A chaotic study on Heisenberg ferromagnetic spin chain using Dzyaloshinski–Moriya interactions

Abstract

The chaotic dynamics of a one-dimensional Heisenberg ferromagnetic spin chain incorporating Dzyaloshinski–Moriya (D–M) interaction, dipole–dipole and quadrupole–quadrupole interactions has been investigated. The studies are carried out by plotting phase diagrams and chaotic trajectories. We then analyse the stability of the system using the Lyapunov stability analysis.

Appendices

Appendix A

The coefficients of eq. (14) are

$$\begin{aligned} E_{1}= & {} h^{2}Ja^{{2}\,}+{2}hJxa^{{2}}-{2}\,Ax^{{2}\,} a^{{2}\,}+{2}\,Jx^{{2}\,}a^{{2}\,}\\&+\,{2}Ap_{x}b^{{2}\,} -{2}Jp_{x}^{{2}\,}b^{{2}\,}+{2}\,h^{{2}\,}a^{{2}} J^{\prime }+4\,hxa^{{2}}J^{\prime }\\&+\,4\,x^{{2}\,} a^{{2}}J^{\prime }{-}4\,p_{x}^{{2}\,}b^{{2}}J^{\prime }{-}4\,x^{{2}\,} a^{{2}\,}A^{\prime }{+}4\,p_{x}^{{2}\,}b^{{2}}A^{\prime },\\ E_{{2}\,}= & {} -h^{{2}}Jx^{{2}\,}a^{4}-{2}\,hJx^{3}a^{4} +Ax^{4}a^{4}-Jx^{4}a^{4}\\&+\,h^{{2}}Jp_{x}^{{2}\,}a^{{2}\,}b^{{2}\,}+{2}\,hJp_{x}^{{2}\,} xa^{{2}\,}b^{{2}\,}-{2}Ap_{x}^{{2}\,}a^{{2}\,}b^{{2}\,}\\&+\,{2}Jp_{x}^{{2}\,} x^{{2}\,}a^{{2}\,}b^{{2}\,}+Ap_{x}^{4}\,b^{4}-Jp_{x}^{4}\,b^{4}-h^{4}a^{4}J^{\prime }\\&-\,4\,h^{3}x a^{4}J^{\prime }-10\,h^{{2}\,} x^{{2}\,}a^{4}J^{\prime }-12 \,hx^{3}a^{4}J^{\prime }\\&-\,6\,x^{4}a^{4}J^{\prime } {-}6\,h^{{2}\,}p_{x}^{{2}\,}a^{{2}\,}b^{{2}}J^{\prime }{+}1 2\, hp_{x}^{{2}\,}xa^{{2}\,}b^{{2}}J^{\prime }\\&+\,1 2p_{x}^{{2}\,} x^{{2}\,}a^{{2}\,}b^{{2}}J^{\prime }-6\, p_{x}^{4}\,b^{4} J^{\prime }+6\,x^{4}a^{4}A^{\prime }\\&+\,1 2\,p_{x}^{{2}\,} x^{{2}\,}a^{{2}\,}b^{{2}}A^{\prime }+6\,p_{x}^{4}\,b^ {4\,}J^{\prime },\\ E_{3}= & {} 2\,h^{4}x^{{2}\,}a^{6}J^{\prime }+8\,h^{3}x^{3}a^{6}J^{\prime } +14\,h^{2}\,x^{4}a^{6}J^{\prime }\\&+\,12\,hx^{5}a^{6}J^{\prime }+4\,x^{6} a^{6}J^{\prime }+{2}\,h^{4}p_{x}^{{2}\,}a^{4}b^{{2}} J^{\prime }\\&+\,8\,h^{3}p_{x}^{{2}\,}xa^{4}b^{{2}}J^{\prime }+20\,h^{{2}\,} p_{x}^{{2}\,}x^{{2}\,}a^{4}b^{{2}}J^{\prime }\\&+\,2 4\, hp_{x}^{{2}\,}x^{3}a^{4}b^{{2}\,}J^{\prime }+1 2\,p_{x}^{{2}\,} x^{4}a^{4}b^{{2}\,}J^{\prime }\\&+\,6\,h^{{2}}p_{x}^{4}\,b^{4}J^{\prime }+1 2\, hp_{x}^{4}\,xa^{{2}\,}b^{4}J^{\prime }\\&+\,1 2\,p_{x}^{4}x^{{2}\,}a^{{2}\,}b^{4} J^{\prime }-\,4\,p_{x}^{6}\,b^{6}J^{\prime }-4\,x^{6}a^{6}\,A^{\prime }\\&-\,1 2\,p_{x}^{{2}\,} x^{4}a^{4}b^{{2}\,}A^{\prime }-1 2\,p_{x}^{4}\,x^{{2}\,}b^{4} A^{\prime }\\&-\,4\,p_{x}^{6}\,b^{6}A^{\prime },\\ E_{4}= & {} -h^{4}x^{4}a^{8}J^{\prime }-4\,h^{{3 }}x^{5}a^{8}J^{\prime }-6\,h^{{2}\,}x^{6}a^{8}J^{\prime }\\&-\,4\,hx^{{7 }}a^{8}J^{\prime }-x^{8}a^{8}J^{\prime }-{2}\,h^{4\,}p_{x}^{{2}\,} x^{{2}\,}a^{6}b^{{2}}J^{\prime }\\&-\,{8\, }h^{{3}}p_{x}^{{2}\,}x^{{3 }}a^{6}b^{{2}}J^{\prime }-1 4\,h^{{2}\,}p_{x}^{{2}\,}x^{4\,} a^{6}b^{{2}}J^{\prime }\\&-\,1 2\,hp_{x}^{{2}\,}x^{5}a^{6}b^{{2}\,}J^{\prime } -4\,p_{x}^{{2}\,}x^{6}a^{6}b^{{2}\,}J^{\prime }\\&-\,h^{4}p_{x}^{4\,}a^{4\,} b^{4\,}J^{\prime }-4\,h^{{3}}p_{x}^{4\,}xa^{4\,}b^{4}J^{\prime }\\&-\,{10}h^{{2}}p_{x}^{4\,}x^{{2}\,}a^{4\,}b^{4}J^{\prime }-1 2\,hp_{x}^{4\,}x^{{3}}a^{4\,}b^{4}J^{\prime }\\&-\,6p_{x}^{4\,}x^{4\,}a^{4\,}b^{4} J^{\prime }-{2}\,h^{{2}}p_{x}^{6}\,a^{{2}\,}b^{6}J^{\prime }\\&-\,4\,hp_{x}^{6}\,xa^{{2}\,} b^{6}J^{\prime }-4p_{x}^{6}\,x^{{2}\,}a^{{2}\,}b^{6}J^{\prime }-p_{x}^{8}\,b^{8}J^{\prime }\\&+\,x^{8}a^{8}A^{\prime }+4p_{x}^{{3}}\,x^{6}a^{6}b^{{2}}A^{\prime }+6\,p_{x}^{4\,} x^{4\,}a^{4\,}b^{4}A^{\prime }\\&+\,4\,p_{x}^{6}\,x^{{2}\,}a^{{2}\,}b^{6}A^{\prime }+p_{x}^{8}\,b^{8}A^{\prime },\\ F_{1}= & {} -{2}\,hJxa^{{2}\,}{-}{2}\,Jx^{{2}\,}a^{{2}\,}+{2}\,Dhp_{x} ab-{2}\,Jp_{x}^{{2}\,}b^{{2}\,}\\&-\,4\,hxa^{{2}\,}f_{i}^{\prime }J^{\prime } -4\,x^{{2}\,}a^{{2}\,}J^{\prime }+4\,p_{x}^{{2}\,}b^{{2}\,}J^{\prime },\\ F_{{2}\,}= & {} 4\,h^{{3}}xa^{4\,}J^{\prime }+1 2\,h^{{2}\,}x^{{2}\,} a^{4\,}J^{\prime }+1 6\,hx^{{3}}a^{4\,}J^{\prime }\\&+\,{8\,}x^{4\,}a^{4\,}J^{\prime } {-}4\,h^{{2}\,}p_{x}^{{2}\,}a^{{2}\,}b^{{2}\,}J^{\prime } {-}16\,hp_{x}^{{2}\,}xa^{{2}\,}b^{{2}\,}J^{\prime }\\&-\,1 6\,p_{x}^{{2}\,} x^{{2}\,}a^{{2}\,}b^{{2}\,}J^{\prime }+{8\, }p_{x}^{4\,}b^{4\,}J^{\prime },\\ F_{3}= & {} -\,4\,h^{{3 }}x^{{3 }}a^{6}J^{\prime }-12\,h^{{2}\,}x^{4\,} a^{6}J^{\prime }-12\,hx^{5}a^{6}J^{\prime }\\&-\,4\,x^{6}a^{6}J^{\prime }+4\,h^{{3}} p_{x}^{{2}\,}xa^{4\,}b^{{2}\,}J^{\prime }\\&+\,16\,h^{{2}\,}i^{{2}\,} p_{x}^{{2}\,}x^{{2}\,}a^{4\,}b^{{2}}J^{\prime }+{2\, 4\,} hp_{x}^{{2}\,}x^{{3}}a^{4\,}b^{{2}}J^{\prime }\\&+\,12\,p_{x}^{{2}\,} x^{{3 }}a^{4\,}b^{{2}}J^{\prime }-4\,h^{{2}\,}p_{x}^{4\,}a^{{2}\,} b^{4}J^{\prime }\\&\left. {-}\,12\,hp_{x}^{4\,}xa^{{2}\,}b^{4}J^{\prime } {-}12\,p_{x}^{4\,}x^{{2}\,}a^{{2}\,}b^{4}J^{\prime }\right. \\&{+}\,4\,p_{x}^{6}\,b^{6}J^{\prime },\\ G_{1}= & {} 2\,hJx^{{3}}a^{4\,}+Jx^{4\,}a^{4\,}-Dh^{{2}\,}p_{x}xa^{{3}} b\\&{-}Dhp_{x}x^{{2}\,}a^{{3}}b{+}{2}\,hi^{{2}}Jp_{x}^{{2}\,}xa^{{2}\,} b^{{2}\,}\\&+\,{2}Jp_{x}^{{2}\,}x^{{2}\,}a^{{2}\,}b^{{2}\,} -Dhp_{x}^{{3 }}ab^{{3}}+Jp_{x}^{4\,}b^{4\,}\\&+\,h^{{3 }}xa^{4\,} J^{\prime }-h^{{2}\,}x^{{2}\,}a^{4}J^{\prime }-4\,hx^{{3 }}a^{4}J^{\prime }\\&-\,{2}\,x^{4\,}a^{4}J^{\prime }+h^{{2}}p_{x}^{{2}\,}a^{{2}\,}b^{{2}} J^{\prime }-4\,hp_{x}^{{2}\,}xa^{{2}\,}b^{{2}}J^{\prime }\\&-\,4\,p_{x}^{{2}\,} x^{{2}\,}a^{{2}\,}b^{{2}\,}J^{\prime }-{2}p_{x}^{4}b^{4}J^{\prime }\\&+\frac{1}{{2}\,}\Big (h^{{3 }}Ja^{4\,}x+3\,h^{{2}}Jx^{{2}\,} a^{4\,}\\&-\,Dh^{{3}}p_{x}a^{{3 }}b+h^{{2}\,}i^{{2}\,}Jp_{x}^{{2}\,} a^{{2}\,}b^{{2}\,}\Big ),\\ G_{{2}\,}= & {} -h^{5}xa^{6}J^{\prime }-{5\,}h^{4\,}x^{{2}\,}a^{6}J^{\prime }\\&-\,12\,h^{{3}}x^{{3 }}a^{6}J^{\prime }-16\,h^{{2}\,}x^{4\,}a^{6} J^{\prime }\\&-\,12\,hx^{6}a^{6}J^{\prime }-4\,x^{6}a^{6}J^{\prime }-h^{4\,}p_{x}^{{2}\,} a^{4\,}b^{{2}}J^{\prime }\\&-\,{8\, }h^{{3}}p_{x}^{{2}\,}xa^{4\,}b^{{2}\,} J^{\prime }-{2\, 0\,}h^{{2}\,}p_{x}^{{2}\,}x^{{2}\,}a^{4\,}b^{{2}}J^{\prime }\\&-\,{2\, 4\,}hp_{x}^{{2}\,}x^{{3}}a^{4\,}b^{{2}\,}J^{\prime }-12\,p_{x}^{{2}\,}x^{4\,}a^{4\,}b^{{2}\,}J^{\prime }\\&-\,4\,h^{{2}} p_{x}^{4\,}a^{{2}\,}b^{4}J^{\prime }-12\,hp_{x}^{4\,} xa^{{2}\,}b^{4\,}J^{\prime }\\&-\,12p_{x}^{4\,}x^{{2}\,}a^{{2}\,} b^{4}J^{\prime }-4\,p_{x}^{6}\,b^{6}J^{\prime },\\ G_{3}= & {} h^{5}x^{{3 }}a^{8}J^{\prime }+{5 }h^{4\,}x^{4\,}a^{8}J^{\prime } +11\,h^{{3}}x^{5}a^{8}J^{\prime }\\&+\,13\,h^{{2}\,}x^{6}a^{8}J^{\prime }+{8\, } hx^{{7}}a^{8}J^{\prime }+{2}\,x^{8}a^{8}J^{\prime }\\&-\,h^{5}p_{x}^{{2}\,} xa^{6}b^{{2}}J^{\prime }+6\,h^{4\,}p_{x}^{{2}\,}x^{{2}\,}a^{6}b^{{2}\,} J^{\prime }\\&+\,18\,h^{{3}}p_{x}^{{2}\,}x^{{3}}a^{6}b^{{2}}J^{\prime } +2\, 9\,h^{{2}\,}p_{x}^{{2}\,}x^{4\,}a^{6}b^{{2}}J^{\prime }\\&+\,{2\, 4\,} hp_{x}^{{2}\,}x^{5}a^{5}b^{{2}}J^{\prime }+{8\, }p_{x}^{{2}\,}x^{6} a^{6}b^{{2}}J^{\prime }\\&+\,h^{4}p_{x}^{4\,}a^{4\,}b^{4}J^{\prime }+7\,h^{{3}} p_{x}^{4\,}xa^{4\,}b^{4}J^{\prime }\\&+\,19\,h^{{2}}p_{x}^{4\,} x^{{2}\,}a^{4\,}b^{4\,}J^{\prime }+{2\, 4\,}hp_{x}^{4\,}x^{{3}} a^{4\,}b^{4}J^{\prime }\\&+\,12\,p_{x}^{4\,}x^{4\,}a^{4\,}b^{4}J^{\prime } +3\,h^{{2}\,}p_{x}^{6}\,a^{{2}\,}b^{6}J^{\prime }\\&\left. \left. +\,{8\,}hp_{x}^{6}\,xa^{{2}\,}b^{6}J^{\prime } {+}{8\,}p_{x}^{6}\,x^{{2}\,}a^{{2}\,}b^{6}J^{\prime }\right. \right. \\&+\,{2}\,p_{x}^{8}\,b^{8}J^{\prime }. \end{aligned}$$

Appendix B

$$\begin{aligned} \frac{\mathrm {d}x}{\mathrm {d}t}= & {} \frac{1}{S}\Big (-\,4Ap_{x}b^{{2}\,} +4\,Jp_{x}b^{{2}\,}+{8}p_{x}b^{{2}\,}J^{\prime }\nonumber \\&-\,{8}p_{x}b^{{2}\,} A^{\prime }\Big )-\frac{1}{S^{{2}\,}}\Big (2\,h^{{2}}Jp_{x}a^{{2}\,}b^{{2}\,}\nonumber \\&-\,4hJp_{x}xa^{{2}\,}b^{{2}\,}+4Ap_{x}x^{{2}\,}a^{{2}\,} b^{{2}\,}\nonumber \\&-\,4Jp_{x}x^{{2}\,}a^{{2}\,}b^{{2}\,}+4\,Ap_{x}^{3}\,b^{4\,}\nonumber \\&-\,4 Jp_{x}^{3}\,b^{4\,}-{12\,}h^{{2}}p_{x}a^{{2}\,}b^{{2}}J^{\prime }\nonumber \\&-\,{24\,}hp_{x}xa^{{2}\,}b^{{2}}J^{\prime }-{24\,}p_{x}x^{{2}\,}a^{{2}\,} b^{{2}}J^{\prime }\nonumber \\&-\,24\,p_{x}^{3}\,b^{4}J^{\prime }+{24\,}p_{x}x^{{2}\,}a^{{2}\,}b^{{2}} A^{\prime }+{24\, } p_{x}^{3}\,b^{4}A^{\prime }\Big )\nonumber \\&+\,\frac{1}{S^{3}}\Big (4\,h^{4\,}p_{x}a^{4\,} b^{{2}}J^{\prime }+16\,h^{3}p_{x}xa^{4\,}b^{{2}}J^{\prime }\nonumber \\&+\,40\,h^{{2}}p_{x}x^{{2}\,}a^{4\,} b^{{2}}J^{\prime }+48\,hp_{x}x^{3}a^{4\,}b^{{2}}J^{\prime }\nonumber \\&+\,{24}p_{x}x^{4\,}a^{4\,} b^{{2}}J^{\prime }+{24\,}h^{{2}}p_{x}^{3}\,a^{{2}\,}b^{4}J^{\prime }\nonumber \\&+\,48\,hp_{x}^{3}xa^{{2}\,}b^{4}J^{\prime }+48p_{x}^{3}\,x^{{2}\,}a^{{2}\,} b^{4}J^{\prime }\nonumber \\&+\,{24 }p_{x}^{{5\,}}b^{6}J^{\prime }-24\,p_{x}x^{4\,}a^{4\,}b^{{2}\,}A^{\prime }\nonumber \\&\left. -\,48 p_{x}^{3} x^{{2}\,}a^{{2}\,}b^{4\,}A^{\prime }-{24}p_{x}^{{5\,}}b^{6}A^{\prime }\right) \nonumber \\&+\,\frac{1}{S^{4\,}}\Big (-\,4\,h^{4}p_{x}x^{{2}\,}a^{6}b^{{2}}J^{\prime } -16\,h^{3}p_{x}x^{3}a^{6}b^{{2}}J^{\prime }\nonumber \\&-\,28\,h^{{2}\,}p_{x}x^{4\,} a^{6}b^{{2}\,}J^{\prime }-{24\, }hp_{x}x^{{5\, }} a^{6}b^{{2}}J^{\prime }\nonumber \\&-\,{8\, }p_{x}x^{6}a^{6}b^{{2}}J^{\prime }-4\,h^{4} p_{x}^{3}\,a^{4\,}b^{4}J^{\prime }\nonumber \\&-\,16\,h^{3}p_{x}^{3}\,xa^{4\,}b^{4}J^{\prime } -40\,h^{{2}\,} p_{x}^{3}x^{{2}\,}a^{4\,}b^{4}J^{\prime }\nonumber \\&-\,48\,hp_{x}^{3}\,x^{3}a^{4\,} b^{4}J^{\prime }-{24 }p_{x}^{3}\,x^{4\,}a^{4\,}b^{4}J^{\prime }\nonumber \\&-\,{12\, } h^{{2}}p_{x}^{{5\, }}a^{{2}\,}b^{6}J^{\prime } -{24\, }hp_{x}^{{5\, }}xa^{{2}\,}b^{6}J^{\prime }\nonumber \\&-\,{24\,}p_{x}^{{5\, }} x^{{2}\,}a^{{2}\,}b^{6}J^{\prime }-{8}p_{x}^{{7}}b^{{8\, }}J^{\prime }\nonumber \\&\left. +\,{8\,} p_{x}x^{6}a^{6}b^{{2}\,}A^{\prime }+{24\, }p_{x}^{3}x^{4\,}a^{4\,}b^{4\,}A^{\prime }\right. \nonumber \\&+\,{24\, }p_{x}^{{5\,}} x^{{2}\,}a^{{2}\,}b^{6}A^{\prime }+{8\, }p_{x}^{7}\,b^{{8 }}A^{\prime }\Big )\nonumber \\&+\,\epsilon ^{{2}\,}\Big (2\, \textit{Dhab}-4\,J\,p_{x}b^{{2}\,}-{8\,}p_{x}b^{{2}\,}J^{\prime }\nonumber \\&+\,\frac{1}{S}\Big (8\,h^{{2}}p_{x} a^{{2}\,}b^{{2}}J^{\prime }+32\,hp_{x}xa^{{2}\,}b^{{2}\,}J^{\prime }\nonumber \\&+\,32\,p_{x}x^{{2}\,}a^{2}\, b^{{2}}J^{\prime }+32\,p_{x}^{3}\,b^{4}J^{\prime }\Big )\nonumber \\&+\,\frac{1}{S^{{2}\,}}\Big (-{8\,} h^{3}p_{x}xa^{4\,}b^{{2}\,}J^{\prime }-32\,h^{{2}\,}p_{x}x^{{2}\,}a^{4\,} b^{{2}\,}J^{\prime }\nonumber \\&-\,48\,hp_{x}\,x^{3}\,a^{4\,}b^{{2}\,}J^{\prime }-{24\,}p_{x}x^{4\,}a^{4\,} b^{{2}\,}J^{\prime }\nonumber \\&-\,16\,h^{{2}}p_{x}^{3}\,a^{{2}\,}b^{4\,}J^{\prime }-48\,hp_{x}^{3}\,xa^{{2}\,}b^{4\,} J^{\prime }\nonumber \\&-\,48\,p_{x}^{3}\,x^{{2}\,}a^{{2}\,}b^{4\,}J^{\prime }-{24\, } p_{x}^{{5\,}}b^{6}J^{\prime }\Big )\Big )\nonumber \\&+\,\epsilon ^{4\,}\left( -\frac{1}{{2}\,}Dh^{3}a^{3}\,b-Dh^{{2}\,} xa^{3}b-Dhx^{{2}\,}a^{3}\,b\right. \nonumber \\&+\,h^{{2}\,}Jp_{x}a^{{2}\,}b^{{2}\,}+4\,hJp_{x}xa^{{2}\,} b^{{2}\,}+4\,Jp_{x}x^{{2}\,}a^{{2}\,}b^{{2}\,}\nonumber \\&-\,3Dh p_{x}^{{2}\,}ab^{3}+4\,Jp_{x}^{3}\,b^{4\,}+{2}\,h^{{2}}p_{x}a^{{2}\,} b^{{2}}J^{\prime }\nonumber \\&-\,{8\,}hp_{x}xa^{{2}\,}b^{{2}\,}J^{\prime }-{8\, }p_{x}x^{{2}\,} a^{{2}\,}b^{{2}\,}J^{\prime } -{8}p_{x}^{3}\,b^{4}J^{\prime }\nonumber \\&+\,\frac{1}{S}\Big (-{2}\,h^{4\,}p_{x}a^{4\,} b^{{2}}J^{\prime }-16\,h^{3}p_{x}xa^{4\,}b^{{2}}J^{\prime }\nonumber \\&-\,40\,h^{{2}} p_{x}x^{{2}\,}a^{4\,} b^{{2}}J^{\prime }-48\,hp_{x}x^{3}a^{4\,}b^{{2}}J^{\prime }\nonumber \\&-\,{24\,}p_{x}x^{4\,} a^{4\,}b^{{2}}J^{\prime }-16\,h^{{2}\,}p_{x}^{3}\,a^{{2}\,}b^{4}J^{\prime }\nonumber \\&\left. -\,48\,h\,p_{x}^{3}\,xa^{{2}\,}b^{4}J^{\prime }{-}48\,p_{x}^{3}\,x^{{2}\,}a^{{2}\,} b^{4}J^{\prime }\right. \nonumber \\&\left. {-}{24\, }p_{x}^{{5}}\,b^{6}J^{\prime }\right) \nonumber \\&+\,\frac{1}{S^{{2}}} \left( 2\,h^{{5}}p_{x}xa^{6}b^{{2}}J^{\prime } +{12\, }h^{4\,}p_{x}x^{{2}\,}a^{6}b^{{2}}J^{\prime }\right. \nonumber \\&+\,36\,h^{3}p_{x}x^{3} a^{6}b^{{2}}J^{\prime }+58\,h^{{2}\,}p_{x}x^{4\,}a^{6}b^{{2}}J^{\prime }\nonumber \\&+\,48\,hp_{x} x^{{5\, }}a^{6}b^{{2}}J^{\prime }+16\,p_{x}x^{6}a^{6}b^{{2}}J^{\prime }\nonumber \\&+\,4\,h^{4\,}p_{x}^{3}a^{4\,}b^{4}J^{\prime }+28\,h^{3}p_{x}^{3}xa^{4\,} b^{4}J^{\prime }\nonumber \\&+\,76\,h^{{2}\,}p_{x}^{3}x^{{2}\,}a^{4\,}b^{4}J^{\prime }+96\,hp_{x}^{3}x^{3} a^{4\,}b^{4}J^{\prime }\nonumber \\&+\,48\,p_{x}^{3}x^{4\,}a^{4}b^{4\,}J^{\prime }+18\,h^{{2}\,} p_{x}^{{5\, }} a^{{2}\,}b^{6}J^{\prime }\nonumber \\&+\,48\,hp_{x}^{{5\, }}xa^{{2}}b^{6}J^{\prime }+48\,p_{x}^{{5\, }}x^{{2}\,}a^{{2}\,}b^{6}J^{\prime }\nonumber \\&\left. +\,16\,p_{x}^{7}b^{{8\, }}J^{\prime }\right) \Bigg ), \end{aligned}$$

(15)

$$\begin{aligned} \frac{\mathrm {d}p_{x}}{\mathrm {d}t}= & {} \frac{1}{S}\Big (2\,hJa^{{2}\,} -4Axa^{{2}\,}+4Jxa^{{2}\,}+4\,ha^{{2}\,}J^{\prime }\nonumber \\&+\,{8\,}xa^{{2}\,}J^{\prime }+{8\, }xa^{{2}\,}A^{\prime }\Big )\nonumber \\&+\,\frac{1}{S^{{2}\,}}\Big (-{2}\,h^{{2}\,}Jxa^{4\,}-6\,hJx^{{2}\,}a^{4\,} +4\,Ax^{3}a^{4\,}\nonumber \\&-\,4\,Jx^{3}a^{4\,}-{2}hJp_{x}^{{2}\,}a^{{2}\,}b^{{2}\,} +4\,Ap_{x}^{{2}\,}xa^{{2}\,}b^{{2}\,}\nonumber \\&-\,4\,Jp_{x}^{{2}\,} xa^{{2}\,}b^{{2}\,}-4\,h^{3}a^{4\,}J^{\prime } -{2\, 0\, }h^{{2}\,}xa^{4\,}J^{\prime }\nonumber \\&-\,36\,hx^{{2}\,}a^{4\,} J^{\prime }-{2\, 4\, }x^{3}a^{4\,}J^{\prime }-{12\, }hp_{x}^{{2}\,}a^{{2}\,}b^{{2}\,}J^{\prime }\nonumber \\&\left. -\,{24\,}p_{x}^{{2}\,}xa^{{2}\,}b^{{2}\,}J^{\prime }{-}{24\, }x^{3}a^{4\,}A^{\prime }\right. \nonumber \\&{+}{24\, }p_{x}^{{2}\,}xa^{{2}\,}b^{{2}\,}A^{\prime }\Big )\nonumber \\&+\,\frac{1}{S^{3}}\Big (4\,h^{4\,}xa^{6}J^{\prime }{+}{24\,}h^{3}x^{3}a^{6}J^{\prime } {+}56\,h^{{2}\,}x^{3}a^{6}J^{\prime }\nonumber \\&+\,60\,hx^{4\,}a^{6}J^{\prime }+{24\, }x^{{5\,}} a^{6}J^{\prime }+{8\,}h^{3}p_{x}^{{2}\,}a^{4\,}b^{{2}}J^{\prime }\nonumber \\&+\,40\,h^{{2}} p_{x}^{{2}\,}xa^{4\,}b^{{2}}J^{\prime }+72\,hp_{x}^{{2}\,}x^{{2}\,}a^{4\,} b^{{2}}J^{\prime }\nonumber \\&+\,48p_{x}^{{2}\,}x^{3}a^{4\,}b^{{2}}J^{\prime }+{12\, } hp_{x}^{4\,}a^{{2}\,}b^{4}J^{\prime }\nonumber \\&+\,{24}p_{x}^{4\,}xa^{{2}\,} b^{4}J^{\prime }{24\, }x^{{5\,}}a^{6}A^{\prime }-48\,p_{x}^{{2}\,}x^{3}a^{4\,} b^{{2}\,}A^{\prime }\nonumber \\&+\,{24\, }p_{x}^{4\,}xa^{{2}\,}b^{4\,}A^{\prime }\Big ) +\frac{1}{S^{4\,}}\Big (-\,4\,h^{4\,}x^{3}a^{{8}}J^{\prime }\nonumber \\&-\,20\,h^{3}x^{4\,}a^{{8}} J^{\prime }-36\,h^{{2}\,}x^{{5\, }}a^{{8}}J^{\prime }-28\,hx^{6}a^{{8 }}J^{\prime }\nonumber \\&-\,{8\,}x^{{7 }}a^{{8\,}}J^{\prime }{-}4\,h^{4}p_{x}^{{2}\,}xa^{6}\,b^{{2}\,}J^{\prime }\nonumber \\&{-}{24\, }h^{3}p_{x}^{{2}\,}x^{{2}\,}a^{6}\,b^{{2}}J^{\prime }\nonumber \\&-\,56\,h^{{2}\,} p_{x}^{{2}\,}x^{3}a^{6}b^{{2}}J^{\prime }-60\,hp_{x}^{{2}\,}x^{4\,} a^{6}b^{{2}}J^{\prime }\nonumber \\&-\,{24\, }p_{x}^{{2}\,}x^{{5\, }}a^{6}b^{{2}}J^{\prime } -4\,h^{3}p_{x}^{4\,}a^{4\,}b^{4}J^{\prime }\nonumber \\&-\,20\,h^{{2}\,}p_{x}^{4\,}xa^{4\,} b^{4}J^{\prime }-36\,hp_{x}^{4\,}x^{{2}\,}a^{4\,}b^{4}J^{\prime }\nonumber \\&-\,24\,p_{x}^{4\,}x^{3}a^{4\,}b^{4}J^{\prime }-4\,hp_{x}^{6}a^{{2}\,}b^{6}J^{\prime }\nonumber \\&-\,{8\,}p_{x}^{6}xa^{{2}\,}b^{6}J^{\prime }-{8\,}x^{{7 }}a^{{8\,}}A^{\prime }+{2\, 4\, } p_{x}^{{2}\,}x^{{5\, }}a^{6}b^{{2}\,}A^{\prime }\nonumber \\&-\,{24\,}p_{x}^{4\,}x^{3} a^{4\,}b^{4}A^{\prime }+{8\, }p_{x}^{6}xa^{{2}\,}b^{6}\,A^{\prime }\Big )\nonumber \\&-\,\epsilon ^{{2}\,}\Big [-{2}\,hJa^{{2}\,}-4Jxa^{{2}\,}-4\,ha^{{2}}J^{\prime }\nonumber \nonumber \\&-\,{8\, }xa^{{2}}J^{\prime }+\frac{1}{S}\Big (4\,h^{3}a^{4}J^{\prime }+{24\, }h^{{2}\,}xa^{4\,}J^{\prime }\nonumber \\&+\,48\,hx^{{2}\,}a^{4}J^{\prime }+\,32\,x^{3}a^{4}J^{\prime }+16\,hp_{x}^{{2}\,} a^{{2}\,}b^{{2}}J^{\prime }\nonumber \\&+\,32\,p_{x}^{{2}\,}xa^{{2}}b^{{2}}J^{\prime }\Big ) +\frac{1}{S^{{2}\,}}\Big (-12\,h^{3}x^{{2}}a^{6}J^{\prime }\nonumber \\&-\,48\,h^{{2}\,}x^{3}a^{6}J^{\prime }-60\,hx^{4\,}a^{6}J^{\prime }-{24\, }x^{{5\,}} a^{6}J^{\prime }\nonumber \\&-\,4\,h^{3}p_{x}^{{2}\,}a^{4\,}b^{{2}}J^{\prime }-32\,h^{{2}\,} p_{x}^{{2}\,}xa^{4\,}b^{{2}}J^{\prime }\nonumber \\&-\,72\,hp_{x}^{{2}\,}x^{{2}\,}a^{4\,} b^{{2}}J^{\prime }-48\,p_{x}^{{2}\,}x^{3}a^{4\,}b^{{2}}J^{\prime }\nonumber \\&-\,{12\, }hp_{x}^{4\,}a^{{2}\,}b^{4}J^{\prime }-24\,p_{x}^{4\,}xa^{{2}\,} b^{4}J^{\prime }\Big )\Big ]\nonumber \\&-\,\epsilon ^{4\,}\left[ \frac{1}{{2}\,}h^{3}Ja^{4\,}+3\,h^{{2}\,} Jxa^{4\,}+6\,hJx^{{2}\,}a^{4\,}\right. \nonumber \\&+\,4\,Jx^{3}a^{4\,}-Dh^{{2}\,}p_{x}a^{3}b-{2} \,Dhp_{x}a^{3}b\nonumber \\&+\,{2}\,hJp_{x}^{{2}\,}a^{{2}\,}b^{{2}\,}+4\,Jp_{x}^{{2}\,} xa^{{2}\,}b^{{2}\,}+h^{3}a^{4}J^{\prime }\nonumber \\&-\,{2}\,h^{{2}\,}xa^{4}J^{\prime } -{12\, }hx^{{2}\,}a^{4}J^{\prime }-{8\,}x^{3}a^{4}J^{\prime }\nonumber \\&-\,4\,hp_{x}^{{2}\,} a^{{2}\,}b^{{2}}J^{\prime }-{8\, }p_{x}^{{2}\,}xa^{{2}\,}b^{{2}}J^{\prime }\nonumber \\&+\,\frac{1}{S}\Big (-h^{{5\,}}a^{6}J^{\prime }-10\,h^{4\,}xa^{6}J^{\prime } -36\,h^{3}x^{{2}\,}a^{6}J^{\prime }\nonumber \\&-\,64\,h^{{2}\,}x^{3}a^{6}J^{\prime }-60\,h\,x^{4\,} a^{6}J^{\prime }-{24\, }x^{{5\,}}a^{6}J^{\prime }\nonumber \\&-\,{8\,}h^{3}p_{x}^{{2}\,}a^{4\,} b^{{2}}J^{\prime }-40\,h^{{2}\,}p_{x}^{{2}\,}xa^{4\,}b^{{2}}J^{\prime }\nonumber \\&+\,72\,hp_{x}^{{2}\,}x^{{2}\,}a^{4\,}b^{{2}}J^{\prime }-48\,p_{x}^{{2}\,} x^{3}a^{4\,}b^{{2}}J^{\prime }\nonumber \\&-\,{12\, }hp_{x}^{4\,}a^{{2}\,}b^{4} J^{\prime }-{24\, }p_{x}^{4\,}xa^{{2}\,}b^{4}J^{\prime }\Big )\nonumber \\&+\,\frac{1}{S^{{2}\,}} \left( 3\,h^{{5\, }}x^{{2}\,}a^{{8\,}}J^{\prime }+20\,h^{4\,}x^{3}a^{{8}}J^{\prime }\right. \nonumber \\&+\,55\,h^{3}x^{4\,}a^{{8}}J^{\prime }+78\,h^{{2}\,}x^{{5\, }}a^{{8}} J^{\prime }\nonumber \\&+\,56\,hx^{6}a^{{8 }}J^{\prime }+16\,x^{{7}}a^{{8}}J^{\prime }+h^{{5\,}} p_{x}^{{2}\,}a^{6}b^{{2}}J^{\prime }\nonumber \\&+\,{12\,}h^{4\,}p_{x}^{{2}\,}xa^{6}b^{{2}} J^{\prime }+54\,h^{3}p_{x}^{{2}\,}x^{{2}\,}a^{6}b^{{2}}J^{\prime }\nonumber \\&+\,116\,h^{{2}\,}p_{x}^{{2}\,}x^{3}a^{6}b^{{2}}J^{\prime }+120\,hp_{x}^{{2}\,} x^{4\,}a^{6}b^{{2}}J^{\prime }\nonumber \\&+\,48\,p_{x}^{{2}\,}x^{{5\, }} a^{6}b^{{2}}J^{\prime }+7\,h^{3}p_{x}^{4\,}a^{4\,}b^{4}J^{\prime }\nonumber \\&+\,38\,h^{{2}\,}p_{x}^{4\,}xa^{4\,}b^{4}J^{\prime }+72\,hp_{x}^{4\,}x^{{2}\,} a^{4\,}b^{4}J^{\prime }\nonumber \\&+\,48\,p_{x}^{4\,}x^{3}a^{4\,}b^{4\,}J^{\prime } +{8\, }hp_{x}^{6}a^{{2}\,}b^{6}J^{\prime }\nonumber \\&+\,\left. 16\,p_{x}^{6}xa^{{2}\,}b^{6}J^{\prime }\Big )\right] . \end{aligned}$$

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