{x} = X + \mu\_1(X, t) + \mu\_2(X, t) + \dots + \mu\_5(X, t) + \dots,\tag{119}$$

which is the approximate solution of system (117) and, hence, of the transformed system (118). The functions *ui* and *Fi* can be found one by one after differentiating (119) with respect to time, substituting there expressions (117) and (118), expanding the functions *fi* around *X* into Taylor's series, and collecting there terms of the same order.

Averaging operator is defined as follows

28 Will-be-set-by-IN-TECH

constant angular speed equal to the speed of the excitation. We showed that one family of the solutions is stable, while the other one is unstable. These exact solutions allowed us to obtain the approximate solutions for the case of an elliptic trajectory of the waist. An interesting effect of inverse twirling was described when the waist moves in opposite direction to the hula-hoop rotation. It is shown that the approximate analytical solutions agree with the results

The authors express their gratitude to Angelo Luongo for his contribution to dynamics of a

*Institute of Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria*

The standard averaging method is a straightforward procedure and in the literature [9, 18] usually only two first approximations are written down. We use the averaging method down

The standard averaging method finds the transformation of the equation system with small

into autonomous system (also called averaged system) which could be solved analytically

where the lower index denotes the order of smallness with respect to one. A new variable *X*

which is the approximate solution of system (117) and, hence, of the transformed system (118). The functions *ui* and *Fi* can be found one by one after differentiating (119) with respect to time, substituting there expressions (117) and (118), expanding the functions *fi* around *X* into

*x*˙ = *f*1(*x*, *t*) + *f*2(*x*, *t*) + ... + *f*6(*x*, *t*) (117)

*X*˙ = *F*1(*X*) + *F*2(*X*) + ... + *F*6(*X*) + ..., (118)

*x* = *X* + *u*1(*X*, *t*) + *u*2(*X*, *t*) + ... + *u*5(*X*, *t*) + ..., (119)

pendulum with variable length and fruitful discussions on nonlinear mechanics.

*Institute of Mechanics, Lomonosov Moscow State University (MSU), Moscow, Russia*

*Institute of Mechanics, Lomonosov Moscow State University (MSU), Moscow, Russia*

to the fifth approximation and feel obliged to present the resulting formulas.

**Appendix. The standard method of averaging**

Taylor's series, and collecting there terms of the same order.

of numerical simulation.

**Acknowledgements**

**Author details**

Belyakov Anton

Seyranian Alexander P.

and 2*π* time periodic right hand side

is presented in the following series

$$\langle \cdot \rangle = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T \cdot \vert\_{\mathbf{x} = X} d\tau = \frac{1}{2\pi} \int\_0^{2\pi} \cdot \vert\_{\mathbf{x} = X} d\tau.$$

We also define integral operator {·} with the following expression

$$\{f(\mathbf{x},\tau)\} = \int \left(f(\mathbf{x},\tau) - \langle f(\mathbf{x},\tau) \rangle\right) d\tau\_{\mathbf{x}}$$

which is such an antiderivative that satisfies the condition �{ *f*(*x*, *τ*)}� = {�*f*(*x*, *τ*)�} = 0. Latter condition is necessary to obviate an ambiguity. We define following vector product operators

$$f\mu\_k = \sum\_{i=1}^n \frac{\partial f}{\partial \boldsymbol{\omega}\_i} \boldsymbol{u}\_{k\prime}^i \quad f\mu\_{k,m} = \sum\_{i,j=1}^n \frac{\partial^2 f}{\partial \boldsymbol{\omega}\_i \partial \boldsymbol{\omega}\_j} \boldsymbol{u}\_k^i \boldsymbol{u}\_{m\prime}^j \quad \dots \dots$$

and so on, where *i* and *j* are the indices of vector components placed in *u<sup>i</sup> <sup>k</sup>* and *<sup>u</sup><sup>j</sup> <sup>m</sup>* on the top not to confuse it with smallness order indices *k* and *m*; *n* is the length of vectors *f* , *uk* and *um*. Hence, we can write recurrent expressions

$$F\_1(\mathbf{X}) = \left< f\_1(\mathbf{x}, t) \right> ,\tag{120}$$

$$\mu\_1(\mathbf{X}, t) = \left\{ f\_1(\mathbf{x}, t) \right\} + \mathcal{U}\_1(\mathbf{X}), \quad \mathcal{F}\_2(\mathbf{X}) = \left\langle f\_2(\mathbf{x}, t) + f\_1(\mathbf{x}, t)\mu\_1(\mathbf{X}, t) \right\rangle,\tag{121}$$

