Local and global bifurcations in magnetic resonance force microscopy

Abstract

The focus of this paper is on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for measurement of electron spin via magnetic resonance force microscopy (MRFM). The resonator model, derived using the extended Hamilton’s principle incorporating the Bloch equations for magnetization, is investigated analytically and numerically. Analysis of both adiabatic and non-adiabatic equilibrium configurations enables formulation of the dynamical system bifurcation structure and identification of the parameter space required for stable MRFM operation. A multiple-scales analysis of the limiting adiabatic model enables estimation of the local bifurcation thresholds for bistable solutions and prediction of the frequency shift that enables spin detection. Orbital instabilities of the adiabatic model reveal a global bifurcation structure where lengthy chaotic transients occur below a homoclinic jump-to-contact threshold which is determined via a Melnikov–Holmes analysis. Both local and global bifurcations are verified numerically in the non-adiabatic model and reveal a dense power spectra for the magnetic moments. The computation of the parameter space governing the model orbital instabilities enables a consistent estimation of robust MRFM operation conditions.

Appendices

Appendix A

The integral expressions in Eq. (24) are:

$$\begin{aligned} I_1= & {} \int \limits _0^1 {\phi ^{2}\mathrm{d}s} =1 \\ I_2= & {} \int \limits _0^1 {\phi \phi _{ssss} \mathrm{d}s} \\ I_3= & {} \int \limits _0^1 {\phi \left( {\phi _s \left( {\phi _{ss} \phi _s } \right) _s } \right) _s \mathrm{d}s} \\ I_4= & {} \int \limits _0^1 {\phi \left( {\phi _s \int \limits _1^s {\int \limits _0^{{s}'} {\phi _{{s}''}^2 \mathrm{d}{s}''} \mathrm{d}{s}'} } \right) _s \mathrm{d}s} \\ I_5= & {} \int \limits _0^1 {\phi \mathrm{d}s} \end{aligned}$$

Appendix B

The coefficients of the sixteenth-order equilibrium equation are:

$$\begin{aligned} a_0= & {} \left( {3\delta -\chi } \right) ^{2}+9\omega _M^2 \\ a_1= & {} 14\left( {\left( {3\delta -\chi } \right) ^{2}+9\omega _M^2 } \right) \\ a_2= & {} 91\left( {\left( {3\delta -\chi } \right) ^{2}+9\omega _M^2 } \right) \\ a_3= & {} 3276\delta ^{2}-2178\delta \chi +362\chi ^{2}+3276\omega _M^2 \\ a_4= & {} 9009\delta ^{2}-5940\delta \chi +979\chi ^{2}+9009\omega _M^2 \\ a_5= & {} 18018\delta ^{2}-11682\delta \chi +1892\chi ^{2}+18018\omega _M^2 \\ a_6= & {} 27027\delta ^{2}-17028\delta \chi +2674\chi ^{2}+27027\omega _M^2 \\ a_7= & {} 30888\delta ^{2}-18612\delta \chi +2780\chi ^{2}+30888\omega _M^2 \\ a_8= & {} 27027\delta ^{2}-15246\delta \chi +2107\chi ^{2}+27027\omega _M^2 \\ a_9= & {} 18018\delta ^{2}-9240\delta \chi +1134\chi ^{2}+18018\omega _M^2 \\ a_{10}= & {} 9009\delta ^{2}-4026\delta \chi +411\chi ^{2}\\&+\,9009\omega _M^2 -\Gamma ^{2}\left( {3\delta -\chi } \right) ^{2} \\ a_{11}= & {} 3276\delta ^{2}-1194\delta \chi +90\chi ^{2}\\&+\,3276\omega _M^2 -6\Gamma ^{2}\left( {3\delta -\chi } \right) ^{2} \\ a_{12}= & {} 819\delta ^{2}-216\delta \chi +9\chi ^{2}+819\omega _M^2 \\&-\,15\Gamma ^{2}\left( {3\delta -\chi } \right) ^{2} \\ a_{13}= & {} 126\delta ^{2}-18\delta \chi +126\omega _M^2 \\&-\,6\Gamma ^{2}\left( {30\delta ^{2}-19\delta \chi +3\chi ^{2}} \right) \\ a_{14}= & {} 9\left( {\delta ^{2}+\omega _M^2 -\Gamma ^{2}\left( {15\delta ^{2}-8\delta \chi +\chi ^{2}} \right) } \right) \\ a_{15}= & {} -18\Gamma ^{2}\delta \left( {3\delta -\chi } \right) \\ a_{16}= & {} -9\Gamma ^{2}\delta ^{2} \end{aligned}$$

