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Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point

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Abstract

We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge \(q\) . The pendulum oscillates above a fixed point with a charge \(Q.\) The dynamics is studied as a system in the small parameter \(\epsilon\) given by the amplitude of the suspension point. The system depends on two other parameters, \(\alpha\) and \(\beta,\) the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the Deprit – Hori method to construct the boundary surfaces of the stability/instability regions.

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ACKNOWLEDGMENTS

The authors would like to thank Dieter Schmidt and Adecarlos Carvalho for discussions on our implemention of the algorithm to compute the coefficients of the stability surfaces. The authors thank the referee for the enquiries and suggestions which were very helpful in improving the presentation of the manuscript.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Sergipe, 49100-000 Sao Cristovao, SE, Brazil

    Gerson Cruz Araujo

  2. Departamento de Matemática, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil

    Hildeberto E. Cabral

Authors
  1. Gerson Cruz Araujo
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  2. Hildeberto E. Cabral
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Correspondence to Gerson Cruz Araujo or Hildeberto E. Cabral.

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MSC2010

37N05, 70H14, 70J40, 70J25

APPENDIX

Coefficients of Equation (6.1) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha^{(20)}_{1}(\beta)=-\frac{4+13\beta}{8\left(1+2\beta\right)^{2}};\\ \alpha_{2}^{(20)}(\beta)=\frac{-16+8(13+4\sqrt{2})\beta+(-169+232\sqrt{2})\beta^{2}}{128\sqrt{2}\left(1+2\beta\right)^{3}};\\ \alpha^{(20)}_{3}(\beta)=\left(\frac{64\sqrt{2}+48(56+93\sqrt{2})\beta+4(4728+6907\sqrt{2})\beta^{2}+(33072+22037\sqrt{2})\beta^{3})}{4096\left(1+2\beta\right)^{4}}\right);\\ \alpha^{(20)}_{4}(\beta)=\frac{-256+512(91+246\sqrt{2})\beta+96(1105+9584\sqrt{2})\beta^{2}+64(-12028+6681\sqrt{2})\beta^{3}-(2089837+1491552\sqrt{2})\beta^{4}}{196608\sqrt{2}\left(1+2\beta)\right)^{5}};\\ \alpha^{(20)}_{5}(\beta)=-\left(\frac{11264\sqrt{2}-256(36896+43123\sqrt{2})\beta-640(90976+100605\sqrt{2})\beta^{2}-32(2950080+3174643\sqrt{2})\beta^{3}}{18874368\sqrt{2}(1+2\beta)^{6}}\right)\\ \ \ \ \ \ \ \ \ \ \ \ =-\left(\frac{-4(7088704+16704377\sqrt{2})\beta^{4}-(38443616+29551575\sqrt{2})\beta^{5}}{18874368\sqrt{2}(1+2\beta)^{6}}\right).\end{array}\right.$$
(A.1)

Coefficients of Equation (6.2) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha_{1}^{(20)}(\beta)=\left(\frac{-4+13\beta}{8\left(1-2\beta\right)^{2}}\right),\\ \alpha_{2}^{(20)}(\beta)=\frac{16+8(-13+2\sqrt{2})\beta+(169-232\sqrt{2})\beta^{2}}{128\sqrt{2}\left(2\beta-1\right)^{3}},\\ \alpha_{3}^{(20)}(\beta)=\left(\frac{64\sqrt{2}-48(56+93\sqrt{2})\beta+4(4728+6907\sqrt{2})\beta^{2}-(33072+22037\sqrt{2})\beta^{3}}{4096\sqrt{2}(1-2\beta)^{4}}\right),\\ \alpha_{4}^{(20)}(\beta)=\frac{256+256(347+534\sqrt{2})\beta-96(5399+8832\sqrt{2})\beta^{2}-80(3377+11562\sqrt{2})\beta^{3}+(3339409+2257776\sqrt{2})\beta^{4}}{196608\sqrt{2}(2\beta-1)^{5}},\\ \alpha_{5}^{(20)}(\beta)=-\left(\frac{11264\sqrt{2}+256(97392+124181\sqrt{2})\beta-128(1386456+1671329\sqrt{2})\beta^{2}+32(7124760+391730\sqrt{2})\beta^{3}}{18874368\sqrt{2}(1-2\beta)^{6}}\right)\\ \ \ \ \ \ \ \ \ \ \ \ =-\left(\frac{4(62281968+24034855\sqrt{2})\beta^{4}+(185088288+47906305\sqrt{2})\beta^{5})}{18874368\sqrt{2}(1-2\beta)^{6}}\right);\\ \alpha_{6}^{(20)}(\beta)=\frac{1}{150994944\left(\beta-1\right)^{7}}\left(-9208137\beta^{6}+9386136\beta^{5}+6134672\beta^{4}\right)\\ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{150994944\left(\beta-1\right)^{7}}\left(24800000\beta^{3}-52645632\beta^{2}+22132736\beta-200704\right).\end{array}\right.$$
(A.2)

