Stability Analysis of Resonant Rotation of a Gyrostat in an Elliptic Orbit Under Third-and Fourth-Order Resonances

Abstract

This paper presents the stability of resonant rotation of a symmetric gyrostat under third- and fourth-order resonances, whose center of mass moves in an elliptic orbit in a central Newtonian gravitational field. The resonant rotation is a special planar periodic motion of the gyrostat about its center of mass, i. e., the body performs one rotation in absolute space during two orbital revolutions of its center of mass. The equations of motion of the gyrostat are derived as a periodic Hamiltonian system with three degrees of freedom and a constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. By analyzing the Floquet multipliers of the linearized equations of perturbed motion, the unstable region of the resonant rotation and the region of stability in the first-order approximation are determined in the dimensionless parameter plane. In addition, the third- and fourth-order resonances are obtained in the linear stability region and further nonlinear stability analysis is performed in the third- and fourth-order resonant cases.

APPENDIX

The coefficient $w_{{l_{1}}{l_{2}}{l_{3}}{l_{1}}{l_{2}}{l_{3}}}$ in Eqs. (4.52) and (4.54) includes ${w_{200200}}$, ${w_{020020}}$, ${w_{002002}}$, ${w_{110110}}$, ${w_{101101}}$ and ${w_{011011}}$, and can be computed by the following formulae:

$$\displaystyle{w_{200200}}=f_{200200}^{*}+\frac{1}{8}\left({4{c_{200100}}+3c_{200100300000}^{-}+{c_{110100}}+c_{200010200010}^{-}+{c_{101100}}+c_{200001200001}^{-}}\right)$$

$$\displaystyle+\frac{1}{{16}}\left[{9\cot\left({3\pi{\sigma_{1}}}\right)\tilde{c}_{300000}^{+}+3\cot\left({\pi{\sigma_{1}}}\right)\tilde{c}_{200100}^{+}+\cot\left({2\pi{\sigma_{1}}+\pi{\sigma_{2}}}\right)\tilde{c}_{210000}^{+}}\right.$$

$$\displaystyle+\cot\left({\pi{\sigma_{2}}}\right)\tilde{c}_{110100}^{+}-\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\tilde{c}_{200010}^{+}+\cot\left({2\pi{\sigma_{1}}+\pi{\sigma_{3}}}\right)\tilde{c}_{201000}^{+}$$

$$\displaystyle\left.{+\cot\left({\pi{\sigma_{3}}}\right)\tilde{c}_{101100}^{+}-\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)\tilde{c}_{200001}^{+}}\right],$$

$$\displaystyle{w_{110110}}=f_{110110}^{*}+\frac{1}{4}\left({c_{110100210000}^{-}+c_{110010120000}^{-}+c_{110100200010}^{+}+c_{020100110010}^{+}+c_{110010200100}^{+}}\right.$$

$$\displaystyle\left.{+c_{020010110100}^{+}}\right)+\frac{1}{8}\left({c_{101100011010}^{+}+c_{110001111000}^{-}+c_{011100101010}^{+}}\right)$$

$$\displaystyle+\frac{1}{4}\left[{\cot\left({\pi{\sigma_{1}}+2\pi{\sigma_{2}}}\right)\tilde{c}_{120000}^{+}+\cot\left({2\pi{\sigma_{1}}+\pi{\sigma_{2}}}\right)\tilde{c}_{210000}^{+}+\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\tilde{c}_{200010}^{+}}\right.$$

$$\displaystyle\left.{-\cot\left({\pi{\sigma_{1}}-2\pi{\sigma_{2}}}\right)\tilde{c}_{100020}^{+}+\cot\left({\pi{\sigma_{1}}}\right)\hat{c}_{200100110010}^{+}+\cot\left({\pi{\sigma_{2}}}\right)\hat{c}_{110100020010}^{+}}\right]$$

$$\displaystyle+\frac{1}{{16}}\left[{2\cot\left({\pi{\sigma_{3}}}\right)\hat{c}_{101100011010}^{+}+\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}+\pi{\sigma_{3}}}\right)\tilde{c}_{111000}^{+}}\right.$$

$$\displaystyle-\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)\tilde{c}_{110001}^{+}+\cot\left({\pi{\sigma_{1}}-\pi{\sigma_{2}}+\pi{\sigma_{3}}}\right)\tilde{c}_{101010}^{+}$$

$$\displaystyle\left.{-\cot\left({\pi{\sigma_{1}}-\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)\tilde{c}_{011100}^{+}}\right].$$