The coefficients of the fourteenth-order equilibrium equation are:

$$\begin{aligned} b_0= & {} -9\Gamma ^{2}+\omega _B^2 \\ b_1= & {} -18\Gamma ^{2}+14\omega _B^2 \\ b_2= & {} -15\Gamma ^{2}+91\omega _B^2 +9 \\ b_3= & {} -6\Gamma ^{2}+364\omega _B^2 +90 \\ b_4= & {} -\Gamma ^{2}+1001\omega _B^2 +411 \\ b_5= & {} 2002\omega _B^2 +1134 \\ b_6= & {} 3003\omega _B^2 +2107\\ b_7= & {} 3432\omega _B^2 +2780 \\ b_8= & {} 3003\omega _B^2 +2674 \\ b_9= & {} 2002\omega _B^2 +1892 \\ b_{10}= & {} 1001\omega _B^2 +979 \\ b_{11}= & {} 364\omega _B^2 +362 \\ b_{12}= & {} 91\omega _B^2 +91 \\ b_{13}= & {} 14\omega _B^2 +14 \\ b_{14}= & {} \omega _B^2 +1 \end{aligned}$$

where $\omega _B =\frac{3\omega _M }{\chi }=\frac{2\pi d^{3}B_1 }{\mu _0 m_f }$.

Appendix C

The discriminant of the 14-order polynomial is as follows:

$$\begin{aligned} DES= & {} c_{20} \Gamma ^{2}+c_{40} \Gamma ^{4}+c_{60} \Gamma ^{6}+\cdots +c_{16,0} \Gamma ^{16}+ \\&+\,c_{02} \omega _B^2 +c_{22} \Gamma ^{2}\omega _B^2 +c_{42} \Gamma ^{4}\omega _B^2 +\cdots \\&+\,c_{16,2} \Gamma ^{16}\omega _B^2 + \\&+\,c_{04} \omega _B^4 +c_{24} \Gamma ^{2}\omega _B^4 +c_{44} \Gamma ^{4}\omega _B^4 +\cdots \\&+\,c_{16,4} \Gamma ^{16}\omega _B^4 + \\&+\,c_{06} \omega _B^6 +c_{26} \Gamma ^{2}\omega _B^6 +c_{46} \Gamma ^{4}\omega _B^6 +\cdots \\&+\,c_{16,6} \Gamma ^{16}\omega _B^6 + \\&+\,c_{28} \Gamma ^{2}\omega _B^8 +c_{48} \Gamma ^{4}\omega _B^8 +\cdots +c_{16,8} \Gamma ^{16}\omega _B^8 + \\&+\,c_{4,10} \Gamma ^{4}\omega _B^{10} +\cdots \\&+\,c_{16,10} \Gamma ^{16}\omega _B^{10} + \\&+\,\cdots +c_{12,18} \Gamma ^{12}\omega _B^{18} +c_{14,18} \Gamma ^{14}\omega _B^{18} \\&+\,c_{16,18} \Gamma ^{16}\omega _B^{18} + \\&+\,c_{12,20} \Gamma ^{12}\omega _B^{20} +c_{14,20} \Gamma ^{14}\omega _B^{20} \\&+\,c_{12,22} \Gamma ^{12}\omega _B^{22} \end{aligned}$$

The respective coefficients ${c}_{{i,j}}$ are as follows:

$$\begin{aligned} c_{20}= & {} -2^{16}\cdot 3^{8} \\ c_{40}= & {} 3^{8}\cdot \left( {9634553} \right) \\ c_{60}= & {} 2\cdot 3^{7}\cdot 5\cdot \left( {403} \right) \cdot \left( {56569} \right) \\ c_{80}= & {} 3^{5}\cdot 5\cdot \left( {8863} \right) \cdot \left( {9887} \right) \\ c_{10,0}= & {} 2^{2}\cdot 3^{9}\cdot 5\cdot 37\cdot \left( {3253} \right) \\ c_{12,0}= & {} 3^{4}\cdot \left( {2039} \right) \cdot \left( {71777} \right) \\ c_{14,0}= & {} 2\cdot 3^{2}\cdot \left( {87857857} \right) \\ c_{16,0}= & {} 3^{2}\cdot 5^{10} \\ c_{02}= & {} 2^{16}\cdot 3^{6} \\ c_{22}= & {} -3^{6}\cdot \left( {163} \right) \cdot \left( {80819} \right) \\ c_{42}= & {} 2^{2}\cdot 3^{5}\cdot \left( {29} \right) \cdot \left( {10699387} \right) \\ c_{62}= & {} -3^{3}\cdot \left( {59} \right) \cdot \left( {269500697} \right) \\ c_{82}= & {} 2\cdot 3^{5}\cdot 5\cdot \left( {5249639} \right) \\ c_{10,2}= & {} 2\cdot 3^{2}\cdot 7\cdot \left( {103} \right) \cdot \left( {211} \right) \cdot \left( {158419} \right) \\ c_{12,2}= & {} \left( {167} \right) \cdot \left( {1255182899} \right) \\ c_{14,2}= & {} 2^{2}\cdot 5\cdot 7^{2}\cdot \left( {83} \right) \cdot \left( {389189} \right) \\ c_{16,2}= & {} 3^{4}\cdot 5^{10} \end{aligned}$$

$$\begin{aligned} c_{04}= & {} 2^{17}\cdot 3^{6} \\ c_{24}= & {} -2\cdot 3^{6}\cdot \left( {19} \right) ^{2}\cdot \left( {27409} \right) \\ c_{44}= & {} 3^{5}\cdot \left( {13} \right) \cdot \left( {236262581} \right) \\ c_{64}= & {} -2^{2}\cdot 3^{5}\cdot 7\cdot \left( {572365993} \right) \\ c_{84}= & {} 3^{6}\cdot 5\cdot \left( {241} \right) \cdot \left( {4337717} \right) \\ c_{10,4}= & {} -2\cdot 3^{2}\cdot \left( {89} \right) \cdot \left( {471} \right) \cdot \left( {1156751} \right) \\ c_{12,4}= & {} \left( {29} \right) \cdot \left( {77255361353} \right) \\ c_{14,4}= & {} 2^{2}\cdot 3^{2}\cdot 5\cdot \left( {107} \right) \cdot \left( {10715717} \right) \\ c_{16,4}= & {} 2^{2}\cdot 3^{4}\cdot 5^{10} \\ c_{06}= & {} 2^{16}3^{6} \\ c_{26}= & {} -3^{6}\cdot 7\cdot \left( {41} \right) \cdot \left( {8423} \right) \\ c_{46}= & {} 2\cdot 3^{5}\cdot 5\cdot \left( {1051} \right) \cdot \left( {328667} \right) \\ c_{66}= & {} 2^{4}\cdot 3^{5}\cdot \left( {21227} \right) \cdot \left( {116989} \right) \\ c_{86}= & {} 2^{2}\cdot 3^{3}\cdot 7^{2}\cdot \left( {37} \right) \cdot \left( {2273} \right) \cdot \left( {51941} \right) \\ c_{10,6}= & {} -3^{2}\cdot 7\cdot \left( {557} \right) \cdot \left( {14221} \right) \cdot \left( {45737} \right) \\ c_{12,6}= & {} 2^{3}\cdot 3^{2}\cdot \left( {17} \right) \cdot \left( {139} \right) \cdot \left( {72728059} \right) \\ c_{14,6}= & {} 2\cdot 3^{2}\cdot 5\cdot \left( {8032392593} \right) \\ c_{16,6}= & {} 2^{2}\cdot 3^{3}\cdot 5^{10}\cdot 7 \\ c_{28}= & {} 2^{12}\cdot 3^{6}\cdot 7\cdot \left( {167} \right) \\ c_{48}= & {} 2\cdot 3^{7}\cdot \left( {79} \right) \cdot \left( {955127} \right) \\ c_{68}= & {} -3^{6}\cdot \left( {11} \right) \cdot \left( {7949} \right) \cdot \left( {169079} \right) \\ c_{88}= & {} 2\cdot 3^{5}\cdot 5\cdot \left( {53} \right) \cdot \left( {157} \right) \cdot \left( {2533291} \right) \\ c_{10,8}= & {} -3^{2}\cdot \left( {31} \right) \cdot \left( {457} \right) \cdot \left( {574110233} \right) \\ c_{12,8}= & {} 2^{2}\cdot 3\cdot \left( {3307245364807} \right) \\ c_{14,8}= & {} 2\cdot 3\cdot 5\cdot 7\cdot \left( {11} \right) \cdot \left( {675106291} \right) \\ c_{16,8}= & {} 2\cdot 3^{4}\cdot 5^{10}\cdot 7 \\ c_{4,10}= & {} -2^{6}\cdot 3^{5}\cdot 7\left( {11} \right) \cdot \left( {1153} \right) \\ c_{6,10}= & {} -3^{5}\cdot \left( {11} \right) \cdot \left( {13} \right) \cdot \left( {79} \right) \cdot \left( {1918811} \right) \\ c_{8,10}= & {} 3^{4}\cdot \left( {686002603513} \right) \\ c_{10,10}= & {} -2^{2}\cdot 3^{2}\cdot \left( {1511} \right) \cdot \left( {6959} \right) \cdot \left( {327553} \right) \\ c_{12,10}= & {} 3\cdot \left( {419} \right) \cdot \left( {39233} \right) \cdot \left( {1642279} \right) \end{aligned}$$