Coefficients of Equation (6.3) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha_{2}^{(20)}(\beta)=-\dfrac{5+11\beta+83\beta^{2}}{12(-1+2\beta)^{3}},\\ \alpha_{2}^{(02)}(\beta)=\dfrac{1-41\beta+103\beta^{2}}{12(-1+2\beta)^{3}},\\ \alpha_{4}^{(20)}(\beta)=\left(\dfrac{763-33526\beta+39957\beta^{2}-279082\beta^{3}+381211\beta^{4}}{3456(-1+2\beta)^{5}}\right),\\ \alpha_{4}^{(02)}(\beta)=-\left(\dfrac{5+12022\beta-63573\beta^{2}+50218\beta^{3}+90725\beta^{4}}{3456(-1+2\beta)^{5}}\right),\\ \alpha_{6}^{(20)}(\beta)=\dfrac{199459\beta^{6}-884361\beta^{5}+2800788\beta^{4}-4124041\beta^{3}+2621868\beta^{2}-59713\beta+289}{4976640\left(\beta-1\right)^{7}},\\ \alpha_{6}^{(02)}(\beta)=\dfrac{1}{4976640\left(\beta-1\right)^{7}}\left(-6980281\beta^{6}+30452649\beta^{5}-51751092\beta^{4}+46431209\beta^{3}\right)\\ \ \ \ \ \ \ \ \ \ \ \ -\dfrac{1}{4976640\left(\beta-1\right)^{7}}\left(23921292\beta^{2}+6715617\beta-1002401\right).\end{array}\right.$$
(A.3)

Coefficients of Equation (6.4) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha_{1}^{(20)}(\beta)=-\left(\dfrac{27(-108+7\beta)}{8(27-2\beta)^{2}}\right);\\ \alpha_{2}^{(20)}(\beta)=-\dfrac{(-108+23\beta)(-108+(23+8\sqrt{2})\beta)}{128\sqrt{2}(-9+2\beta)^{3}};\\ \alpha_{3}^{(20)}(\beta)=\left(\frac{-24794911296\sqrt{2}+25509168(-408+1333\sqrt{2})\beta-236196(-23544+66739\sqrt{2})\beta^{2}+2187(-503088+1564231\sqrt{2})\beta^{3}}{4096\sqrt{2}(27-2\beta)^{4}(-9+2\beta)^{3}}\right),\\ \alpha_{4}^{(20)}(\beta)=\frac{892616806656+11019960576(173+18\sqrt{2})\beta-17006112(94085+35664\sqrt{2})\beta^{2}+1574640(296705+207702\sqrt{2})\beta^{3}}{65536\sqrt{2}(-27+2\beta)^{5}(-9+2q)^{3}}\\ \alpha_{5}^{(20)}(\beta)=-\left(\frac{31812862989\sqrt{2}-29753893(624+2369\sqrt{2})\beta-1836660(11587+61479\sqrt{2})\beta^{2}+10183(2912+1598\sqrt{2})\beta^{3}}{18874368\sqrt{2}(27-2\beta)^{6}(-9+2\beta)^{3}}\right).\\ \end{array}\right.$$
(A.4)

