The extended 16th Hilbert problem for a class of discontinuous piecewise differential systems

Abstract

In order to understand the dynamics of the planar differential systems, the limit cycles play a main role, but in general their study is not easy. These last years, an increasing interest appeared for studying the limit cycles of some classes of piecewise differential systems, due to the rich applications of this kind of differential systems. This paper solves the extended 16th Hilbert problem for a family of discontinuous planar differential systems with two regions separated by the straight line $x=0$. By using the first integrals, we prove that the maximum number of crossing limit cycles in the family of systems formed by a linear center and a class of Hamiltonian isochronous global center with a polynomial first integral of degree 2n is 5.

Appendix

The formulas of this appendix are now in the paper only for its revision; if the paper is accepted, then the appendix will come as a data attached file.

Here we provide the expressions $A_i$, with $i=0,\dots ,10$.

$$\begin{aligned} A_0= & {} \dfrac{1}{(a^2+\omega ^2\big )^{11}} \big ( b_1 c_1 a^{22}+\beta _1 \gamma _1 a^{22}\\&+11 b_1 c_1 \omega ^2 a^{20} +b d \beta _1^2 a^{20}+b b_1^2 d a^{20}+11 \\&\times \omega ^2 \beta _1 a^{20}+55 b_1 c_1 \omega ^4 a^{18} +10 b b_1^2 d \omega ^2 a^{18}\\&+10 b d \omega ^2 \beta _1^2 a^{18} +55 \omega ^4 \beta _1 \gamma _1 a^{18}\\&+165b_1 c_1\omega ^6 a^{16}+45 b b_1^2 d \omega ^4 a^{16}\\&+45 b d \omega ^4 \beta _1^2 a^{16}+165 \omega ^6 \beta _1 \gamma _1 a^{16}\\&+330 b_1 c_1 \omega ^8 a^{14}+120 b b_1^2 d \omega ^6 a^{14}\\&+120 b d \omega ^6 \beta _1^2 a^{14} +330 \omega ^8 \beta _1 \gamma _1 a^{14}\\&+462 b_1 c_1 \omega ^{10} a^{12} +210 b b_1^2 d \omega ^8 a^{12}\\&+210 b d \omega ^8 \beta _1^2 a^{12}+462 \omega ^{10} \beta _1 \gamma _1 a^{12}\\&+462 b_1 c_1 \omega ^{12} a^{10}+252b b_1^2 d \omega ^{10}a^{10}+252 b d \\&\times \omega ^{10} \beta _1^2 a^{10}+462 \omega ^{12}\beta _1 \gamma _1 a^{10}\\&+330 b_1 c_1\omega ^{14} a^8+210 b b_1^2 d\omega ^{12} a^8\\&+210 b d \omega ^{12} \beta _1^2 a^8 +330 \omega ^{14} \beta _1 \gamma _1 a^8\\&+165 b_1 c_1 \omega ^{16} a^6 +120 b b_1^2 d \omega ^{14} a^6\\&+120 b d\omega ^{14}+55 b_1 c_1 \omega ^{18} a^4\\&+165 \omega ^{16} \beta _1 \gamma _1 a^6+45 b b_1^2 d \omega ^{16} a^4\\&+45 b d \omega ^{16} \beta _1^2 a^4+55 \omega ^{18}\beta _1 \gamma _1 a^4\\&+11 b_1 c_1 \omega ^{20} a^2 +10 b b_1^2 d \omega ^{18} a^2\\&+10 b d \omega ^{18} \beta _1^2 a^2 +11\omega ^{20} \beta _1 \gamma _1a^2+b_1 c_1 \omega ^{22}\\&+b b_1^2 d \omega ^{20} +b d \omega ^{20} \beta _1^2 +2 \beta _1 (512 b^{11} d^{11} \beta _1^{11}\\&+3072 b^{10} d^{10} (a^2+\omega ^2) \gamma _1 \beta _1^{10} \\&+8448 b^9 d^9(a^2+\omega ^2)^2\gamma _1^2 \beta _1^9\\&+14080 b^8 d^8(a^2+\omega ^2)^3 \gamma _1^3 \beta _1^8\\&+15840b^7 d^7 (a^2+\omega ^2)^4 \gamma _1^4 \beta _1^7+12672 b^6 d^6 \\&\times (a^2 +\omega ^2)^5\gamma _1^5 \beta _1^6 +7392 b^5 d^5 (a^2+\omega ^2)^6\gamma _1^6\\&\times \beta _1^5 +3168 b^4 d^4 (a^2+\omega ^2)^7\gamma _1^7 \beta _1^4+990 b^3 d^3\\&(a^2 +\omega ^2)^8\gamma _1^8 \beta _1^3+220 b^2 d^2(a^2+\omega ^2)^9 \gamma _1^9 \beta _1^2\\&+33 b d (a^2+\omega ^2)^{10} \gamma _1^{10} \beta _1+3 (a^2 \\&+\omega ^2)^{11}\gamma _1^{11}) \delta ^2+\omega ^{22} \beta _1 \gamma _1 +(a^2+\omega ^2)^5 \\&\times (32 b^5 d^5 (2 b b_1 d +c_1(a^2+\omega ^2)) \beta _1^6\\&+96 b^4 d^4 (a^2 +\omega ^2) (2 b b_1 d+c_1(a^2\\&+\omega ^2))\gamma _1 \beta _1^5 +120 b^3 d^3(a^2+\omega ^2)^2(2 b b_1 d\\&+c_1 (a^2+\omega ^2) ) \gamma _1^2 \beta _1^4+80 b^2 d^2 (a^2+\omega ^2)^3\\&\times (2 b b_1 d +c_1 (a^2+\omega ^2))\gamma _1^3 \beta _1^3\\&+30 b d (a^2+\omega ^2)^4 (2b b_1 d \\&+c_1 (a^2+\omega ^2)) \gamma _1^4 \beta _1^2+6 (a^2+\omega ^2)^5(2 b b_1 d \\&+c_1(a^2+\omega ^2)) \gamma _1^5 \beta _1+b_1 (a^2+\omega ^2)^6 \gamma _1^6) \delta ) \\ A_1= & {} -\dfrac{1}{(a^2+\omega ^2\big )^{10}}\big (2 b d \beta _1^2 \delta (5 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6\\&+3 b_1 \gamma _1+4 c_1 \beta _1) a^{18}+5 \gamma _1^2 (9 \gamma _1(22 \beta _1 \delta \gamma _1^6\\&+3 b_1 \gamma _1+4 c_1 \beta _1) \omega ^2 +2 b d \beta _1 (99 \beta _1 \delta \gamma _1^6\\&+8 b_1 \gamma _1+6 c_1 \beta _1)) a^{16} \\&+4 \gamma _1 (45 \gamma _1^2(22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1) \omega ^4\\&+20 b d \beta _1 \gamma _1 (99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^2 \\&+9 b^2 d^2\beta _1^2(5 b_1 \gamma _1+2 \beta _1 (66 \delta \gamma _1^6+c_1))) a^{14} \\&+4 (105 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 \\&c_1 \beta _1) \omega ^6 +70 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6 \\&+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^4+63 b^2 d^2 \beta _1^2\gamma _1(5 b_1 \gamma _1 \\&+2\beta _1(66 \delta \gamma _1^6+c_1)) \omega ^2+8 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6\\&+6 b_1 \gamma _1+c_1 \beta _1)) a^{12}+2(315 \\&\gamma _1^3(22\beta _1\delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1) \omega ^8\\&+280 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1+6 c_1 \beta _1)\omega ^6 \\&+378b^2d^2\beta _1^2 \gamma _1 (5 b_1 \gamma _1+2 \beta _1 (66 \delta \gamma _1^6+c_1)) \omega ^4\\&+96 b^3 d^3 \beta _1^3(462 \beta _1 \delta \gamma _1^6+6 b_1 \gamma _1+c_1 \\&\beta _1)\omega ^2+40 b^4 