$$\begin{aligned} c_{14,10}= & {} 2\cdot 3^{2}\cdot 7\cdot \left( {53} \right) \cdot \left( {173} \right) \cdot \left( {661} \right) \cdot \left( {2903} \right) \\ c_{16,10}= & {} 2\cdot 3^{4}\cdot 5^{10}\cdot 7 \\ c_{6,12}= & {} -3^{4}\cdot 7\cdot \left( {11863} \right) \cdot \left( {121727} \right) \\ c_{8,12}= & {} 2\cdot 3^{3}\cdot 7\cdot \left( {4241} \right) \cdot \left( {19107149} \right) \\ c_{10,12}= & {} -2\cdot 3^{2}\cdot \left( {83} \right) \cdot \left( {264127} \right) \cdot \left( {314693} \right) \\ c_{12,12}= & {} 2\cdot 3^{2}\cdot \left( {101} \right) \cdot \left( {30829} \right) \cdot \left( {1971521} \right) \\ c_{14,12}= & {} 2\cdot 3^{2}\cdot 5\cdot 7\cdot \left( {1901} \right) \cdot \left( {1778417} \right) \\ c_{16,12}= & {} 2^{2}\cdot 3^{3}\cdot 5^{10}\cdot 7 \\ c_{6,14}= & {} -2^{9}\cdot 3^{3}\cdot 7^{7} \\ c_{8,14}= & {} 3^{3}\cdot 5^{2}\cdot 7\cdot \left( {11} \right) \cdot \left( {144217123} \right) \\ c_{10,14}= & {} -2\cdot 3^{2}\cdot \left( {2857} \right) \cdot \left( {1443655937} \right) \\ c_{12,14}= & {} 3\cdot \left( {73} \right) \cdot \left( {89003} \right) \cdot \left( {5269151} \right) \\ c_{14,14}= & {} 2\cdot 3\cdot 5\cdot \left( {11} \right) \cdot \left( {13} \right) \cdot \left( {937} \right) \cdot \left( {342319} \right) \\ c_{16,14}= & {} 2^{2}\cdot 3^{4}\cdot 5^{10} \\ c_{8,16}= & {} 2^{4}\cdot 3^{3}\cdot 7^{8}\cdot \left( {13} \right) ^{2} \\ c_{10,16}= & {} -3^{2}\cdot 7\cdot \left( {11} \right) ^{2}\cdot \left( {3259} \right) \cdot \left( {991031} \right) \\ c_{12,16}= & {} 3\cdot \left( {43} \right) \cdot \left( {1801} \right) \cdot \left( {3169} \right) \cdot \left( {87649} \right) \\ c_{14,16}= & {} 2^{2}\cdot 3^{2}\cdot 5\cdot 7^{2}\cdot \left( {2657} \right) \left( {24533} \right) \\ c_{16,16}= & {} 3^{4}\cdot 5^{10} \\ c_{10,18}= & {} -3^{2}\cdot 7^{8}\cdot \left( {67447} \right) \\ c_{12,18}= & {} 2^{2}\cdot 3^{2}\cdot 7\cdot \left( {419} \right) \cdot \left( {709} \right) \cdot \left( {351707} \right) \\ c_{14,18}= & {} 2\cdot 3^{2}\cdot 5\cdot \left( {1871} \right) \cdot \left( {835859} \right) \\ c_{16,18}= & {} 3^{2}\cdot 5^{10} \\ c_{12,20}= & {} 7^{8}\cdot \left( {1095781} \right) \\ c_{14,20}= & {} 2\cdot \left( {19} \right) \cdot \left( {37} \right) \cdot \left( {619} \right) \cdot \left( {17659} \right) \\ c_{12,22}= & {} 7^{14} \\ \end{aligned}$$