Coefficients of Equation (6.5) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha_{2}^{(20)}(\beta)=-\dfrac{9(-3645+189\beta+\beta^{2})}{4(-27+2\beta)^{3}},\\ \alpha_{2}^{(02)}(\beta)=-\dfrac{27(243-27\beta+\beta^{2})}{4(-27+2\beta)^{3}},\\ \alpha_{4}^{(20)}(\beta)=\dfrac{-1216468449+266862114\beta-18510039\beta^{2}+439398\beta^{3}-1513\beta^{4}}{384(-27+2\beta)^{5}},\\ \alpha_{4}^{(02)}(\beta)=\dfrac{7971615-196830\beta+12393\beta^{2}-2970\beta^{3}+151\beta^{4}}{384(-27+2\beta)^{5}},\\ \alpha_{6}^{(20)}(\beta)=\dfrac{1165052056782267-387069098616477\beta+50678424210636\beta^{2}-3305202565221\beta^{3}}{552960\left(-27+2\beta\right)^{7}}\\ \ \ \ \ \ \ \ \ \ \ \ +\dfrac{110868795684\beta^{4}-1737425349\beta^{5}+8893913\beta^{6}}{552960\left(-27+2\beta\right)^{7}},\\ \alpha_{6}^{(02)}(\beta)=\dfrac{-12440502369-1807430841\beta+487272348\beta^{2}-71483553\beta^{3}}{20480\left(-27+2\beta\right)^{7}}\\ \ \ \ \ \ \ \ \ \ \ \ +\dfrac{6497172\beta^{4}-304577\beta^{5}+5349\beta^{6}}{20480(-27+2\beta)^{7}}.\\ \end{array}\right.$$
(A.5)

Coefficients of Equation (6.6) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha^{(20)}_{1}(\beta)=-\dfrac{2994003(-1328007994668+1332667\beta)}{8(997002999-1000\beta)^{2}},\\ \alpha_{2}^{(20)}(\beta)=\frac{2997(-5290815701706368149290672-7976023992(-1331334333+2663330\sqrt{2})\beta)}{128\sqrt{2}(-997002999+1000\beta)^{3}}\\ \ \ \ \ \ \ \ \ \ \ \ +\frac{20979(-761143428381+3045331048\sqrt{2})\beta^{2})}{128\sqrt{2}(-997002999+1000\beta)^{3}},\\ \alpha^{(20)}_{3}(\beta)=-\left[\frac{-210787366505431236951145553080744\sqrt{2}+159042396802399040(31944000024+39900729638\sqrt{2})\beta}{4096\sqrt{2}(997002999-1000\beta)^{4}}\right]\\ \ \ \ \ \ \ \ \ \ \ \ -\left[\frac{39880119(2556799840719+159683693169545\sqrt{2})\beta^{2}+(5116160640959360+213017068184190912\sqrt{2})\beta^{3})}{4096\sqrt{2}(997002999-1000\beta)^{4}}\right].\\ \end{array}\right.$$
(A.6)

Coefficients of Equation (6.7) of the Stability/Instability Surfaces

$$\left\{\begin{array}[]{l}\alpha_{2}^{(20)}(\beta)=\dfrac{2997(-40562920379748822477895152-23928071976(-3406741071+4004\sqrt{2})\beta}{128\sqrt{2}(-997002999+1000\beta)^{3}}\\ +\dfrac{(-40954777403147+96096000\sqrt{2})\beta^{2}}{128\sqrt{2}(-997002999+1000\beta)^{3}},\\ \alpha_{2}^{(02)}(\beta)=\dfrac{2997(1763605233902122716430224-7976023992(446443443+4004\sqrt{2})\beta}{128\sqrt{2}(-997002999+1000\beta)^{3}}\\ \ \ \ \ \ \ \ \ \ \ \ +\dfrac{(1797350682229+32032000\sqrt{2})\beta^{2}}{128\sqrt{2}(-997002999+1000\beta)^{3}},\\ \alpha_{4}^{(20)}(\beta)=\dfrac{-256+768(-455+1126\sqrt{2})\beta-3168(41+240\sqrt{2})\beta^{2}-48(-28713+36082\sqrt{2})\beta^{3}}{128\sqrt{2}(-997002999+1000\beta)^{3}},\\ \ \ \ \ \ \ \ \ \ \ \ +\dfrac{27(-32515+54704\sqrt{2})\beta^{4}}{128\sqrt{2}(-997002999+1000\beta)^{3}}\\ \alpha_{4}^{(02)}(\beta)=\dfrac{7971615-196830\beta+12393\beta^{2}-2970\beta^{3}+151\beta^{4}}{128\sqrt{2}(-997002999+1000\beta)^{3}}.\\ \end{array}\right.$$
(A.7)

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Araujo, G.C., Cabral, H.E. Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point. Regul. Chaot. Dyn. 26, 39–60 (2021). https://doi.org/10.1134/S1560354721010032

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