$$\mathcal{U}\_2 = \left\{f\_2 + f\_1 u\_1 - u\_1 F\_1\right\} + \mathcal{U}\_2, \quad F\_3 = \left\langle f\_3 + f\_1 u\_2 + f\_2 u\_1 + \frac{1}{2} f\_1 u\_{1,1} \right\rangle,\tag{122}$$

$$u\_3 = \left\{f\_3 + f\_1 u\_2 + f\_2 u\_1 + \frac{1}{2} f\_1 u\_{1,1} - u\_1 F\_2 - u\_2 F\_1\right\} + \mathcal{U}\_{3\prime} \tag{123}$$

$$F\_4 = \left\langle f\_4 + f\_1 u\_3 + f\_2 u\_2 + f\_3 u\_1 + f\_1 u\_{1,2} + \frac{1}{2} f\_2 u\_{1,1} + \frac{1}{6} f\_1 u\_{1,1,1} \right\rangle,\tag{124}$$

$$u\_4 = \left\{f\_4 + f\_1 u\_3 + f\_2 u\_2 + f\_3 u\_1 + f\_1 u\_{1,2} + \frac{1}{2} f\_2 u\_{1,1} + \frac{1}{6} f\_1 u\_{1,1,1}\right.$$

$$-u\_1 F\_3 - u\_2 F\_2 - u\_3 F\_1\right\} + \mathcal{U}\_{4\nu} \tag{125}$$

$$F\_5 = \left\langle f\_5 + f\_1 u\_4 + f\_2 u\_3 + f\_3 u\_2 + f\_4 u\_1 + f\_1 u\_{1,3} + f\_2 u\_{1,2} \right\rangle$$

$$+ \frac{1}{2} f\_1 u\_{2,2} + \frac{1}{2} f\_3 u\_{1,1} + \frac{1}{2} f\_1 u\_{1,1,2} + \frac{1}{6} f\_2 u\_{1,1,1} + \frac{1}{24} f\_1 u\_{1,1,1,1} \right\rangle\_{\prime} \tag{126}$$

$$\begin{aligned} u\_5 &= \left\{ f\_5 + f\_1 u\_4 + f\_2 u\_3 + f\_3 u\_2 + f\_4 u\_1 + f\_1 u\_{1,3} + f\_2 u\_{1,2} \\ &+ \frac{1}{2} f\_1 u\_{2,2} + \frac{1}{2} f\_3 u\_{1,1} + \frac{1}{2} f\_1 u\_{1,1,2} + \frac{1}{6} f\_2 u\_{1,1,1} + \frac{1}{24} f\_1 u\_{1,1,1,1} \\ &- u\_1 F\_4 - u\_2 F\_3 - u\_3 F\_2 - u\_4 F\_1 \right\} + \mathcal{U}\_5 \end{aligned} \tag{127}$$

$$\begin{aligned} F\_6 &= \left\langle f\_6 + f\_1 u\_5 + f\_2 u\_4 + f\_3 u\_3 + f\_4 u\_2 + f\_5 u\_1 + f\_1 u\_{1,4} + f\_1 u\_{2,3} + f\_2 u\_{1,3} + f\_3 u\_{1,2} \right\rangle \\ &+ \frac{1}{2} f\_2 u\_{2,2} + \frac{1}{2} f\_4 u\_{1,1} + \frac{1}{2} f\_1 u\_{1,2,2} + \frac{1}{2} f\_2 u\_{1,1,2} + \frac{1}{6} f\_3 u\_{1,1,1} + \frac{1}{6} f\_1 u\_{1,1,1,2} \\ &+ \frac{1}{24} f\_2 u\_{1,1,1,1} + \frac{1}{120} f\_1 u\_{1,1,1,1} \right\rangle, \end{aligned} \tag{128}$$

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where we similarly denote *ukFm* = ∑*<sup>n</sup> i*=1 *∂uk ∂xi Fi <sup>m</sup>*. Functions *Uk* = *Uk*(*X*) can be chosen arbitrarily, so for convenience we set *Uk*(*X*) ≡ 0. Thus, knowing the functions *Fi* from expressions (120)-(128) we can write averaged system (118), which is simpler than the original system (117). If we solve averaged system (118) we are able to write the approximate solution (119) of system (117) substituting slow variables *X*(*t*) into the functions *ui*(*X*, *t*) obtained from (121)-(127). Since the functions *ui*(*X*, *t*) are periodic with respect to time *t* the behavior of slow variables determines the behavior of the approximate solution. It means that we can study stability of the approximate solutions by stability of the regular solutions of averaged system (118).