Appendix D

The substitution of Eqs. (52), (54)–(56) in (51) yields the following:

$$\begin{aligned}&D_0^2 Z_3 +\bar{{\omega }}_1^2 Z_3 \\&\quad =-2i\bar{{\omega }}_1 \left( {D_2 A} \right) e^{i\bar{{\omega }}_1 T_0 } \\&\quad \quad +\,2i\bar{{\omega }}_1 \left( {D_2 \bar{{A}}} \right) e^{-i\bar{{\omega }}_1 T_0 }-i\bar{{\omega }}_1 \bar{{\beta }}Ae^{i\bar{{\omega }}_1 T_0 }\\&\qquad +\,i\bar{{\omega }}_1 \bar{{\beta }}\bar{{A}}e^{-i\bar{{\omega }}_1 T_0 } \\&\qquad -\,2\alpha _2 \left( {Ae^{i\bar{{\omega }}_1 T_0 }+\bar{{A}}e^{-i\bar{{\omega }}_1 T_0 }} \right) \\&\qquad \times \,\left( {\frac{\alpha _2 A^{2}}{3\bar{{\omega }}_1^2 }e^{2i\bar{{\omega }}_1 T_0 }-\frac{2\alpha _2 A\bar{{A}}}{\bar{{\omega }}_1^2 }+\frac{\alpha _2 \bar{{A}}^{2}}{3\bar{{\omega }}_1^2 }e^{-2i\bar{{\omega }}_1 T_0 }} \right) \\&\qquad -\,\alpha _3 \left( A^{3}e^{3i\bar{{\omega }}_1 T_0 }+3A^{2}\bar{{A}}e^{i\bar{{\omega }}_1 T_0 }\right. \\&\left. \qquad +\,3A\bar{{A}}^{2}e^{-i\bar{{\omega }}_1 T_0 }+\bar{{A}}^{3}e^{-3i\bar{{\omega }}_1 T_0 } \right) \\&\quad \quad +\,\frac{\bar{{\alpha }}F_0 \left( \Omega \right) }{2}\left( {e^{i\left( {\sigma T_2 +\psi _0 +\bar{{\omega }}_1 T_0 } \right) }+e^{-i\left( {\sigma T_2 +\psi _0 +\bar{{\omega }}_1 T_0 } \right) }} \right) \end{aligned}$$

Appendix E

The coefficients of the fourth-order polynomial are as follows:

$$\begin{aligned} \bar{{A}}= & {} \left( {\frac{\alpha \eta _1 }{2\bar{{\omega }}_1 \varepsilon b}} \right) \\ \bar{{C}}= & {} \left( {\beta ^{2}+2\eta _1 } \right) \left[ {\left( {\frac{\alpha }{2\bar{{\omega }}_1 \varepsilon b}} \right) ^{2}+1} \right] \\ \bar{{D}}= & {} -2\left( {p\left( {\varepsilon b} \right) ^{2}-\bar{{\omega }}_1 } \right) \\ \bar{{E}}= & {} \left( {\frac{\alpha }{2\bar{{\omega }}_1 \varepsilon b}} \right) ^{2}-\left( {\frac{\beta }{2}} \right) ^{2}-\left( {p\left( {\varepsilon b} \right) ^{2}-\bar{{\omega }}_1 } \right) ^{2} \end{aligned}$$

where $p=\left( {10\alpha _2^2 -9\alpha _3 \bar{{\omega }}_1^2 } \right) /24\bar{{\omega }}_1^3$.