The coefficients ${\mu_{{k_{1}}{k_{2}}{k_{3}}000}}$ and ${\nu_{{k_{1}}{k_{2}}{k_{3}}000}}$ shown in Eq. (4.56) include $\mu_{400000}$, $\nu_{400000}$, $\mu_{040000}$, $\nu_{040000}$, $\mu_{004000}$, $\nu_{004000}$, $\mu_{310000}$, $\nu_{310000}$, $\mu_{301000}$, $\nu_{301000}$, $\mu_{031000}$, $\nu_{031000}$, $\mu_{130000}$, $\nu_{130000}$, $\mu_{103000}$, $\nu_{103000}$, $\mu_{013000}$, $\nu_{013000}$, $\mu_{220000}$, $\nu_{220000}$, $\mu_{202000}$, $\nu_{202000}$, $\mu_{022000}$, $\nu_{022000}$, $\mu_{211000}$, $\nu_{211000}$, $\mu_{121000}$, $\nu_{121000}$, $\mu_{112000}$ and $\nu_{112000}$, and can be computed by the following formulae:

$$\displaystyle{\mu_{400000}}=\frac{1}{4}{a_{400000}}+\frac{1}{{16}}\left[{9{c_{300000}}-{c_{200100}}+{c_{210000}}+{c_{201000}}-{c_{200010}}-{c_{200001}}}\right.$$

$$\displaystyle\left.{-3\cot\left({\pi{\sigma_{1}}}\right)\hat{c}_{300000200100}^{-}-\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\hat{c}_{210000200010}^{-}-\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)\hat{c}_{201000200001}^{-}}\right],$$

$$\displaystyle{\nu_{400000}}=-\frac{1}{4}{b_{400000}}+\frac{1}{{32}}\left[{9\tilde{c}_{300000}^{-}-\tilde{c}_{200100}^{-}+\tilde{c}_{210000}^{-}+\tilde{c}_{201000}^{-}-\tilde{c}_{200010}^{-}-\tilde{c}_{200001}^{-}}\right.$$

$$\displaystyle+6\cot\left({\pi{\sigma_{1}}}\right)c_{200100300000}^{+}+2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)c_{200010210000}^{+}$$

$$\displaystyle\left.{+2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)c_{200001201000}^{+}}\right],$$

$$\displaystyle{\mu_{310000}}=\frac{1}{4}{a_{310000}}+\frac{1}{{16}}\left[{6c_{210000300000}^{+}+2c_{120000210000}^{+}-c_{200100110100}^{+}-c_{110010200010}^{+}+c_{201000111000}^{+}}\right.$$

$$\displaystyle-c_{200001110001}^{+}-3\cot\left({\pi{\sigma_{2}}}\right)\hat{c}_{300000110100}^{-}-\cot\left({\pi{\sigma_{1}}}\right)\left({2\hat{c}_{210000200100}^{-}+\hat{c}_{210000110010}^{-}}\right)$$

$$\displaystyle-2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\hat{c}_{120000200010}^{-}-\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)\hat{c}_{201000110001}^{-}$$

$$\displaystyle\left.{-\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)\hat{c}_{200001111000}^{-}}\right],$$

$$\displaystyle{\nu_{310000}}=-\frac{1}{4}{b_{310000}}+\frac{1}{{16}}\left[{6\hat{c}_{210000300000}^{-}+2\hat{c}_{120000210000}^{-}-\hat{c}_{200100110100}^{-}-\hat{c}_{110010200010}^{-}+\hat{c}_{201000111000}^{-}}\right.$$

