“Duffing Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Duffing Differential Equation

释义

Duffing Differential Equation

The most general forced form of the Duffing equation is

(1)

If there is no forcing, the right side vanishes, leaving

(2)

and we take the plus sign,

(3)

This equation can display chaotic behavior. For

, the equation represents a ``hard spring,'' and for

, it represents a ``soft spring.'' If

, the phase portrait curves are closed.Returning to (1),take

, and use the minus sign. Then the equation is

(4)

(Ott 1993, p. 3). This can be written as a system of first-orderordinary differential equations by writing

			(5)
			(6)

The fixed points of these differential equations

(7)

, and

(8)

giving

. Differentiating,

			(9)
			(10)

(11)

Examine the stability of the point (0,0):

(12)

(13)

But

, so

is real. Since

, there will always be onePositive Root, so this fixed point is unstable. Now look at (

, 0).

(14)

(15)

For

, so the point is asymptotically stable. If

, so the point is linearly stable. If

, the radical gives anImaginary Part and the Real Part is

, so the point is unstable. If

, which has a Positive Real Root, so the point is unstable. If

, then

, so both Roots are Positive and the point is unstable. The following table summarizes these results.

	asymptotically stable
	linearly stable (superstable)
	unstable

Now specialize to the case , which can be integrated by quadratures.In this case, the equations become

			(16)
			(17)

Differentiating (16) and plugging in (17) gives

(18)

Multiplying both sides by

gives

(19)

(20)

so we have an invariant of motion

(21)

Solving for

gives

(22)

(23)

(24)

Note that the invariant of motion satisfies

(25)

(26)

so the equations of the Duffing oscillator are given by the Hamiltonian System

(27)

References

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.

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