Appendix F

We approximate the homoclinic orbit by expanding the force in the second equation of Eq. (64) up to quadratic order:

$$\begin{aligned} R(x_1 )= & {} -x_1 \mp \frac{\Gamma }{\left( {1+x_1 } \right) ^{4}\left( {1+\left[ {\omega _M /f(x_1 )} \right] ^{2}} \right) ^{0.5}}\nonumber \\&\cong c_1 +c_2 x_1 +c_3 x_1 ^{2}=\tilde{R}(x_1 ) \end{aligned}$$

(71)

We integrate Eq. (71) to yield the potential energy function:

$$\begin{aligned} \tilde{V}(x_1 )=c_1 x_1 +\frac{c_2 x_1 ^{2}}{2}+\frac{c_3 x_1 ^{3}}{3} \end{aligned}$$

(72)

In order to find the constants ${c}_{{1}}$, ${c}_{{2}}$, ${c}_{{3}}$, we demand that the saddle point ${x}_{{{s}}}$ is an equilibrium point, therefore:

$$\begin{aligned} \tilde{R}(x_s )=c_1 +c_2 x_s +c_3 x_s ^{2}=0 \end{aligned}$$

(73)

Moreover, the points ${x}_{{{s}}}$ and ${x}_{{{E}}}$ pass through the same homoclinic orbit whose Hamiltonian is the same. Therefore, their potential energy is equal.

$$\begin{aligned} c_1 x_s +\frac{c_2 x_s ^{2}}{2}+\frac{c_3 x_s ^{3}}{3}=c_1 x_E +\frac{c_2 x_E ^{2}}{2}+\frac{c_3 x_E ^{3}}{3} \end{aligned}$$

(74)

We also choose an arbitrary point on the exact homoclinic orbit (${x}_{{A}}$, ${y}_{A})$.

$$\begin{aligned} c_1 x_s +\frac{c_2 x_s ^{2}}{2}+\frac{c_3 x_s ^{3}}{3}=c_1 x_A +\frac{c_2 x_A ^{2}}{2}+\frac{c_3 x_A ^{3}}{3}+\frac{y_A ^{2}}{2}\nonumber \\ \end{aligned}$$

(75)

Equations (73)–(75) can be represented in a matrix form:

$$\begin{aligned} \left( {{\begin{array}{lll} 1&{} {x_s }&{} {x_s ^{2}} \\ {x_s -x_E }&{} {\frac{x_s ^{2}-x_E ^{2}}{2}}&{} {\frac{x_s ^{3}-x_E ^{3}}{3}} \\ {x_s -x_A }&{} {\frac{x_s ^{2}-x_A ^{2}}{2}}&{} {\frac{x_s ^{3}-x_A ^{3}}{3}} \\ \end{array} }} \right) \left( {{\begin{array}{l} {c_1 } \\ {c_2 } \\ {c_3 } \\ \end{array} }} \right) =\left( {{\begin{array}{l} 0 \\ 0 \\ {\frac{y_A ^{2}}{2}} \\ \end{array} }} \right) \end{aligned}$$

(76)

The solution of (76) yields:

$$\begin{aligned} \left( {{\begin{array}{l} {c_1 } \\ {c_2 } \\ {c_3 } \\ \end{array} }} \right) =\frac{y_A ^{2}}{2\left( {x_A -x_s } \right) ^{2}\left( {x_A -x_E } \right) }\left( {{\begin{array}{l} {-x_s \left( {2x_E +x_s } \right) } \\ {2\left( {x_E +2x_s } \right) } \\ {-3} \\ \end{array} }} \right) \nonumber \\ \end{aligned}$$

(77)

The approximated potential energy has the following form:

$$\begin{aligned} \tilde{V}\left( {x_1 } \right)= & {} \frac{y_A ^{2}}{2\left( {x_A -x_s } \right) ^{2}\left( {x_A -x_E } \right) }\left( -x_s x_1 \left( {2x_E +x_s } \right) \nonumber \right. \\&\left. +\,\left( {x_E +2x_s } \right) x_1 ^{2}-x_1 ^{3} \right) \end{aligned}$$

(78)

and the approximated Hamiltonian has the following form:

$$\begin{aligned} \tilde{H}\left( {x_1 ,x_2 } \right) =\frac{x_2 ^{2}}{2}+\tilde{V}(x_1 ) \end{aligned}$$

(79)

where $\tilde{V}( {x_1 })$ is defined in Eq. (72). Throughout the entire homoclinic orbit, its value is constant is equal to its value in the saddle point:

$$\begin{aligned} h_s =\tilde{H}\left( {x_s ,0} \right) =-\frac{y_A ^{2}x_s ^{2}x_E }{\left( {x_A -x_s } \right) ^{2}\left( {x_A -x_E } \right) } \end{aligned}$$