$$\displaystyle-\hat{c}_{200001110001}^{-}+3\cot\left({\pi{\sigma_{2}}}\right)c_{300000110100}^{+}+\cot\left({\pi{\sigma_{1}}}\right)\left({2c_{210000200100}^{+}+c_{210000110010}^{+}}\right)$$

$$\displaystyle+2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)c_{200010120000}^{+}+\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)c_{201000110001}^{+}$$

$$\displaystyle\left.{+\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)c_{200001111000}^{+}}\right],$$

$$\displaystyle{\mu_{220000}}=\frac{1}{4}{a_{220000}}+\frac{1}{{16}}\left[{4\left({{c_{210000}}+{c_{120000}}}\right)+3\left({c_{120000300000}^{+}+c_{030000210000}^{+}}\right)-{c_{110100}}-{c_{110010}}}\right.$$

$$\displaystyle+c_{201000021000}^{+}-c_{200001020001}^{+}+{c_{111000}}-{c_{110001}}-c_{200100020100}^{+}-c_{200010020010}^{+}$$

$$\displaystyle-\cot\left({\pi{\sigma_{1}}}\right)\left({2\hat{c}_{110010120000}^{-}+\hat{c}_{120000200100}^{-}}\right)-\cot\left({\pi{\sigma_{2}}}\right)\left({2\hat{c}_{110100210000}^{-}+\hat{c}_{210000020010}^{-}}\right)$$

$$\displaystyle-3\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\hat{c}_{030000200010}^{-}-3\cot\left({2\pi{\sigma_{2}}-\pi{\sigma_{1}}}\right)\hat{c}_{300000020100}^{-}$$

$$\displaystyle-\left({\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)+\cot\left({2\pi{\sigma_{\rm{2}}}-\pi{\sigma_{3}}}\right)}\right)\hat{c}_{200001020001}^{-}$$

$$\displaystyle\left.{-\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)\hat{c}_{110001111000}^{-}}\right],$$

$$\displaystyle{v_{220000}}=-\frac{1}{4}{b_{220000}}+\frac{1}{{32}}\left[{6\left({\hat{c}_{120000300000}^{-}+\hat{c}_{030000210000}^{-}}\right)+4\left({\tilde{c}_{210000}^{-}+\tilde{c}_{120000}^{-}}\right)-\tilde{c}_{110100}^{-}-\tilde{c}_{110010}^{-}}\right.$$

$$\displaystyle-2\left({\hat{c}_{200100020100}^{-}+\hat{c}_{020020200010}^{-}}\right)+2\left({\hat{c}_{201000021000}^{-}-\hat{c}_{200001020001}^{-}}\right)+\tilde{c}_{111000}^{-}-\tilde{c}_{110001}^{-}$$

$$\displaystyle+2\cot\left({\pi{\sigma_{1}}}\right)\left({2c_{110010120000}^{+}+c_{120000200100}^{+}}\right)+2\cot\left({\pi{\sigma_{2}}}\right)\left({2c_{110010012000}^{+}+c_{210000020010}^{+}}\right)$$

$$\displaystyle+6\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)c_{030000200010}^{+}+2\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)c_{110001111000}^{+}$$

$$\displaystyle\left.{+6\cot\left({2\pi{\sigma_{2}}-\pi{\sigma_{1}}}\right)c_{300000020100}^{+}+2\left({\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)+\cot\left({2\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)}\right)c_{200001020001}^{+}}\right],$$

$$\displaystyle{\mu_{211000}}=\frac{1}{4}{a_{211000}}+\frac{1}{{16}}\left[{4c_{210000201000}^{+}+3c_{300000111000}^{+}-c_{110100101100}^{+}-c_{110010110010}^{+}-c_{101001110001}^{+}}\right.$$

$$\displaystyle-c_{200010011010}^{+}+2\left({c_{300000111000}^{+}+c_{201000012000}^{+}+c_{120000111000}^{+}+c_{102000111000}^{+}}\right)$$

