1 Erratum to: Artif Life Robotics (2012) 17(2):275–286 DOI 10.1007/s10015-012-0056-y

Among the quasi-stable size ratios of \( \varepsilon = r_{di} /r_{dj} \,[i, j = 1,2,\;i \ne j] \) for two spheroid-like parcels connected, i.e., super-magic numbers in Table 1, the ratios related to the convection force for \( m \ne 1 \) and \( \varepsilon \ne 1 \) appear with the statistic indeterminacy of the contact position of two parcels, whereas the other ratios can exist even when the statistic indeterminacy effect due to small number of sub-particles inside a parcel is absent.

For details, the relation between dimensionless deformation rate \( \gamma_{k} ( \equiv a_{k} /b_{k} \;[k = 1,\;2]) \) of each parcel dependent on dimensionless time and the size ratio of the two parcels \( \varepsilon = r_{di} /r_{dj} \,[i, j = 1,2,\;i \ne j] \), i.e., Eq. 2, should be described as

$$ \left.\frac{{{\text{d}}^{2} }}{{{\text{d}}\bar{t}_{i}^{2} }}\gamma_{i} = \left\{ {m_{ci} \left( {\frac{\text{d}}{{{\text{d}}\bar{t}_{i} }}\gamma_{i} } \right)^{2} \,+\, m_{cj} \left( {\frac{\text{d}}{{{\text{d}}\bar{t}_{j} }}\gamma_{j} } \right)^{2}\, +\, m_{si} \;\gamma_{i}^{{\frac{5}{3} - \frac{2}{3}m}} +\, m_{sj} \;\gamma_{j}^{{\frac{5}{3} - \frac{2}{3}m}} } \right\}\right/{\text{Det}} + \delta_{st} \quad \left[ {{\text{for }}i = 1, \, 2.\; j = 1, \, 2. \, i \ne j} \right] $$
(2)

with

$$ \begin{aligned} m_{ci} & = \left[ {\left( { - \varepsilon - \varepsilon^{4} + \frac{2}{3}\varepsilon E_{0j} \gamma_{j}^{ - 1/3} } \right)B_{0i} + \frac{2}{9}\varepsilon^{4 - 2\Updelta m} E_{0i} \gamma_{i}^{ - 4/3} } \right] \\ m_{cj} & = \left[ {\frac{2}{3}\varepsilon^{2 + m} E_{0i} \gamma_{j}^{ - 1/3} B_{0j} - \frac{2}{9}\varepsilon^{2 + m} E_{0i} \gamma_{j}^{ - 4/3} } \right] \\ m_{si} & = \left( { - \varepsilon - \varepsilon^{4} + \frac{2}{3}\varepsilon E_{0j} \gamma_{j}^{ - 1/3} } \right)C_{0i} \\ m_{sj} & = \frac{2}{3}\varepsilon^{2 + m} E_{0i} \gamma_{j}^{ - 1/3} C_{0j} \\ {\text{Det}} & = - \varepsilon - \varepsilon^{4} + \frac{2}{3}\varepsilon^{4} E_{0i} \gamma_{i}^{ - 1/3} + \frac{2}{3}\varepsilon^{{}} E_{0j} \gamma_{j}^{ - 1/3} ,\;B_{0k} = \frac{1}{{3\gamma_{k} }}\frac{{\gamma_{k}^{2} - 2}}{{\gamma_{k}^{2} - 1/2}},\; \\ C_{0k} & = \frac{3}{8}\frac{{2\gamma_{k}^{2m} - 1/\gamma_{k}^{m} - \gamma_{k}^{m} }}{{\gamma_{k}^{2} - 1/2}},\;{\text{and }}E_{0k} = 3\frac{{\gamma_{k}^{7/3} }}{{\gamma_{k}^{2} - 1/2}} \quad [{\text{for }}k = 1,\;2] \\ \end{aligned} $$

where the parameter \( \delta_{st} \) and \( \Updelta m\, \)denote the random force due to small number of sub-particles inside a parcel and the indeterminacy effect (stochastic variation) of the contact surface position between two parcels connected, respectively. When we assume the relation of \( \Updelta m\, = (1 - m)/2 \), Table 1 can be obtained.