(80)

Substituting Eqs. (78) and (80) into (79) yields:

$$\begin{aligned} x_2 \left( {t_N } \right)= & {} \frac{\mathrm{d}x_1 }{\mathrm{d}t_N }=\pm \sqrt{2\left( {h_s -\tilde{V}\left( {x_1 } \right) } \right) }\nonumber \\= & {} \frac{y_A \left( {x_1 -x_s } \right) \sqrt{\left( {x_1 -x_E } \right) }}{\left( {x_A -x_s } \right) \sqrt{\left( {x_A -x_E } \right) }} \end{aligned}$$

(81)

We solve Eq. (81) using separation of variables:

$$\begin{aligned}&\pm \frac{y_A }{\left( {x_A -x_s } \right) \sqrt{\left( {x_A -x_E } \right) }}dt_N \nonumber \\&\quad =\frac{dx_1 }{\left( {x_1 -x_s } \right) \sqrt{\left( {x_1 -x_E } \right) }} \end{aligned}$$

(82)

We define the following transformation:

$$\begin{aligned} z=x_1 -x_s ; \mathrm{d}z=\mathrm{d}x_1 \end{aligned}$$

(83)

Substitution of Eqs. (83) into (82) yields:

$$\begin{aligned}&\pm \frac{y_A \left( {t_N -t_0 } \right) }{\left( {x_A -x_s } \right) \sqrt{\left( {x_A -x_E } \right) }}\nonumber \\&\quad =\int _{z\left( {t_0 } \right) }^z {\frac{\mathrm{d}z}{z\sqrt{z+x_s -x_E }}} = \nonumber \\&\quad =\frac{1}{\sqrt{x_s -x_E }}\cdot \ln \left( {\frac{\sqrt{z+x_s -x_E }-\sqrt{x_s -x_E }}{\sqrt{z+x_s -x_E }+\sqrt{x_s -x_E }}} \right) \left| {_{_{_{_{z\left( {t_0 } \right) } } } }^{^{^{^{z}}}} } \right. \nonumber \\ \end{aligned}$$

(84)

We note that $z\left( {t_0 } \right) =x_1 \left( {t_0 } \right) -x_s =x_E -x_s$ and therefore we get :

$$\begin{aligned} \pm \hat{{p}}\hat{{t}}_N +\pi i=\ln \left( {\frac{\sqrt{z+x_s -x_E }-\sqrt{x_s -x_E }}{\sqrt{z+x_s -x_E }+\sqrt{x_s -x_E }}} \right) \nonumber \\ \end{aligned}$$

(85)

where $t_N^*=t_N -t_0$ and:

$$\begin{aligned} \hat{{p}}=\frac{y_A \sqrt{x_s -x_E }}{\left( {x_A -x_s } \right) \sqrt{x_A -x_E }} \end{aligned}$$

(86)

We define:

$$\begin{aligned} \phi =x_s -x_E \end{aligned}$$

(87)

Taking an exponent from both sides of Eq. (85) yields:

$$\begin{aligned} \frac{\sqrt{z+\phi }-\sqrt{\phi }}{\sqrt{z+\phi }+\sqrt{\phi }}=-e^{\pm \hat{{p}}t_N^*} \end{aligned}$$

(88)

Or equivalently:

$$\begin{aligned} z=-\phi +\phi \tanh ^{2}\left( {\pm \frac{\hat{{p}}t_N^*}{2}} \right) \end{aligned}$$

(89)

Finally, we substitute Eqs. (83) and (87) in (89) to yield:

$$\begin{aligned} x_1 \left( {t_N^*} \right) =x_E -\left( {x_E -x_s } \right) \tanh ^{2}\left( {\pm \frac{\hat{{p}}t_N^*}{2}} \right) \end{aligned}$$

(90)

The derivation of Eq. (90) with respect to $t_N^*$ yields:

$$\begin{aligned} x_2 \left( {t_N^*} \right) =\pm \hat{{p}}\left( {x_E -x_s } \right) \tanh \left( {\pm \frac{\hat{{p}}t_N^*}{2}} \right) \hbox {sech}^{2}\left( {\pm \frac{pt_N^*}{2}} \right) \nonumber \\ \end{aligned}$$

(91)