$$\displaystyle-c_{200001011001}^{+}-c_{200100011100}^{+}-\cot\left({\pi{\sigma_{1}}}\right)\left({\hat{c}_{200100111000}^{-}+\hat{c}_{111000110010}^{-}+\hat{c}_{111000101001}^{-}}\right)$$

$$\displaystyle-\cot\left({\pi{\sigma_{2}}}\right)\left({2\hat{c}_{201000110100}^{-}+\hat{c}_{201000011001}^{-}}\right)-\cot\left({\pi{\sigma_{3}}}\right)\left({2\hat{c}_{210000101100}^{-}+\hat{c}_{210000011010}^{-}}\right)$$

$$\displaystyle-2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)\hat{c}_{200010021000}^{-}-2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)\hat{c}_{200001012000}^{-}$$

$$\displaystyle-3\cot\left({\pi{\sigma_{2}}+\pi{\sigma_{3}}-\pi{\sigma_{1}}}\right)\hat{c}_{300000011100}^{-}-2\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{3}}-\pi{\sigma_{2}}}\right)\hat{c}_{120000110010}^{-}$$

$$\displaystyle\left.{-2\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)\hat{c}_{102000110001}^{-}}\right],$$

$$\displaystyle{v_{211000}}=-\frac{1}{4}{b_{211000}}\!+\!\frac{1}{{16}}\left[{4\left({\hat{c}_{210000201000}^{-}\!-\!\hat{c}_{110100101100}^{-}}\right)\!+\!3\hat{c}_{300000111000}^{-}-\hat{c}_{200100211100}^{-}-\hat{c}_{200010011010}^{-}}\right.$$

$$\displaystyle-\hat{c}_{200001011001}^{-}+2\left({\hat{c}_{210000021000}^{-}+\hat{c}_{201000012000}^{-}+\hat{c}_{120000111000}^{-}+\hat{c}_{102000111000}^{-}}\right)$$

$$\displaystyle-\hat{c}_{110010101010}^{-}-\hat{c}_{101001110001}^{-}+\cot\left({\pi{\sigma_{1}}}\right)\left({c_{200100111000}^{+}+c_{111000110010}^{+}+c_{111000101001}^{+}}\right)$$

$$\displaystyle+\cot\left({\pi{\sigma_{2}}}\right)\left({2c_{201000110100}^{+}+c_{201000011001}^{+}}\right)+\cot\left({\pi{\sigma_{3}}}\right)\left({2c_{110000101100}^{+}+c_{210000011010}^{+}}\right)$$

$$\displaystyle+2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{2}}}\right)c_{200010021000}^{+}+2\cot\left({2\pi{\sigma_{1}}-\pi{\sigma_{3}}}\right)c_{200001012000}^{+}$$

$$\displaystyle+3\cot\left({\pi{\sigma_{2}}+\pi{\sigma_{3}}-\pi{\sigma_{1}}}\right)c_{300000011100}^{+}+2\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{3}}-\pi{\sigma_{2}}}\right)c_{120000101010}^{+}$$

$$\displaystyle\left.{+2\cot\left({\pi{\sigma_{1}}+\pi{\sigma_{2}}-\pi{\sigma_{3}}}\right)c_{102000110001}^{+}}\right],$$

where

$$\displaystyle{c_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}}={a_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}}{b_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}},\quad\tilde{c}_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}^{\pm}=a_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}^{2}\pm b_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}^{2},$$

$$\displaystyle c_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}^{\pm}={a_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}}}{b_{{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}}\pm{a_{{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}}{b_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}}},$$

$$\displaystyle\hat{c}_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}^{\pm}={a_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}}}{a_{{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}}\pm{b_{{m_{1}}{m_{2}}{m_{3}}{n_{1}}{n_{2}}{n_{3}}}}{b_{{r_{1}}{r_{2}}{r_{3}}{s_{1}}{s_{2}}{s_{3}}}}.$$

