We consider a series–resonance circuit; an inductor *L* and a capacitor *C* are connected in series under impression of an alternating voltage *E* sin ω *t*. Inductance of the inductor with saturable–core is not constant but a function of magnetic flux φ in the core and a resistor *R* is paralleled with the capacitor, so that the circuit is dissipative. We assume that saturation curve of the inductor is represented by

n i = | a_{1} φ + a_{3} φ^{3} (a_{1} ≥ 0, a_{3} > 0), | ⋅⋅⋅(1) |

where *n* is the number of turns of the inductor coil, and *i* is current in the inductor coil. Then, equations for the circuit are written as

v_{L} + v_{C} = | E sin ω t, | ⋅⋅⋅(2) | |

v_{L} = | n dφ ⁄ dt, | ⋅⋅⋅(3) | |

dv_{C} ⁄ dt = | i_{C} ⁄ C, | ⋅⋅⋅(4) | |

v_{C} = | R i_{R}, | ⋅⋅⋅(5) | |

i = | i_{C} + i_{R}. | ⋅⋅⋅(6) |

By differentiating equations (2) and (3) with respect to *t*, and by using equation (4), we get

n d^{2}φ ⁄ dt^{2} + i_{C} ⁄ C = | ω E cos ω t. | ⋅⋅⋅(7) |

Substituting equations (3), (5), and (6) into equation (2), we have

n dφ ⁄ dt + R (i – i_{C}) = | E sin ω t. | ⋅⋅⋅(8) |

We define parameters ν and *k*, and the variable τ by

ν = | ω ⁄ ω_{0}, | ⋅⋅⋅(9) | |

k = | 1 ⁄ (ω_{0} C R), | ⋅⋅⋅(10) | |

τ = | ω_{0} t – α, | ⋅⋅⋅(11) | |

α = | (1 ⁄ ν) tan^{-1}(k ⁄ ν)
| ||

(cos ν α = | ν ⁄ (ν^{2} + k^{2})^{1/2}, sin ν α = k ⁄ (ν^{2} + k^{2})^{1/2}), | ⋅⋅⋅(12) |

where ω_{0} is base quantity of angular frequency, and elimination of *i*_{C} from equations (7) and (8) yields

n d^{2}φ ⁄ dt^{2} + n k ω_{0} dφ ⁄ dt + i ⁄ C = | (ν^{2} + k^{2})^{1/2} ω_{0} E cos ν τ. | ⋅⋅⋅(13) |

We also introduce dimensionless variables *x* and *u*, defined by

x = | φ ⁄ Φ_{n}, | ⋅⋅⋅(14) | |

u = | i ⁄ I_{n}, | ⋅⋅⋅(15) |

where Φ_{n} and *I*_{n} are appropriate base quantities of flux and current respectively. Although the base quantities can be chosen quite arbitrarily, it is preferable, for brevity of expression, to fix them by the relation

n ω_{0}^{2} C Φ_{n} = | I_{n}. | ⋅⋅⋅(16) |

Then, equation (1) becomes

u = | c_{1} x + c_{3} x^{3} | ⋅⋅⋅(17) | |

(c_{1} = | a_{1}Φ_{n} ⁄ (n I_{n}), c_{3} = a_{3}Φ_{n}^{3} ⁄ (n I_{n})), |

and from equations (11) and (14), we have

dφ ⁄ dt = | ω_{0} Φ_{n} dx ⁄ dτ, d^{2}φ ⁄ dt^{2} = ω_{0}^{2} Φ_{n} d^{2}x ⁄ dτ^{2}. | ⋅⋅⋅(18) |

We also replace *E* by the parameter

B = | E (ν^{2} + k^{2})^{1/2} ⁄ (n ω_{0} Φ_{n}). | ⋅⋅⋅(19) |

Dividing equation (13) by *n* ω_{0}^{2} Φ_{n}, and using equations (14)-(19), Duffing's equation is derived as

d^{2}x ⁄ dτ^{2} + k dx ⁄ dτ + c_{1} x + c_{3} x^{3} = | B cos ν τ. | ⋅⋅⋅(20) |

Note:

In equation (20), if we replace ν τ by *t*', *k* ⁄ ν by *k*', *c*_{1} ⁄ ν^{2} by *c*'_{1}, *c*_{3} ⁄ ν^{2} by *c*'_{3}, and *B* ⁄ ν^{2} by *B*', and remove primes ('), equation (20) is rewritten as

d^{2}x ⁄ dt^{2} + k dx ⁄ dt + c_{1} x + c_{3} x^{3} = | B cos t. | ⋅⋅⋅(21) |

References

1. "The Road to Chaos – II," Yoshisuke Ueda, Aerial Press, Inc., 2001

2. "Kaosu Gensho —Hisenkei no Kakutei-kei ni shojiru Kakuritsu-gensho—," Yoshisuke Ueda, Denki Hyoron (in Japanese), 1994-9