d^4\beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1)) a^{10}\\&+10(63 \gamma _1^3(22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1) \\&\omega ^{10}+70b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6+8b_1 \gamma _1\\&+6 c_1 \beta _1) \omega ^8+126 b^2 d^2 \beta _1^2 \gamma _1 (5 b_1 \gamma _1+2 \beta _1 (66 \delta \\&\gamma _1^6+c_1)) \omega ^6 +48 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6+6 b_1 \gamma _1\\&+c_1 \beta _1) \omega ^4 +40 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5\\&+ b_1)\omega ^2 +4752b^5 d^5 \beta _1^6 \gamma _1^4 \delta ) a^8\\&+20 (21 \gamma _1^3(22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1\\&+4 c_1\beta _1) \omega ^{12}+28 bd \beta _1 \gamma _1^2(99 \beta _1 \delta \gamma _1^6 \\&+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^{10}+63 b^2 d^2 \beta _1^2\gamma _1 (5 b_1 \gamma _1\\&+2 \beta _1 (66 \delta \gamma _1^6+c_1))\omega ^8 \\&+32 b^3 d^3 \beta _1^3(462 \beta _1 \delta \gamma _1^6+6 b_1 \gamma _1\\&+c_1 \beta _1) \omega ^6+40 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1)\omega ^4\\&+9504 b^5 d^5 \beta _1^6\gamma _1^4 \delta \omega ^2 +2464 b^6 d^6 \beta _1^7\gamma _1^3 \delta ) a^6\\&+4 (45 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1)\omega ^{14}\\&+70 b d \beta _1\gamma _1^2 (99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1\\&+6 c_1 \beta _1) \omega ^{12}+189 b^2 d^2 \beta _1^2 \gamma _1 (5 b_1 \gamma _1 +2 \beta _1\\&\times (66 \delta \gamma _1^6+c_1)) \omega ^{10}+120 b^3 d^3 \beta _1^3(462 \beta _1 \delta \gamma _1^6\\&+6 b_1 \gamma _1 +c_1 \beta _1) \omega ^8+200 b^4 d^4 \beta _1^4(396\beta _1 \delta \gamma _1^5\\&+b_1) \omega ^6 +71280 b^5 d^5 \beta _1^6 \gamma _1^4 \delta \omega ^4\\&+36960 b^6 d^6 \beta _1^7 \gamma _1^3 \delta \omega ^2 +8448 b^7 d^7 \beta _1^8\gamma _1^2 \delta )a^4\\&\times (45 \gamma _1^3+(22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1)\omega ^{16}\\&+80 b d \beta _1 \gamma _1^2(99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1 \\&+6c_1\beta _1) \omega ^{14}+252b^2 d^2 \\&\beta _1^2 \gamma _1 (5 b_1 \gamma _1+2 \beta _1 (66 \delta \gamma _1^6+c_1))omega^{12}\\&+192 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6 +6 b_1 \gamma _1+c_1 \\&\beta _1) \omega ^{10}+400 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1) \omega ^8\\&+190080 b^5d^5\beta _1^6 \gamma _1^4 \delta \omega ^6+147840 b^6 d^6 \beta _1^7 \\&\gamma _1^3 \delta \omega ^4+67584 b^7 d^7 \beta _1^8 \gamma _1^2 \delta \omega ^2\\&+13824 b^8 d^8 \beta _1^9 \gamma _1 \delta ) a^2+2560 b^9 d^9 \beta _1^{10} \delta \\&+47520 b^5 d^5 \omega ^8 \beta _1^6 \gamma _1^4 \delta +49280 b^6 d^6\omega ^6 \beta _1^7 \gamma _1^3 \delta \\&+33792 b^7 d^7 \omega ^4 \beta _1^8\gamma _1^2 \delta +13824 b^8 d^8 \omega ^2 \beta _1^9 \gamma _1 \delta \\&+80 b^4 d^4\omega ^{10}\beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1)\\&+5 \omega ^{18} \gamma _1^3 (22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1)\\&+10 b d \omega ^{16}\beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6\\&+8 b_1 \gamma _1+6 c_1 \beta _1)+32 b^3 d^3 \omega ^{12} \beta _1^3\\&+6 b_1 \gamma _1+c_1 \beta _1)(462 \beta _1 \delta \gamma _1^6 +36 b^2 d^2\\&\omega ^{14} \beta _1^2 \gamma _1 (5 b_1 \gamma _1\\&+2 \beta _1 (66 \delta \gamma _1^6+c_1))\big ),\\ A_2= & {} \dfrac{1}{(a^2+\omega ^2)^9}\beta _1^2 \delta (5 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6\\&+3 b_1 \gamma _1+4 c_1 \beta _1) a^{18}+5 \gamma _1^2 (9 \gamma _1 (22 \beta _1 \delta \gamma _1^6+3 \\&b_1 \gamma _1+4 c_1 \beta _1) \omega ^2+2 b d \beta _1 (99 \beta _1 \delta \gamma _1^6\\&+8 b_1 \gamma _1+6 c_1 \beta _1)) a^{16}+4 \gamma _1 (45 \gamma _1^2(22\beta _1 \\&\delta \gamma _1^6+3 b_1 \gamma _1 +4 c_1 \beta _1) \omega ^4 +20 b d \beta _1 \gamma _1 (99 \beta _1 \\&\delta \gamma _1^6 +8 b_1 \gamma _1 +6 c_1 \beta _1)\omega ^2+12 b^2 d^2\beta _1^2 (5 b_1 \gamma _1\\&+2 \beta _1 (66 \delta \gamma _1^6+c_1))) a^{14} +28 (15 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6\\&+3 b_1 \gamma _1+4 c_1 \beta _1)\omega ^6+10 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6\\&+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^4+12 b^2 d^2 \beta _1^2 \gamma _1 (5 b_1 \gamma _1\\&+2 \beta _1 (66\delta \gamma _1^6+c_1)) \omega ^2+2 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6\\&+6 b_1 \gamma _1+c_1 \beta _1)) a^{12}+2(315 \gamma _1^3 (22 \\&\beta _1\delta \gamma _1^6+3b_1 \gamma _1+4 c_1 \beta _1) \omega ^8\\&+280 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6 +8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^6\\&+504 b^2 d^2 \beta _1^2 \gamma _1(5 b_1 \gamma _1 +2 \beta _1 (66 \delta \gamma _1^6+c_1)) \omega ^4 \\&+168 b^3 d^3 \beta _1^3(462 \beta _1 \delta \gamma _1^6+6 b_1 \gamma _1+c_1 \beta _1) \omega ^2\\&+88 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1)) a^{10}\\&+10 (63 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1 +4 c_1\beta _1)\omega ^{10} \\&+70 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^8\\&+168 b^2 d^2 \beta _1^2 \gamma _1(5 b_1 \gamma _1+2 \beta _1 (66 \delta \gamma _1^6+c_1))\omega ^6\\&+84 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6 +6 b_1 \gamma _1+c_1\beta _1)\omega ^4\\&+88 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1) \omega ^2\\&+12672 b^5 d^5 \beta _1^6 \gamma _1^4 \delta ) a^8 +20(21 \gamma _1^3 (22 \beta _1 \delta \gamma _1^6\\&+3 b_1 \gamma _1+4 c_1 \beta _1) \omega ^{12}\\&+28 b d \beta _1\gamma _1^2 (99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^{10}\\&+84 b^2 d^2 \beta _1^2 \gamma _1 (5 b_1\gamma _1+2 \beta _1 (66 \delta \\&\gamma _1^6+c_1)) \omega ^8+56 b^3 d^3\beta _1^3 (462 \beta _1 \delta \gamma _1^6\\&+6 b_1 \gamma _1+c_1 \beta _1) \omega ^6+88 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5\\&+b_1)\omega ^4+25344 b^5 d^5 \beta _1^6 \gamma _1^4 \delta \omega ^2\\&+7744 b^6 d^6 \beta _1^7 \gamma _1^3 \delta ) a^6+4 (45 \gamma _1^3(22 \beta _1 \delta \gamma _1^6\\&+3 b_1\gamma _1 +4 c_1 \beta _1) \omega ^{14}+70 b d \beta _1 \gamma _1^2 (99 \beta _1 \delta \gamma _1^6\\&+8 b_1 \gamma _1+6 c_1 \beta _1) \omega ^{12}+252 b^2 d^2 \beta _1^2 \gamma _1(5 b_1\gamma _1 \\&+2 \beta _1 (66 \delta \gamma _1^6+c_1)) \omega ^{10}\\&+210 b^3 d^3 \beta _1^3 (462 \beta _1 \delta \gamma _1^6+6 b_1 \gamma _1+c_1 \beta _1)\omega ^8\\&+440 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1) \omega ^6\\&+190080 b^5 d^5 \beta _1^6 \gamma _1^4 \delta \omega ^4\\&+116160 b^6 d^6 \beta _1^7 \gamma _1^3 \delta \omega ^2+30624 b^7 d^7 \\&\beta _1^8 \gamma _1^2 \delta ) a^4+(45 \gamma _1^3(22 \beta _1 \delta \gamma _1^6 +3 b_1 \gamma _1\\&+4 c_1 \beta _1) \omega ^{16}+80 b d \beta _1\gamma _1^2(99 \beta _1 \delta \gamma _1^6+8 b_1 \gamma _1 \\&+6 c_1 \beta _1) \omega ^{14}+336 b^2 d^2 \beta _1^2 \gamma _1(5 b_1 \gamma _1\\&+2 \beta _1 (66 \delta \gamma _1^6+c_1))\omega ^{12}+336 b^3 d^3 \beta _1^3(462 \beta _1 \delta \\&\gamma _1^6 +6 b_1 \gamma _1+c_1 \beta _1) \omega ^{10}\\&+880 b^4 d^4 \beta _1^4 (396 \beta _1 \delta \gamma _1^5+b_1) \omega ^8\\&+506880 b^5 d^5 \beta _1^6 \gamma _1^4 \delta \omega ^6 +464640 b^6 d^6\beta _1^7 \\&\gamma _1^3\delta \omega ^4 +244992 b^7 d^7 \beta _1^8 \gamma _1^2 \delta \omega ^2 +56832 b^8 d^8 \\&\beta _1^9 \gamma _1 \delta ) a^2 +11776 b^9 d^9 \beta _1^{10}\delta +126720 b^5 d^5 \\&\omega ^8 \beta _1^6 \gamma _1^4 \delta +154880 b^6 d^6 \omega ^6 \beta _1^7 \gamma _1^3 \delta \\&+122496 b^7 d^7 \omega ^4 \beta _1^8 \gamma _1^2 \delta +56832 b^8 d^8 \omega ^2 \beta _1^9 \gamma _1 \delta \\&+176 b^4 d^4 \omega ^{10} \beta _1^4 (396 \beta _1\delta \gamma _1^5+b_1) \\&+5 \omega ^{18} \gamma _1^3 (22 \beta _1 \delta \gamma _1^6+3 b_1 \gamma _1+4 c_1 \beta _1)\\&+10 b d \omega ^{16} \beta _1 \gamma _1^2 (99 \beta _1+8 b_1 \gamma _1+6 c_1 \beta _1)\delta \gamma _1^6\\&+56 b^3 d^3 \omega ^{12} \beta _1^3(462 \beta _1 \delta \gamma _1^6 +6 b_1 \gamma _1+c_1 \beta _1)\\&+48 b^2 d^2 \omega ^{14}beta_1^2 \gamma _1 \\&(5 b_1 \gamma _1 +2 \beta _1 (66 \delta \gamma _1^6+c_1))),\\ A_3= & {} -\dfrac{1}{\big (a^2+\omega ^2\big )^8}(4 b \beta _1^4 d \delta (15 a^{14} b_1 \gamma _1^2\\&+6 a^{14} \beta _1 c_1 \gamma _1+36 a^{12} b b_1 \beta _1 \gamma _1 d+6 a^{12} b \\&\beta _1^2 c_1 d+105 a^{12} b_1 \gamma _1^2 \omega ^2+42 a^{12} \beta _1 c_1 \gamma _1 \omega ^2\\&+26 a^{10} b^2 b_1 \beta _1^2 d^2+216 a^{10} b b_1 \beta _1\\&\gamma _1 d \omega ^2+36 a^{10} b \beta _1^2 c_1 d \omega ^2\\&+315 a^{10} b_1 \gamma _1^2 \omega ^4+126 a^{10} \beta _1 c_1 \gamma _1 \omega ^4\\&+130a^8 b^2 b_1 \beta _1^2 d^2 \omega ^2+540 a^8 b b_1 \beta _1 \gamma _1 d \omega ^4\\&+90 a^8 b \beta _1^2 c_1 d \omega ^4+525 a^8 b_1 \gamma _1^2 \omega ^6\\&+210 a^8 \beta _1 c_1 \gamma _1 \omega ^6+260 a^6 b^2 b_1 \beta _1^2 d^2 \omega ^4\\&+720 a^6 b b_1\beta _1 \gamma _1 d \omega ^6 +120 a^6b \beta _1^2 c_1 d \omega ^6\\&+525 a^6 b_1\gamma _1^2 \omega ^8+210 a^6 \beta _1 c_1 \gamma _1 \omega ^8\\&+260 a^4 b^2 b_1 \beta _1^2 d^2 \omega ^6 +540 a^4 b b_1 \beta _1\gamma _1 d \omega ^8\\&+90 a^4 b \beta _1^2 c_1 d \omega ^8+315 a^4 b_1 \gamma _1^2 \omega ^{10}\\&+126 a^4 \beta _1 c_1 \gamma _1 \omega ^{10}+130 a^2 b^2 b_1 \beta _1^2 d^2 \omega ^8\\&+4 \beta _1 \delta (4368 b^6 \beta _1^6 \gamma _1 d^6(a^2+\omega ^2)\\&+8184b^5 \beta _1^5 \gamma _1^2 d^5 (a^2+\omega ^2)^2 +8800 b^4 \beta _1^4 \gamma _1^3 d^4 \\&(a^2+\omega ^2)^3 +5940 b^3 \beta _1^3 \gamma _1^4 d^3 (a^2+\omega ^2)^4\\&+2574 b^2 \beta _1^2 \gamma _1^5 d^2 (a^2+\omega ^2)^5+693 b \beta _1 \gamma _1^6 \\&d (a^2 +\omega ^2)^6+99 \gamma _1^7 (a^2+\omega ^2)^7+2574\\&+1024 b^7 \beta _1^7 d^7)+216 a^2 b b_1 \beta _1 \gamma _1 d \omega ^{10} b \beta _1^2c_1 d\\&\omega ^{10}+105 a^2 b_1 \gamma _1^2 \omega ^{12} +42 a^2 \beta _1 c_1 \gamma _1 \omega ^{12}\\&+26 b^2 b_1 \beta _1^2 d^2 \omega ^{10}+36 a^2 +36 b b_1\beta _1\gamma _1 d \omega ^{12}\\&+6 b \beta _1^2 c_1 d \omega ^{12} +15 b_1 \gamma _1^2 \omega ^{14}+6 \beta _1 c_1 \gamma _1 \omega ^{14})),\\ A_4= & {} -\dfrac{1}{(a^2 + \omega ^2)^7}\big (\beta _1^4 \delta (15 a^{14} b_1 \gamma _1^2\\&+6 a^{14} \beta _1 c_1 \gamma _1+36 a^{12} b b_1 \beta _1 \gamma _1 d+6 a^{12} b \beta _1^2 c_1 \\&d+105 a^{12} b_1\gamma _1^2 \omega ^2+42 a^{12} \beta _1 c_1 \gamma _1 \omega ^2\\&+36 a^{10} b^2 b_1 \beta _1^2 d^2+216 a^{10} b b_1 \beta _1 \gamma _1 d \\&\omega ^2+36 a^{10} b \beta _1^2 c_1 d \omega ^2 +315 a^{10} b_1 \gamma _1^2 \omega ^4\\&+126 a^{10} \beta _1 c_1 \gamma _1 \omega ^4+180 a^8 b^2 b_1 \beta _1^2 d^2 \omega ^2\\&+540 a^8 b b_1 \beta _1 \gamma _1 d \omega ^4 +90 a^8b \beta _1^2 c_1 d \omega ^4\\&+525a^8 b_1 \gamma _1^2 \omega ^6 +360 a^6 b^2 b_1 \beta _1^2 d^2 \omega ^4\\&+720 a^6 b b_1 \beta _1\gamma _1 d \omega ^6 +120 a^6 b \beta _1^2 c_1 d \omega ^6 \\&+525 a^6 b_1 \gamma _1^2 \omega ^8+210 a^6 \beta _1 c_1 \gamma _1 \omega ^8\\&+360 a^4 b^2 b_1 \beta _1^2 d^2 \omega ^6+540 a^4 b b_1 \beta _1 \gamma _1 d \omega ^8 \\&+90 a^4 b \beta _1^2 c_1 d \omega ^8+315a^4 b_1 \gamma _1^2 \omega ^{10} \\&+126 a^4 \beta _1 c_1 \gamma _1 \omega ^{10} +180 a^2 b^2 b_1 \beta _1^2 d^2 \omega ^8 \\&+4 \beta _1 \delta (14208b^6 \beta _1^6\gamma _1 d^6 (a^2+ \omega ^2) \\&+22704 b^5 \beta _1^5 \gamma _1^2 d^5(a^2+\omega ^2)^2\\&+20240 b^4 \beta _1^4 \gamma _1^3d^4 (a^2+\omega ^2)^3+10890 b^3 \\&\beta _1^3\gamma _1^4 d^3(a^2+\omega ^2)^4+3564 b^2 \beta _1^2 \gamma _1^5 d^2 \\&(a^2+\omega ^2)^5 +693 b \beta _1 \gamma _1^6 d (a^2+\omega ^2)^6\\&+99 \gamma _1^7(a^2+\omega ^2)^7+3824 b^7 \beta _1^7 d^7)\\&+216 a^2 b b_1 \beta _1 \gamma _1 d \omega ^{10} +36 a^2 b \beta _1^2 c_1d \omega ^{10} \\&+105 a^2b_1 \gamma _1^2\omega ^{12}+42 a^2 \beta _1c_1 \gamma _1 \omega ^{12}\\&+36 b^2 b_1 \beta _1^2 d^2 \omega ^{10}+36 b b_1 \beta _1 \gamma _1 d \omega ^{12}+6 \\&b \beta _1^2 c_1 d\omega ^{12}+15b_1 \gamma _1^2 \omega ^{14}+6 \beta _1 c_1 \gamma _1 \omega ^{14})\big ),\\ A_5= & {} -\dfrac{1}{(a^2 + \omega ^2)^6}(2 b \beta _1^6 d \delta (-(a^2\\&+\omega ^2) ((a^2+\omega ^2) (44 \beta _1 \gamma _1^2 \delta (340 b^2 \beta _1^2 \gamma _1 d^2 (a^2\\&+ \omega ^2)+135 b \beta _1\gamma _1^2 d (a^2+\omega ^2)^2\\&+27 \gamma _1^3 (a^2+\omega ^2)^3+480 b^3 \beta _1^3 d^3)+3 b_1 (a^2+\\&\omega ^2)^3)+15936 b^4 \beta _1^5 \gamma _1 d^4 \delta )-5024 b^5 \beta _1^6 d^5 \delta )),\\ A_6= & {} -\dfrac{1}{(a^2 + \omega ^2)^5}\big (\beta _1^6 \delta ((a^2+\omega ^2) \\&((a^2+\omega ^2) (44 \beta _1 \gamma _1^2 \delta (160 b^2 \beta _1^2 \gamma _1 d^2 (a^2+\omega ^2\\&+45 b \beta _1 \gamma _1^2 d (a^2+\omega ^2)^2+9 \gamma _1^3 (a^2+\omega ^2)^3\\&+300 b^3 \beta _1^3 d^3)+b_1 (a^2+\omega ^2)^3)\\&+12480 b^4 \beta _1^5 \gamma _1 d^4 \delta +4736 b^5 \beta _1^6 d^5 \delta \big ),\\ A_7= & {} -\dfrac{1}{(a^2 + \omega ^2)^4}(-80 b \beta _1^9 d \delta ^2 (42 b^2 \beta _1^2 \gamma _1 d^2 (a^2\\&+\omega ^2) +33 b \beta _1 \gamma _1^2 d (a^2+\omega ^2)^2+\\&11\gamma _1^3 (a^2+\omega ^2)^3+20 b^3 \beta _1^3 d^3)),\\ A_8= & {} \dfrac{1}{(a^2 + \omega ^2)^3}\big (10 \beta _1^9 \delta ^2 \big (60 b^2 \beta _1^2 \gamma _1 d^2 \big (a^2+w^2\big )\\&+33 b \beta _1 \gamma _1^2 d \big (a^2+w^2\big )^2+11\gamma _1^3\\&\big (a^2+w^2\big )^3+38 b^3 \beta _1^3 d^3\big )\big ),\\ A_9= & {} -\dfrac{60 b \beta _1^{11} d \delta ^2 \left( \gamma _1 \left( a^2+w^2\right) +b \beta _1 d\right) }{\left( a^2+w^2\right) ^2},\\ A_{10}= & {} \dfrac{6 \beta _1^{11} \delta ^2 \left( \gamma _1 \left( a^2+w^2\right) +b \beta _1 d\right) }{a^2+w^2}. \end{aligned}$$