In addition, the coefficients ${a_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}}$, ${b_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}}$ in Eq. (4.55), and $f_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}^{*}$, the coefficient of $F_{4}^{*}$ in Eq. (4.26), are explicit functions of ${f_{{\nu_{1}}{\nu_{2}}{\nu_{3}}{\mu_{1}}{\mu_{2}}{\mu_{3}}}}$, which are the coefficients of generating function $F_{k}$ shown in Eq. (4.21). They can be represented as

$$\displaystyle{a_{300000}}={f_{300000}}-{f_{100200}},\quad{b_{300000}}={f_{200100}}-{f_{000300}},$$

$$\displaystyle{a_{200100}}={f_{100200}}+3{f_{300000}},\quad{b_{200100}}={f_{200100}}+3{f_{000300}},$$

$$\displaystyle{a_{210000}}={f_{210000}}-{f_{100110}}-{f_{010200}},\quad{b_{210000}}={f_{200010}}+{f_{110100}}-{f_{000210}},$$

$$\displaystyle{a_{200010}}={f_{210000}}+{f_{100110}}-{f_{010200}},\quad{b_{200010}}={f_{000210}}+{f_{110100}}-{f_{200010}},$$

$$\displaystyle{a_{110100}}=2{f_{210000}}+2{f_{010200}},\quad{b_{110100}}=2{f_{200010}}+2{f_{000210}},$$

$$\displaystyle{a_{111000}}={f_{111000}}-{f_{100011}}-{f_{010101}}-{f_{001110}},\quad{b_{111000}}={f_{110001}}+{f_{101010}}+{f_{011100}}-{f_{000111}},$$

$$\displaystyle{a_{110001}}={f_{111000}}+{f_{100011}}+{f_{010101}}-{f_{001110}},\quad{b_{110001}}=-{f_{110001}}+{f_{101010}}+{f_{011100}}+{f_{000111}},$$

$$\displaystyle{a_{011100}}={f_{111000}}-{f_{100011}}+{f_{010101}}+{f_{001110}},\quad{b_{011100}}={f_{110001}}+{f_{101010}}-{f_{011100}}+{f_{000111}},$$

$$\displaystyle{a_{400000}}={f_{200200}}-{f_{400000}}-{f_{000400}},\quad{b_{400000}}={f_{100300}}-{f_{300100}},$$

$$\displaystyle{a_{310000}}={f_{200110}}+{f_{110200}}-{f_{310000}}-{f_{000310}},\quad{b_{310000}}={f_{100210}}+{f_{010300}}-{f_{210100}}-{f_{300010}},$$

$$\displaystyle{a_{130000}}={f_{020110}}+{f_{110020}}-{f_{130000}}-{f_{000130}},\quad{b_{13000}}={f_{010120}}+{f_{100030}}-{f_{120010}}-{f_{030100}},$$

$$\displaystyle{a_{220000}}={f_{110110}}-{f_{220000}}+{f_{200020}}-{f_{000220}}+{f_{020200}},$$

$$\displaystyle{b_{220000}}={f_{100120}}-{f_{120100}}+{f_{010210}}-{f_{210010}},$$

$$\displaystyle{a_{211000}}={f_{200011}}+{f_{011200}}+{f_{110101}}+{f_{101110}}-{f_{211000}}-{f_{000211}},$$

$$\displaystyle{b_{211000}}={f_{010201}}+{f_{001210}}+{f_{100111}}-{f_{111100}}-{f_{201010}}-{f_{210001}},$$

and

$$f_{200200}^{*}=-\frac{1}{2}\left({3{f_{400000}}+{f_{200200}}+3{f_{000400}}}\right),\quad f_{110110}^{*}=-{f_{220000}}-{f_{200020}}-{f_{020200}}-{f_{000220}}.$$

For the expressions of other coefficients, just replace the subscripts of the corresponding coefficients in the formulae given above. For example, the expression of $a_{030000}$ is obtained by exchanging the first and second subscripts of each term in the expression of $a_{300000}$. The same is true for other coefficients.